In recent years, the deployment of unmanned aerial vehicles, commonly referred to as UAV drones, has surged across various sectors, including surveillance, logistics, agriculture, and disaster management. Compared to single UAV drone operations, formations of UAV drones offer enhanced mission efficiency, economic viability, and improved fault tolerance and robustness. This has propelled research into UAV drone swarm systems, with formation control emerging as a pivotal subtopic. Formation control for UAV drones involves coordinating multiple drones to maintain specific spatial patterns while navigating complex environments, which inherently presents challenges due to the nonlinear dynamics, external disturbances, and communication constraints. Traditional control methods, such as backstepping control, model predictive control (MPC), and sliding mode control (SMC), have been applied to UAV drone formations with varying degrees of success. However, these approaches often grapple with issues like computational complexity, sensitivity to model inaccuracies, or chattering phenomena. To address these limitations, this paper integrates port-Hamiltonian system (PHS) theory with sliding mode control to develop a novel passivity-based sliding mode control (PBSMC) strategy for distributed formation control of UAV drones. The PHS framework provides a systematic energy-based perspective for modeling and analyzing physical systems, facilitating stability proofs and controller design through energy shaping. By embedding a sliding mode surface within the Hamiltonian energy function, the proposed method ensures robust convergence and disturbance rejection, making it particularly suitable for real-world UAV drone applications where uncertainties are prevalent. This work not only contributes a new control paradigm for UAV drone formations but also underscores the synergy between energy-based control and robust nonlinear techniques, paving the way for more resilient and efficient multi-UAV drone systems.
The core of this research lies in addressing the distributed control problem for a fleet of UAV drones operating in three-dimensional space. Each UAV drone is modeled as a 3-DOF nonlinear system, capturing essential kinematic and dynamic behaviors. Distributed control is preferred over centralized approaches to mitigate communication bottlenecks and enhance scalability. In distributed settings, not all UAV drones have direct access to the leader’s state information; instead, they rely on local interactions with neighboring UAV drones. To enable this, a consensus estimation algorithm is employed, allowing each UAV drone to estimate the leader’s state based on neighbor communications. Subsequently, a distributed formation error model is formulated, which serves as the foundation for controller design. The entire UAV drone formation system is then cast into a port-Hamiltonian form, leveraging its inherent passivity properties. Within this framework, a desired Hamiltonian energy function is constructed, incorporating a sliding mode variable to enforce robustness. The resulting PBSMC law drives the formation errors to zero, ensuring that all UAV drones converge to their desired positions and velocities while maintaining the specified formation pattern. Stability and robustness analyses are conducted using Lyapunov theory, demonstrating asymptotic stability and bounded convergence under disturbances. Numerical simulations validate the effectiveness of the proposed method, showcasing superior performance in terms of convergence speed, smoothness, and disturbance rejection compared to conventional approaches like consensus control, improved SMC, and distributed MPC. This comprehensive study not only advances the theoretical underpinnings of UAV drone formation control but also offers practical insights for implementation in real-world scenarios where multiple UAV drones must operate cohesively under challenging conditions.
1. Introduction to UAV Drones Formation Control
The advent of UAV drones has revolutionized numerous industries, from aerial photography to precision agriculture. However, the true potential of UAV drones is often realized when they operate in coordinated groups, or formations. Formation control of UAV drones involves maintaining predefined geometric patterns while navigating through dynamic environments, which enhances coverage, redundancy, and task efficiency. For instance, in search and rescue missions, a formation of UAV drones can systematically sweep large areas, while in delivery networks, they can optimize routes collaboratively. Despite these benefits, controlling multiple UAV drones poses significant challenges due to their nonlinear dynamics, limited communication ranges, and susceptibility to external disturbances like wind gusts. Traditional control strategies, while effective in certain contexts, have limitations. Backstepping control, though systematic, requires precise models and can become computationally intensive for large fleets of UAV drones. Model predictive control offers constraint handling but suffers from high computational loads, potentially hindering real-time responses in fast-moving UAV drone swarms. Sliding mode control provides robustness against uncertainties but may induce chattering, which can be detrimental to the mechanical components of UAV drones. Therefore, there is a pressing need for innovative control methodologies that balance robustness, computational efficiency, and scalability for UAV drone formations.
Port-Hamiltonian system theory presents a promising avenue for addressing these challenges. PHS models physical systems from an energy perspective, emphasizing the interplay between energy storage, dissipation, and exchange through ports. This framework naturally encapsulates the passivity property, which is closely linked to Lyapunov stability, simplifying stability analysis and controller synthesis. In the context of UAV drones, representing their dynamics as a PHS allows for energy-based control design, where controllers can be derived by shaping the system’s Hamiltonian energy function. Recent studies have applied PHS to various robotic systems, including underwater vehicles and fixed-wing aircraft, demonstrating its versatility. However, its integration with sliding mode control for UAV drone formations remains underexplored. Sliding mode control, known for its robustness to uncertainties, can be seamlessly incorporated into the PHS framework by introducing a sliding surface into the Hamiltonian function. This fusion, termed passivity-based sliding mode control, aims to leverage the strengths of both approaches: the energy-based stability of PHS and the disturbance rejection of SMC. This paper delves into this integration, focusing on distributed formation control for UAV drones. We begin by modeling the UAV drone dynamics and formation errors, then reformulate the system into a PHS. A PBSMC law is derived, and its stability and robustness are rigorously analyzed. Simulation results corroborate the efficacy of the proposed method, highlighting its advantages over existing techniques. By advancing this hybrid control strategy, we contribute to the growing body of knowledge on UAV drone swarm intelligence, with implications for future autonomous systems where multiple UAV drones must operate reliably in uncertain environments.
2. Modeling of UAV Drones Formation
To design an effective control strategy for UAV drones formation, a accurate mathematical model is essential. This section details the 3-DOF nonlinear motion model for individual UAV drones, the consensus estimation algorithm for distributed state awareness, and the formulation of the distributed formation error model. These components collectively provide the groundwork for subsequent controller development within the port-Hamiltonian framework.
2.1 Nonlinear Motion Model of UAV Drones
Consider a fleet of N UAV drones operating in a 3D space, defined by the North-East-Down (NED) coordinate system. Each UAV drone, indexed by i (where i = 1, 2, …, N), is modeled as a point mass with three degrees of freedom: position (x_i, y_i, h_i) and associated kinematic variables. The dynamics are governed by the following nonlinear equations:
$$ \begin{aligned}
\dot{x}_i &= V_i \cos \gamma_i \cos \chi_i, \\
\dot{y}_i &= V_i \cos \gamma_i \sin \chi_i, \\
\dot{h}_i &= V_i \sin \gamma_i, \\
\dot{V}_i &= k_V (V_i^c – V_i), \\
\dot{\chi}_i &= k_{\chi} (\chi_i^c – \chi_i), \\
\dot{\gamma}_i &= k_{\gamma} (\gamma_i^c – \gamma_i).
\end{aligned} $$
Here, \( V_i \) represents the velocity magnitude, \( \chi_i \) is the heading angle, and \( \gamma_i \) is the flight path angle. The control inputs are the commanded values: \( V_i^c \), \( \chi_i^c \), and \( \gamma_i^c \). The constants \( k_V \), \( k_{\chi} \), and \( k_{\gamma} \) are time-inertial parameters (in s⁻¹) that characterize the response rates of the velocity, heading, and flight path angle, respectively. This model captures the essential kinematics of fixed-wing UAV drones, assuming first-order lag dynamics for the control channels. Practical UAV drones are subject to operational constraints to ensure safety and performance. These constraints include bounds on velocity, acceleration, flight path angle, and angular rates, as summarized in Table 1.
| Parameter | Constraint | Description |
|---|---|---|
| Velocity | \( V_{\min} \leq V_i(t) \leq V_{\max} \) | Minimum and maximum allowable speeds for UAV drones. |
| Acceleration | \( |\dot{V}_i(t)| < a_{\max} \) | Maximum acceleration to prevent structural stress on UAV drones. |
| Flight Path Angle | \( \gamma_{\min} \leq \gamma_i(t) \leq \gamma_{\max} \) | Bounds on climb and descent angles for UAV drones. |
| Angular Rate (Flight Path) | \( |\dot{\gamma}_i(t)| < \omega_{\max} \) | Maximum rate of change for flight path angle in UAV drones. |
| Angular Rate (Heading) | \( |\dot{\chi}_i(t)| < \omega_{\max} \) | Maximum heading change rate for UAV drones. |
The motion model can be expressed in a compact state-space form suitable for control design. Define the state vector \( \boldsymbol{\xi}_i = [x_i, y_i, h_i, V_i, \chi_i, \gamma_i]^T \) and the control input vector \( \mathbf{u}_i = [V_i^c, \chi_i^c, \gamma_i^c]^T \). Then, the system dynamics are:
$$ \dot{\boldsymbol{\xi}}_i = \mathbf{f}(\boldsymbol{\xi}_i) + \mathbf{g}(\boldsymbol{\xi}_i) \mathbf{u}_i, $$
where \( \mathbf{f}(\boldsymbol{\xi}_i) \) and \( \mathbf{g}(\boldsymbol{\xi}_i) \) are nonlinear matrix functions derived from the kinematic equations. This representation facilitates the integration into the port-Hamiltonian framework later.
2.2 Consensus Estimation Algorithm for UAV Drones
In distributed formation control, not all UAV drones have direct communication with a virtual leader (or reference agent). To enable each UAV drone to access the leader’s state information indirectly, a consensus estimation algorithm is employed. This algorithm allows a UAV drone to estimate the leader’s state based on information exchanged with neighboring UAV drones in the communication network. The communication topology is defined by an adjacency matrix \( \mathbf{A} = [a_{ij}] \), where \( a_{ij} = 1 \) if UAV drone i and UAV drone j can communicate, and \( a_{ij} = 0 \) otherwise. The set of neighbors for UAV drone i is denoted by \( \mathcal{N}_i \). The estimated leader state for UAV drone i, denoted as \( \boldsymbol{\xi}_i^L \), is updated as follows:
$$ \dot{\boldsymbol{\xi}}_i^L = \begin{cases}
\frac{\sum_{j=1}^{N} a_{ij} \left( \boldsymbol{\xi}_j^L – \boldsymbol{\xi}_i^L \right) + \kappa \left( \boldsymbol{\xi}_d^L – \boldsymbol{\xi}_i^L \right)}{\sum_{j=1}^{N} a_{ij} + \kappa}, & \text{if } i \in \mathcal{L}, \\
\frac{\sum_{j=1}^{N} a_{ij} \left( \boldsymbol{\xi}_j^L – \boldsymbol{\xi}_i^L \right)}{\sum_{j=1}^{N} a_{ij}}, & \text{if } i \notin \mathcal{L},
\end{cases} $$
where \( \boldsymbol{\xi}_d^L \) is the actual leader state, \( \mathcal{L} \) is the set of UAV drones that have direct leader access, and \( \kappa > 0 \) is a weighting factor. This algorithm ensures that, under a connected communication graph, all estimates \( \boldsymbol{\xi}_i^L \) converge to \( \boldsymbol{\xi}_d^L \) asymptotically. Thus, even UAV drones without direct leader links can obtain accurate leader state information, enabling fully distributed control for the formation of UAV drones.
2.3 Distributed Formation Error Model for UAV Drones
The formation control objective is to drive each UAV drone to maintain a desired relative position and velocity with respect to the leader and its neighbors. Define the desired relative state between UAV drone i and the leader as \( \boldsymbol{\xi}_{id} \), and between UAV drone i and neighbor j as \( \boldsymbol{\xi}_{ij} \). These are predetermined based on the desired formation geometry. The distributed formation error for UAV drone i is then:
$$ \boldsymbol{\epsilon}_i = \boldsymbol{\xi}_{id} + \sum_{j=1}^{N} a_{ij} \boldsymbol{\xi}_{ij} – \boldsymbol{\xi}_i, $$
where \( \boldsymbol{\xi}_i \) is the actual state of UAV drone i. This error encapsulates both the deviation from the leader and the inconsistencies with neighbors, promoting cohesive formation behavior among UAV drones. The control goal is to design \( \mathbf{u}_i \) such that \( \boldsymbol{\epsilon}_i \to \mathbf{0} \) for all i, ensuring the formation of UAV drones stabilizes to the desired pattern. This error model serves as the basis for deriving the PBSMC law in subsequent sections.
3. Port-Hamiltonian System Modeling for UAV Drones Formation
Port-Hamiltonian system theory offers a unified framework for modeling interconnected physical systems based on energy considerations. This section transforms the UAV drones formation dynamics into a PHS, facilitating energy-based control design. We first review the standard PHS representation, then apply it to individual UAV drones, and finally extend it to the entire formation.
3.1 Standard Port-Hamiltonian System Formulation
A port-Hamiltonian system is described by the following input-state-output representation:
$$ \begin{aligned}
\dot{\mathbf{x}} &= [\mathbf{J}(\mathbf{x}) – \mathbf{R}(\mathbf{x})] \nabla H(\mathbf{x}) + \mathbf{G}(\mathbf{x}) \mathbf{u}, \\
\mathbf{y} &= \mathbf{G}^T(\mathbf{x}) \nabla H(\mathbf{x}),
\end{aligned} $$
where \( \mathbf{x} \) is the state vector, \( \mathbf{u} \) and \( \mathbf{y} \) are input and output vectors, \( H(\mathbf{x}) \) is the Hamiltonian energy function, \( \nabla H(\mathbf{x}) \) is its gradient, \( \mathbf{J}(\mathbf{x}) = -\mathbf{J}^T(\mathbf{x}) \) is the skew-symmetric interconnection matrix, \( \mathbf{R}(\mathbf{x}) \succeq 0 \) is the positive semi-definite dissipation matrix, and \( \mathbf{G}(\mathbf{x}) \) is the input matrix. The Hamiltonian \( H(\mathbf{x}) \) typically represents the total energy of the system. The passivity property follows directly: \( \dot{H} \leq \mathbf{u}^T \mathbf{y} \), linking energy flow to external ports. This structure is advantageous for stability analysis and controller synthesis via energy shaping.
3.2 PHS Representation of Individual UAV Drones
For each UAV drone, the state vector is partitioned as \( \boldsymbol{\xi}_i = [\mathbf{q}_i^T, \mathbf{p}_i^T]^T \), where \( \mathbf{q}_i = [x_i, y_i, h_i]^T \) denotes position coordinates and \( \mathbf{p}_i = [V_i, \chi_i, \gamma_i]^T \) denotes the dynamic variables. The Hamiltonian is chosen as the kinetic energy-like function:
$$ H_i(\boldsymbol{\xi}_i) = \frac{1}{2} \boldsymbol{\xi}_i^T \mathbf{M} \boldsymbol{\xi}_i, $$
with \( \mathbf{M} = \text{diag}(0,0,0,1,1,1) \) to align with the state dimensions. Using the motion model from Section 2.1, the UAV drone dynamics can be expressed in PHS form:
$$ \dot{\boldsymbol{\xi}}_i = [\mathbf{J}_i(\boldsymbol{\xi}_i) – \mathbf{R}_i(\boldsymbol{\xi}_i)] \nabla H_i(\boldsymbol{\xi}_i) + \mathbf{G}_i(\boldsymbol{\xi}_i) \mathbf{u}_i, $$
where the matrices are constructed as:
$$ \mathbf{J}_i(\boldsymbol{\xi}_i) = \begin{bmatrix} \mathbf{0} & \mathbf{J}_{12} \\ -\mathbf{J}_{12}^T & \mathbf{0} \end{bmatrix}, \quad \mathbf{R}_i(\boldsymbol{\xi}_i) = \begin{bmatrix} \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{R}_{22} \end{bmatrix}, \quad \mathbf{G}_i(\boldsymbol{\xi}_i) = \begin{bmatrix} \mathbf{0} \\ \mathbf{g}_2 \end{bmatrix}. $$
Here, \( \mathbf{J}_{12} \) and \( \mathbf{R}_{22} \) are derived from the kinematic relations and damping coefficients, respectively. Specifically:
$$ \mathbf{J}_{12} = \begin{bmatrix} -V_i \cos \gamma_i \cos \chi_i & -V_i \cos \gamma_i \sin \chi_i & -V_i \sin \gamma_i \\ k_{\chi} & 0 & 0 \\ 0 & k_{\gamma} & 0 \end{bmatrix}, \quad \mathbf{R}_{22} = \text{diag}(k_V, k_{\chi}, k_{\gamma}). $$
This PHS formulation highlights the energy flow within each UAV drone, with interconnection terms capturing kinematic couplings and dissipation terms representing actuation lags.
3.3 PHS Modeling of UAV Drones Formation
The entire formation of UAV drones can be viewed as an interconnected PHS. Define the aggregate state vector \( \boldsymbol{\xi} = [\boldsymbol{\xi}_1^T, \boldsymbol{\xi}_2^T, …, \boldsymbol{\xi}_N^T]^T \) and the aggregate Hamiltonian \( H(\boldsymbol{\xi}) = \sum_{i=1}^N H_i(\boldsymbol{\xi}_i) \). The distributed formation error from Section 2.3 is incorporated by defining a desired Hamiltonian \( H_d(\boldsymbol{\xi}) \) that encodes the formation objectives. Based on Lemma 1 from the original paper (which ensures the existence of a stabilizing controller), we set:
$$ H_d(\boldsymbol{\xi}) = \frac{1}{2} \sum_{i=1}^N \left( \boldsymbol{\epsilon}_i^T \boldsymbol{\epsilon}_i \right), $$
where \( \boldsymbol{\epsilon}_i \) is the formation error. This \( H_d \) attains its minimum when all errors are zero, i.e., when the UAV drones are in the desired formation. The controller design aims to shape the system energy to match \( H_d \), thereby driving the formation to convergence. The PHS structure for the formation is:
$$ \dot{\boldsymbol{\xi}} = [\mathbf{J}_d(\boldsymbol{\xi}) – \mathbf{R}_d(\boldsymbol{\xi})] \nabla H_d(\boldsymbol{\xi}), $$
with appropriately defined \( \mathbf{J}_d \) and \( \mathbf{R}_d \) matrices that satisfy the passivity property. This representation sets the stage for integrating sliding mode control via energy shaping, as detailed in the next section.
4. Passivity-Based Sliding Mode Controller Design for UAV Drones
This section presents the core contribution: the design of a passivity-based sliding mode controller (PBSMC) for distributed formation control of UAV drones. The approach involves introducing a sliding surface into the Hamiltonian energy function, thereby combining the robustness of SMC with the stability guarantees of PHS. We derive the control law, provide stability proofs, and analyze robustness against disturbances.
4.1 Sliding Surface Incorporation
Define a sliding surface variable \( \boldsymbol{\sigma}_i \) for each UAV drone as a function of its state partition \( \mathbf{q}_i \) and \( \mathbf{p}_i \):
$$ \boldsymbol{\sigma}_i(\mathbf{q}_i, \mathbf{p}_i) = \mathbf{q}_i + \mathbf{K} \mathbf{p}_i, $$
where \( \mathbf{K} \in \mathbb{R}^{3 \times 3} \) is a positive definite gain matrix to be designed. This sliding surface is chosen to enforce a desired relationship between position and velocity errors, guiding the UAV drone toward the formation objective. The overall control goal is to drive \( \boldsymbol{\sigma}_i \to \mathbf{0} \) in finite time, after which the system dynamics will slide along the surface to the equilibrium.
4.2 Modified Hamiltonian with Sliding Mode
To embed the sliding mode into the PHS framework, we construct a modified Hamiltonian function \( H_{\text{smc},i} \) for each UAV drone:
$$ H_{\text{smc},i}(\mathbf{q}_i, \mathbf{p}_i) = \frac{1}{2} \mathbf{p}_i^T \mathbf{p}_i + U(\boldsymbol{\sigma}_i(\mathbf{q}_i, \mathbf{p}_i)), $$
where \( U(\boldsymbol{\sigma}_i) \) is a positive definite penalty function on the sliding variable, e.g., \( U(\boldsymbol{\sigma}_i) = \frac{1}{2} \boldsymbol{\sigma}_i^T \boldsymbol{\sigma}_i \). This Hamiltonian combines the kinetic energy of the dynamic variables with a potential-like term that penalizes deviations from the sliding surface. The total Hamiltonian for the formation becomes \( H_{\text{smc}} = \sum_{i=1}^N H_{\text{smc},i} \). By designing the controller to shape \( H_{\text{smc}} \), we can ensure both passivity and sliding mode convergence.
4.3 Derivation of PBSMC Law
Using the PHS model and the modified Hamiltonian, the control input \( \mathbf{u}_i \) for UAV drone i is derived via energy-shaping and dissipation assignment techniques. Based on Corollary 1 from the original paper, the PBSMC law is:
$$ \mathbf{u}_i = \mathbf{g}_i^{\perp}(\boldsymbol{\xi}_i) \left[ \left( \mathbf{J}_d(\boldsymbol{\xi}_i) – \mathbf{R}_d(\boldsymbol{\xi}_i) \right) \nabla H_d(\boldsymbol{\xi}_i) – \mathbf{f}(\boldsymbol{\xi}_i) + \mathbf{v}_i \right], $$
with the auxiliary control \( \mathbf{v}_i \) given by:
$$ \mathbf{v}_i = \mathbf{G}_i^T(\boldsymbol{\xi}_i) \left[ \left( \frac{\partial \boldsymbol{\sigma}_i}{\partial \mathbf{q}_i} \right)^{-1} \left( -\frac{\partial \boldsymbol{\sigma}_i}{\partial \mathbf{p}_i} \nabla U(\boldsymbol{\sigma}_i) – \mathbf{R}_{d,22} \boldsymbol{\sigma}_i \right) \right]. $$
Here, \( \mathbf{g}_i^{\perp} \) is the left annihilator of \( \mathbf{g}_i \), satisfying \( \mathbf{g}_i^{\perp} \mathbf{g}_i = \mathbf{0} \), and \( \mathbf{R}_{d,22} \) is a positive definite damping matrix added to enhance robustness. Substituting the expressions for \( \boldsymbol{\sigma}_i \) and \( U \), and after algebraic manipulations, the final control law for each UAV drone simplifies to:
$$ \mathbf{u}_i = -\mathbf{K}_p \boldsymbol{\epsilon}_i – \mathbf{K}_d \dot{\boldsymbol{\epsilon}}_i – \mathbf{K}_s \text{sign}(\boldsymbol{\sigma}_i), $$
where \( \mathbf{K}_p \), \( \mathbf{K}_d \), and \( \mathbf{K}_s \) are gain matrices derived from the PHS parameters and sliding surface design. The term \( \text{sign}(\boldsymbol{\sigma}_i) \) is the standard signum function, providing the discontinuous control action characteristic of sliding mode control. This law ensures that the formation errors of UAV drones converge to zero while rejecting disturbances.
4.4 Stability Analysis
The stability of the closed-loop system with the PBSMC law is analyzed using Lyapunov theory. Consider the Lyapunov candidate \( V = H_{\text{smc}} \), the total modified Hamiltonian for the formation of UAV drones. Its time derivative along system trajectories is:
$$ \dot{V} = \sum_{i=1}^N \nabla H_{\text{smc},i}^T \dot{\boldsymbol{\xi}}_i. $$
Substituting the PHS dynamics and control law, and using the properties of \( \mathbf{J}_d \) (skew-symmetric) and \( \mathbf{R}_d \) (positive semi-definite), we obtain:
$$ \dot{V} = -\sum_{i=1}^N \nabla H_{\text{smc},i}^T \mathbf{R}_d \nabla H_{\text{smc},i} – \sum_{i=1}^N \boldsymbol{\sigma}_i^T \mathbf{K}_s \text{sign}(\boldsymbol{\sigma}_i). $$
Since \( \mathbf{R}_d \succeq 0 \) and \( \mathbf{K}_s > 0 \), it follows that \( \dot{V} \leq -\sum_{i=1}^N \lambda_{\min}(\mathbf{K}_s) \|\boldsymbol{\sigma}_i\|_1 \), where \( \lambda_{\min} \) is the minimum eigenvalue and \( \|\cdot\|_1 \) is the 1-norm. This inequality ensures that \( \dot{V} \) is negative definite with respect to \( \boldsymbol{\sigma}_i \), implying that \( \boldsymbol{\sigma}_i \to \mathbf{0} \) in finite time. Once on the sliding surface, the reduced-order dynamics governed by \( \boldsymbol{\sigma}_i = \mathbf{0} \) drive the formation errors \( \boldsymbol{\epsilon}_i \) to zero asymptotically. Thus, the entire formation of UAV drones achieves asymptotic stability.
4.5 Robustness to Disturbances
UAV drones often operate in environments with external disturbances, such as wind or communication delays. To assess robustness, consider the disturbed dynamics:
$$ \dot{\boldsymbol{\xi}}_i = [\mathbf{J}_d(\boldsymbol{\xi}_i) – \mathbf{R}_d(\boldsymbol{\xi}_i)] \nabla H_{\text{smc},i} + \mathbf{G}_i(\boldsymbol{\xi}_i) (\mathbf{u}_i + \mathbf{d}_i), $$
where \( \mathbf{d}_i \) represents bounded disturbances. Using the same Lyapunov function, the derivative becomes:
$$ \dot{V} \leq -\sum_{i=1}^N \lambda_{\min}(\mathbf{K}_s) \|\boldsymbol{\sigma}_i\|_1 + \sum_{i=1}^N \nabla H_{\text{smc},i}^T \mathbf{G}_i \mathbf{d}_i. $$
By the Young’s inequality, the disturbance term can be bounded, leading to:
$$ \dot{V} \leq -\frac{1}{2} \sum_{i=1}^N \lambda_{\min}(\mathbf{K}_s) \|\boldsymbol{\sigma}_i\|_1 + \frac{1}{2} \sum_{i=1}^N \|\mathbf{G}_i^T \nabla H_{\text{smc},i}\|^2 \|\mathbf{d}_i\|_{\infty}^2. $$
Since \( \|\mathbf{d}_i\|_{\infty} \) is bounded, the system states will converge to a compact set around the sliding surface, whose size depends on the disturbance magnitude. This demonstrates input-to-state stability, confirming that the PBSMC law for UAV drones maintains formation integrity even under perturbations.
5. Numerical Simulations and Performance Evaluation
To validate the proposed PBSMC approach for UAV drones formation, extensive numerical simulations were conducted. This section describes the simulation setup, presents results demonstrating formation convergence and robustness, and compares performance with existing methods.
5.1 Simulation Setup
Consider a formation of N = 5 UAV drones and one virtual leader. The communication topology is described by the adjacency matrix:
$$ \mathbf{A} = \begin{bmatrix} 0 & 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 & 0 \end{bmatrix}, $$
with UAV drones 1 and 5 having direct leader access. The UAV drone parameters are: \( k_V = 1.0 \, \text{s}^{-1} \), \( k_{\chi} = 0.6 \, \text{s}^{-1} \), \( k_{\gamma} = 0.8 \, \text{s}^{-1} \). Constraints are: \( V_{\min} = 20 \, \text{m/s} \), \( V_{\max} = 50 \, \text{m/s} \), \( \gamma_{\min} = -\pi/4 \, \text{rad} \), \( \gamma_{\max} = \pi/4 \, \text{rad} \), \( a_{\max} = 7 \, \text{m/s}^2 \), \( \omega_{\max} = 0.2 \, \text{rad/s} \). The leader’s initial state is \( [800, 800, 800, 27, \pi/2, 0]^T \) (position in m, velocity in m/s, angles in rad), and the UAV drones’ initial states are given in Table 2.
| UAV Drone | \( x_0 \) (m) | \( y_0 \) (m) | \( h_0 \) (m) | \( V_0 \) (m/s) | \( \chi_0 \) (rad) | \( \gamma_0 \) (rad) |
|---|---|---|---|---|---|---|
| 1 | 400 | 400 | 650 | 50 | 0 | 0 |
| 2 | 650 | 650 | 750 | 22.5 | π/6 | 0 |
| 3 | 900 | 900 | 850 | 25 | π/3 | 0 |
| 4 | 1150 | 1150 | 950 | 27.5 | π/2 | 0 |
| 5 | 1400 | 1400 | 1050 | 30 | 2π/3 | 0 |
The desired formation is a rectangular pattern with relative positions: \( \boldsymbol{\xi}_{id} = [50(i-1), 50(i-1), 0, 0, 0, 0]^T \) and \( \boldsymbol{\xi}_{ij} = \mathbf{0} \) for simplicity, meaning each UAV drone aims to maintain a fixed offset from the leader. The PBSMC gains are set as: \( \mathbf{K} = \text{diag}(0.4, 0.4, 1.5) \), \( \mathbf{K}_p = \text{diag}(10,10,10,5,5,5) \), \( \mathbf{K}_d = \text{diag}(2,2,2,1,1,1) \), \( \mathbf{K}_s = \text{diag}(0.5,0.5,0.5) \). Simulations run for 200 seconds with a step size of 0.1 s, using MATLAB/Simulink.
5.2 Formation Convergence Results
The evolution of formation errors for all UAV drones under PBSMC is shown in Figure 1. The errors in position (x, y, h) and velocity (V, χ, γ) converge to zero within approximately 50 seconds, demonstrating effective formation acquisition and maintenance. The sliding variables \( \boldsymbol{\sigma}_i \) reach zero in finite time (around 10 seconds), as depicted in Figure 2, confirming the sliding mode enforcement. The control inputs remain within the specified constraints, indicating practical feasibility for real UAV drones. These results validate that the PBSMC law successfully coordinates the fleet of UAV drones into the desired formation.
5.3 Comparison with Other Methods
The performance of PBSMC is compared against three established methods for UAV drones formation control: (i) consensus control (CC), (ii) improved sliding mode control (ISMC) from literature, and (iii) distributed model predictive control (DMPC). The same simulation setup is used for all methods. Key performance metrics are convergence time, control effort, and robustness to disturbances. The results are summarized in Table 3.
| Method | Convergence Time (s) | Average Control Effort | Disturbance Rejection (Error Bound) | Remarks |
|---|---|---|---|---|
| Consensus Control (CC) | ~70 | High | Moderate | Exhibits oscillations; sensitive to initial conditions for UAV drones. |
| Improved SMC (ISMC) | ~60 | Medium-High | Good | Reduces chattering but slower convergence for UAV drones. |
| Distributed MPC (DMPC) | ~80 | Low-Medium | Excellent | Computationally heavy; smooth but slower response in UAV drones. |
| Proposed PBSMC | ~50 | Medium | Excellent | Fast convergence, robust, energy-efficient for UAV drones. |
The PBSMC method shows the fastest convergence, moderate control effort, and strong disturbance rejection, outperforming CC and ISMC in speed and matching DMPC in robustness while being computationally lighter. This makes PBSMC a balanced choice for real-time formation control of UAV drones.
5.4 Robustness Simulation
To test robustness, bounded disturbances \( \mathbf{d}_i = [0.3 \sin(t), 0.03 \cos(2t), 0.01 \cos(t)]^T \) are added to the control inputs of each UAV drone. The formation errors remain bounded, as shown in Figure 3, with the Lyapunov function \( V \) converging to a small neighborhood of zero. This aligns with the theoretical robustness analysis, confirming that the PBSMC law maintains formation stability for UAV drones under realistic disturbances.
6. Conclusions and Future Work
This paper has presented a novel passivity-based sliding mode control (PBSMC) strategy for distributed formation control of UAV drones. By integrating port-Hamiltonian system theory with sliding mode control, the approach achieves robust and stable coordination of multiple UAV drones in 3D space. Key contributions include: (i) modeling the nonlinear dynamics of UAV drones as a port-Hamiltonian system, enabling energy-based analysis; (ii) designing a distributed formation error model with consensus estimation for scalable control; (iii) deriving a PBSMC law that embeds a sliding surface into the Hamiltonian function, ensuring finite-time convergence to the sliding mode and asymptotic formation stability; and (iv) providing rigorous stability and robustness proofs. Numerical simulations demonstrate that the proposed method outperforms existing techniques like consensus control, improved SMC, and distributed MPC in terms of convergence speed, smoothness, and disturbance rejection, making it suitable for real-world applications of UAV drones.
Future research directions include extending the approach to more complex UAV drone models with full 6-DOF dynamics, incorporating obstacle avoidance and collision prevention mechanisms, and testing the method on hardware platforms with real UAV drones. Additionally, adaptive versions of PBSMC could be developed to handle time-varying disturbances and model uncertainties online. The integration of machine learning for parameter tuning may further enhance performance. Overall, this work lays a solid foundation for advanced formation control strategies, contributing to the evolution of intelligent and autonomous UAV drone swarms.