Adaptive Robust Constraint-Following Control of Vector–Rotor UAVs Subject to High-Intensity Time-Varying Water-Jet Disturbances

Abstract

In high-rise firefighting scenarios, unmanned aerial vehicles (UAVs) equipped with water-spraying systems are subjected to high-intensity and rapidly time-varying reaction forces induced by high-speed water jets. These forces introduce mismatched uncertainties with unknown bounds and make stable flight control particularly challenging. To address this problem, this paper proposes an adaptive robust constraint-following control (ARCFC) strategy for vector–rotor UAVs (VRUAVs). The controller is developed directly for the strongly nonlinear dynamics of the VRUAV without resorting to model linearization. Within a constraint-following-based nonlinear regulation framework, water-jet effects are explicitly modeled as rapidly time-varying uncertainties with unknown bounds, and an adaptive law is introduced to estimate conservative uncertainty bounds online for robust compensation. Lyapunov-based analysis is conducted to establish the uniform boundedness and uniform ultimate boundedness of the closed-loop system, and simulation results are presented to verify the effectiveness of the proposed approach. Compared with representative conventional control methods, the proposed ARCFC strategy provides improved disturbance-rejection capability and enhanced flight stability under demanding firefighting conditions.

1. Introduction

Unmanned aerial vehicles (UAVs) have emerged as promising tools for high-rise firefighting because they offer improved operational safety and deployment flexibility compared with conventional ground-based systems [1]. With the rapid growth of urban high-rise buildings, fire suppression in such environments has become increasingly challenging [2], since traditional firefighting approaches are often limited by insufficient reach, restricted accessibility, and slow deployment. UAV-mounted water-spraying systems provide a promising way to alleviate these limitations [3,4]. However, the high-speed water jets used for fire suppression generate strong reaction forces that can significantly disturb UAV flight dynamics and degrade stabilization performance, as illustrated in Figure 1 [5,6].

The control difficulty mainly arises from two aspects of water-jet dynamics. Firstly, conventional underactuated UAV architectures are inherently limited in compensating for water-jet-induced disturbances because their lateral motion is achieved through attitude adjustment [7], rather than through direct force generation along the disturbance direction [8,9]. Secondly, the fluid dynamics of water spraying (Figure 2a) introduce complex uncertainty patterns. These include transient water-hammer effects, which may produce pressure peaks two to three times higher than the steady-state level [10,11], and persistent bubble-induced flow fluctuations during continuous operation [12]. As a result, firefighting UAVs are subject to uncertainties that are not only strongly time-varying but also difficult to bound a priori [13]. These characteristics lead to two coupled control challenges: a mismatch between the vehicle actuation mechanism and the dominant disturbance direction, and the presence of fast time-varying uncertainties with unknown bounds.

Conventional quadrotor UAVs (QUAVs) (Figure 2b) [14] often have difficulty maintaining stable flight and accurate position regulation under the mismatch uncertainties induced by high-speed water jets because of their fixed-rotor architecture. By contrast, the vector–rotor UAV (VRUAV) studied in this paper adopts a fully actuated configuration with independently tiltable rotors (Figure 2c) [15], which enables decoupled control of translational motion and spraying direction. This configuration offers two main advantages. Firstly, it can generate thrust in multiple directions without requiring large body-attitude changes, thereby helping preserve the water-jet orientation during firefighting operations. Secondly, it can substantially alleviate the actuation–disturbance mismatch associated with conventional underactuated UAVs by reducing the dependence of horizontal motion on attitude reconfiguration [16,17]. Although this fully actuated architecture offers clear advantages for firefighting tasks [18,19,20], it also introduces stronger nonlinear coupling and more complex system dynamics, which place higher demands on controller design.

Most existing UAV control strategies are developed on the basis of model linearization. Such methods have achieved good performance in many conventional UAV applications. For example, Sadiq et al. [21] proposed a feedback linearization-based controller with uncertainty observers, while Yuan et al. [22] developed a feedback linearization–model predictive control (MPC) hybrid approach. However, the applicability of linearization-based methods becomes limited for VRUAV firefighting systems. These methods are typically most effective when rotor directions are fixed relative to the body frame and the vehicle operates in small-attitude conditions. In firefighting missions, however, the vehicle is often required to generate substantial horizontal force while operating with rotor tilting and pronounced nonlinear coupling, which weakens the validity and effectiveness of linearized control models.

To address these limitations, constraint-following control (CFC) provides a useful alternative for nonlinear mechanical systems. Pioneered by Chen et al. [23] and further developed in a series of studies by Sun et al. [24,25,26,27,28,29], CFC avoids direct model linearization and incorporates nonlinear motion constraints into controller design. Its control law is established from fundamental mechanical principles, such as Gauss’s principle of least constraint and d’Alembert’s principle, which makes it well suited to systems with strong nonlinearities and coupled dynamics. These features are particularly appealing for the present VRUAV, whose fully actuated yet strongly nonlinear structure is difficult to handle using conventional local linearization-based approaches.

In addition to strong nonlinearity, a second major challenge in VRUAV firefighting lies in the presence of fast time-varying uncertainties with unknown bounds, originating from both water-jet effects and system parameter variations. Such uncertainties can significantly deteriorate control performance if not properly compensated [30]. Existing methods, such as the backstepping sliding-mode controller proposed by Yu et al. [31] and the hybrid predictive control method reported in [32], often rely on constant or known uncertainty bounds. These assumptions, however, are difficult to justify in aerial firefighting scenarios, where the disturbance intensity may vary rapidly and irregularly. Adaptive control methods with online uncertainty estimation, such as the framework introduced by Chen et al. [33], provide an effective way to relax this requirement and have shown potential in a variety of complex systems [34,35,36,37,38].

Motivated by the above observations, this paper develops an adaptive robust constraint-following control (ARCFC) strategy for VRUAV state regulation under high-intensity time-varying water-jet disturbances. The proposed method combines the nonlinear mechanics-based formulation of CFC with online adaptive bound estimation, so that the controller can operate directly on the nonlinear dynamics without resorting to model linearization and without requiring prior knowledge of exact disturbance bounds. In the present study, the role of CFC is not merely to rewrite the regulation task in a constraint form. Rather, by embedding the desired spraying-state regulation objective into servo-constraints on the generalized motion, it provides a unified mechanical framework for constructing the nominal realization term, the stabilizing feedback term, and the uncertainty-compensation term.

The main contributions of this work are summarized as follows.

First, a VRUAV control framework is established for high-rise firefighting scenarios involving high-speed water spraying, in which the influence of water-jet-induced reaction forces and parametric uncertainties on flight stability is explicitly incorporated into the system dynamics.

Second, an ARCFC law is developed for nonlinear VRUAV state regulation under fast time-varying uncertainties with unknown bounds. The proposed controller integrates nominal constraint realization, stabilizing feedback, and adaptive robust compensation within a unified CFC framework.

Third, Lyapunov-based analysis is performed to establish the uniform boundedness and uniform ultimate boundedness of the closed-loop error system, and simulation studies under water-hammer-induced transient disturbance and bubble-induced rapidly varying disturbance are conducted to verify the effectiveness of the proposed method.

2. Problem Formulation

2.1. Dynamic Model of the VRUAV

To resist the complex reaction forces generated by high-intensity water jets, the VRUAV considered in this paper is equipped with a thrust-direction modulation mechanism. The configuration of the VRUAV is illustrated in Figure 3. The motion of the VRUAV is described using two coordinate frames: the inertial frame $\mathrm{E}(O_{\mathrm{E}},X_{\mathrm{E}},Y_{\mathrm{E}},Z_{\mathrm{E}})$ and the body-fixed frame $\mathrm{B}(O_{\mathrm{B}},X_{\mathrm{B}},Y_{\mathrm{B}},Z_{\mathrm{B}})$, whose origin is located at the center of gravity of the vehicle. The position and Euler-angle vectors of the VRUAV with respect to frame $\mathrm{E}$ are defined as $\varsigma ={\left[\begin{array}{ccc}x& y& z\end{array}\right]}^T,ϵ={\left[\begin{array}{ccc}\varphi & \theta & \psi \end{array}\right]}^T,$ respectively. In the firefighting task considered here, maintaining the VRUAV at a prescribed spraying position with a stable attitude is of primary importance.

Based on the above modeling, the uncertain dynamics of the VRUAV can be written as

$$M(p\left(t\right),\zeta \left(t\right),t)\ddot{p}\left(t\right)+C(\dot{p}\left(t\right),p\left(t\right),\zeta \left(t\right),t)\dot{p}\left(t\right)+G(p\left(t\right),\zeta \left(t\right),t)+K(\zeta \left(t\right),t)=u\left(t\right)$$

(1)

where $t\in {\mathbb{R}}_{\ge 0}$ denotes time, and $\zeta \left(t\right)\in {\mathbb{R}}^N$ is a possibly rapid and irregular time-varying uncertainty vector with unknown bounds, where N denotes the dimension of the uncertainty vector. The generalized state, velocity, and acceleration vectors are given by $p,\dot{p},\ddot{p}\in {\mathbb{R}}^6$, respectively, with $p={\left[\begin{array}{cccccc}x& y& z& \varphi & \theta & \psi \end{array}\right]}^T$, $\dot{p}={\left[\begin{array}{cccccc}\dot{x}& \dot{y}& \dot{z}& \dot{\varphi }& \dot{\theta }& \dot{\psi }\end{array}\right]}^T$, and $\ddot{p}={\left[\begin{array}{cccccc}\ddot{x}& \ddot{y}& \ddot{z}& \ddot{\varphi }& \ddot{\theta }& \ddot{\psi }\end{array}\right]}^T$. Moreover, $u\left(t\right)\in {\mathbb{R}}^6$ denotes the generalized control input at the rigid-body level, and $K(\zeta \left(t\right),t)\in {\mathbb{R}}^6$ denotes the generalized disturbance induced by the water-jet reaction. The matrices $M(·)\in {\mathbb{R}}^{6\times 6}$, $C(·)\in {\mathbb{R}}^{6\times 6}$, and $G(·)\in {\mathbb{R}}^6$ represent the inertia matrix, Coriolis/centrifugal matrix, and gravity vector of the system, respectively. The detailed derivation of the dynamic model is provided in Appendix A.

In this paper, $u\left(t\right)$ is generated by the upper-level nonlinear controller, while its realization in terms of individual rotor thrusts and tilting angles is handled by a lower-level control allocation module. Throughout the present analysis and simulations, this allocation is assumed to be feasible within the operating region of interest. Actuator saturation and allocation errors are not explicitly included in the current theoretical analysis and will be addressed in future work.

2.2. Control Objective

After reaching the designated spraying position, the VRUAV is required to maintain both its desired location and a stable attitude despite fluctuations in the reaction forces induced by the water jet. Specifically, the desired position components are $x_d$, $y_d$, and $z_d$ along the x-, y-, and z-axes, respectively, and the desired attitude components are ${\varphi }_d$, ${\theta }_d$, and ${\psi }_d$. Accordingly, the desired equilibrium state is defined as

$$p_d={\left[\begin{array}{cccccc}{x}_d& y_d& z_d& {\varphi }_d& {\theta }_d& {\psi }_d\end{array}\right]}^T.$$

(2)

Problem formulation: Design a control input $u\left(t\right)$ for the uncertain nonlinear system (1) such that the VRUAV achieves bounded state regulation toward the desired equilibrium state (2) in the presence of possibly rapid and irregular time-varying uncertainties $\zeta \left(t\right)$ with unknown bounds.

3. Control Design

3.1. Constraint-Following Error

To simplify the subsequent derivations, the functional arguments of variables are omitted whenever no ambiguity is caused. We first consider the first-order servo-constraint that a general mechanical system is required to satisfy:

$${\displaystyle \sum _{i=1}^n}{A}_{li}(p,t){\dot{p}}_i=c_l(p,t),l=1,\dots ,j,$$

(3)

which can be written in matrix form as

$$A(p,t)\dot{p}=c(p,t),$$

(4)

where $A(p,t)={\left[A_{li}(p,t)\right]}_{j\times n}$, $c(p,t)={\left[\begin{array}{cccc}{c}_1& c_2& \cdots & c_j\end{array}\right]}^T$, and $1\le j\le n$. The functions $A_{li}(·)$ and $c_l(·)$ are assumed to be continuously differentiable with respect to p and t. In general, the constraint in (4) may be non-integrable.

Differentiating (4) with respect to time yields

$${\displaystyle \sum _{i=1}^n}{\displaystyle \frac{d{A}_{li}(p,t)}{dt}}{\dot{p}}_i+{\displaystyle \sum _{i=1}^n}{A}_{li}(p,t){\ddot{p}}_i={\displaystyle \frac{d{c}_l(p,t)}{dt}},$$

(5)

where

$${\displaystyle \frac{d{A}_{li}(p,t)}{dt}}={\displaystyle \sum _{k=1}^n}{\displaystyle \frac{\partial A_{li}(p,t)}{\partial p_k}}{\dot{p}}_k+{\displaystyle \frac{\partial A_{li}(p,t)}{\partial t}},{\displaystyle \frac{d{c}_l(p,t)}{dt}}={\displaystyle \sum _{k=1}^n}{\displaystyle \frac{\partial c_l(p,t)}{\partial p_k}}{\dot{p}}_k+{\displaystyle \frac{\partial c_l(p,t)}{\partial t}}.$$

(6)

Therefore, the second-order form of the constraint can be expressed as

$${\displaystyle \sum _{i=1}^n}{A}_{li}(p,t){\ddot{p}}_i=-{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{d{A}_{li}(p,t)}{dt}}{\dot{p}}_i+{\displaystyle \frac{d{c}_l(p,t)}{dt}}=:b_l(\dot{p},p,t),l=1,\dots ,j,$$

(7)

or, equivalently, in matrix form as

$$A(p,t)\ddot{p}=b(\dot{p},p,t).$$

(8)

For the present spraying-state regulation task, the servo-constraint is constructed directly on the generalized states, and thus, $A={\left[I\right]}_{6\times 6}$. In this way, the desired equilibrium regulation objective is embedded into a unified servo-constraint form. The role of the CFC framework here is not merely to restate the control objective in a different notation; rather, it provides a mechanically interpretable basis for constructing the nominal realization term, the stabilizing feedback term, and the uncertainty-compensation term within a unified design.

By applying a suitable control input u, the controlled system can be made to satisfy the servo-constraint in (8) approximately. Based on (4), the corresponding constraint-following error is defined as

$$\lambda =A\dot{p}-c.$$

(9)

Accordingly, the regulation performance of the constrained motion can be characterized by $\lambda ={\left[\begin{array}{cccc}{\lambda }_1& {\lambda }_2& \cdots & {\lambda }_j\end{array}\right]}^T$. If the control input $u\in {\mathbb{R}}^6$ ensures that $\lambda$ remains uniformly bounded for all admissible uncertainties, then the VRUAV can achieve bounded real-time regulation toward the desired equilibrium state.

3.2. Adaptive Robust Control Design

The system matrices and the water-jet-induced generalized disturbance are decomposed into nominal and uncertain parts as

$$\begin{array}{cc}\hfill M(p,\zeta ,t)& =\overline{M}(p,t)+\Delta M(p,\zeta ,t),\hfill \\ \hfill C(p,\dot{p},\zeta ,t)& =\overline{C}(p,\dot{p},t)+\Delta C(p,\dot{p},\zeta ,t),\hfill \\ \hfill G(p,\zeta ,t)& =\overline{G}(p,t)+\Delta G(p,\zeta ,t),\hfill \\ \hfill K(p,\zeta ,t)& =\overline{K}(p,t)+\Delta K(p,\zeta ,t).\hfill \end{array}$$

(10)

Here, $\overline{M}(·)$, $\overline{C}(·)$, $\overline{G}(·)$, and $\overline{K}(·)$ denote the nominal parts, with $\overline{M}(p,t)>0$, while $\Delta M(·)$, $\Delta C(·)$, $\Delta G(·)$, and $\Delta K(·)$ represent the uncertain parts.

Assumption 1.

For the dynamic model of the VRUAV, the actual inertia matrix $M\left(p\right)$ is assumed to be symmetric and strictly positive definite, i.e., $M\left(p\right)>0$, across the entire physically meaningful state space.

Define

$$D(p,t):={\overline{M}}^{-1}(p,t),\Delta D(p,\zeta ,t):=M^{-1}(p,\zeta ,t)-{\overline{M}}^{-1}(p,t),$$

and

$$E(p,\zeta ,t):=-I+\overline{M}(p,t)M^{-1}(p,\zeta ,t),$$

so that

$$\Delta D(p,\zeta ,t)=D(p,t)E(p,\zeta ,t).$$

Accordingly, the nominal VRUAV model is written as

$$\overline{M}(p,t)\ddot{p}\left(t\right)+\overline{C}(p,\dot{p},t)\dot{p}\left(t\right)+\overline{G}(p,t)+\overline{K}(p,t)=u\left(t\right).$$

(11)

According to the nominal system (11) and the constraint (8), the nominal constraint force is given by

$$\begin{array}{cc}{P}_1(p,\dot{p},t):=\hfill & {\overline{M}}^{\frac{1}{2}}(p,t)\left[A(p,t){\overline{M}}^{-\frac{1}{2}}(p,t){]}^{+}\right\{b(\dot{p},p,t)\hfill \\ \hfill & +A(p,t){\overline{M}}^{-1}(p,t)[\overline{C}(p,\dot{p},t)\dot{p}+\overline{G}(p,t)+\overline{K}(p,t)]\}.\hfill \end{array}$$

(12)

The above term is constructed in the Lagrange form of d’Alembert’s principle so that the nominal system satisfies the imposed servo-constraint. When the model is exactly known, the control law based on (12) can enforce (8). However, in the presence of uncertainties, an additional robust adaptive compensation term is required.

For a given matrix $h\in {\mathbb{R}}^{6\times 6}$ with $h>0$, define

$$U(p,\zeta ,t):=hA(p,t)D(p,t)E(p,\zeta ,t)\overline{M}(p,t)A^T(p,t){\left[A(p,t)A^T(p,t)\right]}^{-1}{h}^{-1}.$$

(13)

Assume that there exists a constant $\rho >-1$ (possibly unknown) such that, for all $(p,t)\in {\mathbb{R}}^6\times \mathbb{R}$,

$${\displaystyle \frac{1}{2}}\underset{\zeta \in \mathrm{\Sigma }}{min}{\varrho }_m\left[U(p,\zeta ,t)+U^T(p,\zeta ,t)\right]\ge \rho ,$$

(14)

where ${\varrho }_m(·)$ denotes the minimum eigenvalue. Since the uncertainty set $\mathrm{\Sigma }$ is unknown, the constant $\rho$ is generally also unknown. When $M=\overline{M}$, one has $E=0$, $U=0$, and thus, $\rho =0$. Hence, (14) characterizes an admissible deviation range between M and $\overline{M}$.

Define further

$$\begin{array}{c}\hfill P_2(p,\dot{p},t):=-v\overline{M}(p,t)A^T(p,t){\left[A(p,t)A^T(p,t)\right]}^{-1}{h}^{-1}\left[A(p,t)\dot{p}-c(p,t)\right],\end{array}$$

(15)

where $v>0$ is a design constant.

We next introduce the following assumptions.

Assumption 2.

There exists an unknown constant vector $\eta \in {(0,\infty )}^k$ and a known function

$$\mathrm{\Pi }(·):{(0,\infty )}^k\times {\mathbb{R}}^6\times {\mathbb{R}}^6\times \mathbb{R}\to {\mathbb{R}}_{+},$$

such that, for all $(p,\dot{p},t)\in {\mathbb{R}}^6\times {\mathbb{R}}^6\times \mathbb{R}$ and all $\zeta \in \mathrm{\Sigma }$,

$$\begin{array}{c}{(1+\rho )}^{-1}\underset{\zeta \in \mathrm{\Sigma }}{max}\parallel hA\Delta D\left[-C\dot{p}-G-K+P_1+P_2\right]\hfill \\ -hAD\left[\Delta C\dot{p}+\Delta G+\Delta K\right]\parallel \le \mathrm{\Pi }(\eta ,p,\dot{p},t).\hfill \end{array}$$

(16)

Assumption 3.

For all $(p,\dot{p},t)\in {\mathbb{R}}^6\times {\mathbb{R}}^6\times \mathbb{R}$, the function $\mathrm{\Pi }(·,p,\dot{p},t):{(0,\infty )}^k\to {\mathbb{R}}_{+}$ satisfies the following criteria:

(i) It is continuously differentiable;

(ii) It is concave with respect to η, i.e., for any ${\eta }_1,{\eta }_2\in {(0,\infty )}^k$,

$$\mathrm{\Pi }({\eta }_1,p,\dot{p},t)-\mathrm{\Pi }({\eta }_2,p,\dot{p},t)\le {\displaystyle \frac{\partial \mathrm{\Pi }}{\partial \eta }}({\eta }_2,p,\dot{p},t)({\eta }_1-{\eta }_2);$$

(17)

(iii) It is monotonically nondecreasing in each component of η.

Since $\rho >-1$, it follows that $1+\rho >0$. The function $\mathrm{\Pi }(·)$ characterizes an upper bound associated with the uncertainties, while the unknown vector $\eta$ is related to the uncertainty set $\mathrm{\Sigma }$.

For some cases covered by Assumption 3, the function $\mathrm{\Pi }(\eta ,p,\dot{p},t)$ may be linear in $\eta$, i.e., there exists a known function $\widehat{\mathrm{\Pi }}(·):{\mathbb{R}}^6\times {\mathbb{R}}^6\times \mathbb{R}\to {\mathbb{R}}_{+}^k$ such that

$$\mathrm{\Pi }(\eta ,p,\dot{p},t)={\eta }^T\widehat{\mathrm{\Pi }}(p,\dot{p},t).$$

(18)

Based on the definition of the constraint-following error, the adaptive law is designed as

$$\dot{\widehat{\eta }}={\varrho }_1{\displaystyle \frac{\partial {\mathrm{\Pi }}^T}{\partial \eta }}(\widehat{\eta },p,\dot{p},t)\parallel \lambda (p,\dot{p},t)\parallel -{\varrho }_2\widehat{\eta },$$

(19)

where ${\varrho }_1>0$ and ${\varrho }_2>0$ are design constants.

The adaptive robust control law is then proposed as

$$u\left(t\right)=P_1(p,\dot{p},t)+P_2(p,\dot{p},t)+P_3(\widehat{\eta },p,\dot{p},t),$$

(20)

with

$$\begin{array}{cc}\hfill P_3(\widehat{\eta },p,\dot{p},t):=& -\left\{\overline{M}(p,t)A^T(p,t){\left[A(p,t)A^T(p,t)\right]}^{-1}{h}^{-1}\right\}\hfill \\ \hfill & ·\gamma (\widehat{\eta },p,\dot{p},t)\mu (\widehat{\eta },p,\dot{p},t)\mathrm{\Pi }(\widehat{\eta },p,\dot{p},t).\hfill \end{array}$$

(21)

where

$$\mu (\widehat{\eta },p,\dot{p},t)=\left[A(p,t)\dot{p}-c(p,t)\right]\mathrm{\Pi }(\widehat{\eta },p,\dot{p},t),$$

(22)

and

$$\gamma (\widehat{\eta },p,\dot{p},t)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\parallel \mu (\widehat{\eta },p,\dot{p},t)\parallel }},\hfill & \parallel \mu (\widehat{\eta },p,\dot{p},t)\parallel >\xi ,\hfill \\ {\displaystyle \frac{1}{\xi }},\hfill & \parallel \mu (\widehat{\eta },p,\dot{p},t)\parallel \le \xi ,\hfill \end{array}\right.$$

(23)

with $\xi >0$ being a small positive constant.

Remark 1.

It is worth noting that the design parameter ξ determines the trade-off between tracking accuracy and control effort in the proposed scheme. A smaller ξ permits a larger adaptation gain γ, resulting in increased control effort $P_3$ but a tighter ultimate error bound R, thereby improving tracking accuracy. Conversely, a larger ξ reduces the control effort while enlarging the tracking error bound. Therefore, $\xi >0$ serves as a key tuning parameter for balancing control performance and control expenditure in practical applications.

In the above controller, $P_3$ serves as the adaptive robust compensation term. Since the unknown parameter vector $\eta$ is unavailable, it is replaced by its estimate $\widehat{\eta }$. In the adaptive law (19), the first term updates the parameter estimate according to the magnitude of the constraint-following error, whereas the second term is a leakage term that prevents parameter drift and improves robustness.

Remark 2.

Because the controlled system is subject to uncertain disturbances, the proposed controller is designed to guarantee the uniform boundedness and uniform ultimate boundedness of the closed-loop error signals rather than exact asymptotic convergence. The control law consists of three coordinated components: $P_1$ is the nominal constraint force used to realize the desired constrained dynamics; $P_2$ is a stabilizing feedback term for reducing the constraint-following error; and $P_3$ is an adaptive robust compensation term for handling uncertainties.

Compared with existing nonlinear control studies on disturbance-affected VRUAVs, the proposed method has the following distinguishing features. First, a VRUAV model subject to strong water-jet-induced disturbances is considered, in which the time-varying reaction effects of water spraying are explicitly taken into account. Second, by introducing the constraint-following error, the spraying-state regulation problem is reformulated as an approximate constraint-following problem, which makes it possible to design the controller directly on the nonlinear dynamics without resorting to model linearization. Third, on this basis, an adaptive robust controller is developed to handle uncertainties with unknown bounds online while maintaining moderate control effort.

Theorem 1.

Let

$$\varpi :={\left[\begin{array}{cc}{\lambda }^T& {(\widehat{\eta }-\eta )}^T\end{array}\right]}^T\in {\mathbb{R}}^{m+k}.$$

Under Assumptions 1–2, the closed-loop VRUAV system (1) with the controller (20) renders ϖ uniformly bounded and uniformly ultimately bounded.

Proof. 

Let

$$V(\lambda ,\widehat{\eta }-\eta )={\lambda }^{\mathrm{T}}h\lambda +{\varrho }_1^{-1}(1+\rho ){(\widehat{\eta }-\eta )}^{\mathrm{T}}(\widehat{\eta }-\eta )$$

(24)

be a legitimate Lyapunov function candidate to prove the stability of the controller. Considering the uncertainties $\zeta (·)$ and the associated trajectories $p(·),\dot{p}(·)$ and $\widehat{\eta }(·)$, the derivative of V is obtained as

$$\dot{V}(\lambda ,\widehat{\eta }-\eta )=2{\lambda }^{\mathrm{T}}h\dot{\lambda }+2{\varrho }_1^{-1}(1+\rho ){(\widehat{\eta }-\eta )}^{\mathrm{T}}\dot{\widehat{\eta }}.$$

(25)

Let the analysis proceed by examining every term in the equation on its own. The first step is to consider

$$2{\lambda }^{\mathrm{T}}h\dot{\lambda }=2{\lambda }^{\mathrm{T}}h(A\ddot{p}-b)=2{\lambda }^{\mathrm{T}}hA\left[M^{-1}(-C\dot{p}-G-K)\right.\left.+M^{-1}\left(P_1+P_2+P_3\right)\right]-2{\lambda }^{\mathrm{T}}hb,$$

(26)

$$A\left[M^{-1}(-C\dot{p}-G-K)+M^{-1}\left(P_1+P_2+P_3\right)\right]-b=:V_1.$$

(27)

Here, $\dot{\lambda }=A\ddot{p}-b$; then, we have

$$\begin{array}{cc}\hfill V_1=& A[(D+\Delta D)(-\overline{C}\dot{p}-\overline{G}-\overline{K}-\Delta C\dot{p}-\Delta G-\Delta K)\left.+(D+\Delta D)\left(P_1+P_2+P_3\right)\right]-b\hfill \\ \hfill =& A\left[\Delta D\left(-C\dot{p}-G-K+P_1+P_2\right)\right.+D(-\Delta C\dot{p}-\Delta G-\Delta K)+D(-\overline{C}\dot{p}-\overline{G}-\overline{K})\hfill \\ \hfill & \left.+D{P}_1+D{P}_2+(D+\Delta D)P_3\right]-b.\hfill \end{array}$$

(28)

$$V_2:=\Delta D\left(-C\dot{p}-G-K+P_1+P_2\right)+D(-\Delta C\dot{p}-\Delta G-\Delta K),$$

(29)

$$V_3:=D(-\overline{C}\dot{p}-\overline{G}-\overline{K})+D{P}_1,$$

(30)

$$V_4:=D{P}_2,$$

(31)

$$V_5:=(D+\Delta D)P_3.$$

(32)

For the nominal case, where $\zeta \equiv 0$ and hence $\Delta M=\Delta C=\Delta G=\Delta K=0$, one has

$$V_3=0.$$

(33)

According to Equation (16),

$$\begin{array}{cc}\hfill 2{\lambda }^{\mathrm{T}}hA{V}_2& \le 2\parallel \lambda \parallel ∥hA{V}_2∥\hfill \\ \hfill & \le 2(1+\rho )\parallel \lambda \parallel \mathrm{\Pi }(\eta ,p,\dot{p},t).\hfill \end{array}$$

(34)

By Equation (15), we have

$$\begin{array}{cc}\hfill 2{\lambda }^{\mathrm{T}}hA{V}_4& =2{\lambda }^{\mathrm{T}}hAD\hfill \\ \hfill & \left[-v\overline{M}{A}^{\mathrm{T}}{\left(A{A}^{\mathrm{T}}\right)}^{-1}{h}^{-1}(A\dot{p}-c)\right]\hfill \\ \hfill & =-2v{\lambda }^{\mathrm{T}}(A\dot{p}-c)\hfill \\ \hfill & =-{2v\parallel \lambda \parallel }^2.\hfill \end{array}$$

(35)

Because $\Delta D=DE$, according to Equation (21), we have

$$2{\lambda }^{\mathrm{T}}hA{V}_5=2{\lambda }^{\mathrm{T}}hAD{P}_3+2{\lambda }^{\mathrm{T}}hADE{P}_3.$$

(36)

Since $P_3=-\left[\overline{M}{A}^{\mathrm{T}}{\left(A{A}^{\mathrm{T}}\right)}^{-1}{h}^{-1}\right]\gamma \mu \mathrm{\Pi }(\widehat{\eta },p,\dot{p},t)$, $\mu =$$\lambda \prod (\widehat{\eta },p,\dot{p},t)$, so

$$2{\lambda }^{\mathrm{T}}hAD{P}_3=-2{\left(\lambda \mathrm{\Pi }(\widehat{\eta },p,\dot{p},t)\right)}^T\gamma \mu =-2\gamma {\parallel \mu \parallel }^2.$$

(37)

From Equation (13), together with Equation (14), and in accordance with the Rayleigh principle, we arrive at

$$\begin{array}{cc}\hfill 2{\lambda }^{\mathrm{T}}hADE{P}_3& =-2{\mu }^{\mathrm{T}}\left[hADE{P}_3\right]\le -2\gamma {\mu }^{\mathrm{T}}{\displaystyle \frac{1}{2}}{\varrho }_m\left(U+U^{\mathrm{T}}\right)\mu \le -2\gamma \rho {\parallel \mu \parallel }^2.\hfill \end{array}$$

(38)

Upon substituting Equation (37) into Equation (38), the following result is derived:

$$2{\lambda }^{\mathrm{T}}hA{V}_5\le -2\gamma (1+\rho ){\parallel \mu \parallel }^2.$$

(39)

According to Equation (23), if $\parallel \mu \parallel >\xi$

$$\begin{array}{cc}\hfill -{2\gamma (1+\rho )\parallel \mu \parallel }^2& =-2(1+\rho ){\displaystyle \frac{1}{\parallel \mu \parallel }}{\parallel \mu \parallel }^2\hfill \\ \hfill & =-2(1+\rho )\parallel \mu \parallel ,\hfill \end{array}$$

(40)

and if $\parallel \mu \parallel \le \xi$

$$\begin{array}{cc}\hfill -{2\gamma (1+\rho )\parallel \mu \parallel }^2& =-2(1+\rho ){\displaystyle \frac{1}{\xi }}{\parallel \mu \parallel }^2\hfill \\ \hfill & =-2(1+\rho ){\displaystyle \frac{{\parallel \mu \parallel }^2}{\xi }}.\hfill \end{array}$$

(41)

Based on Equations (33)–(41), it can be concluded that if $\parallel \mu \parallel >\xi$,

$$\begin{array}{cc}\hfill 2{\lambda }^{\mathrm{T}}h\dot{\lambda }⩽& -{2v\parallel \lambda \parallel }^2+2(1+\rho )\{-\parallel \lambda \parallel \mathrm{\Pi }(\widehat{\eta },p,\dot{p},t)+\parallel \lambda \parallel \mathrm{\Pi }(\eta ,p,\dot{p},t)\},\hfill \end{array}$$

(42)

and if $\parallel \mu \parallel \le \xi$,

$$\begin{array}{cc}\hfill 2{\lambda }^{\mathrm{T}}h\dot{\lambda }\le & -{2v\parallel \lambda \parallel }^2-2\gamma (1+\rho ){\displaystyle \frac{{\parallel \mu \parallel }^2}{\xi }}+2(1+\rho )\parallel \lambda \parallel \mathrm{\Pi }(\eta ,p,\dot{p},t)\hfill \\ \hfill & \le -{2v\parallel \lambda \parallel }^2+(1+\rho ){\displaystyle \frac{\xi }{2}}+2(1+\rho )\{-\parallel \lambda \parallel \mathrm{\Pi }(\widehat{\eta },p,\dot{p},t)+\parallel \lambda \parallel \mathrm{\Pi }(\eta ,p,\dot{p},t)\}.\hfill \end{array}$$

(43)

Next, by Assumption 2,

$$\begin{array}{cc}\hfill -\parallel \lambda \parallel \mathrm{\Pi }(\widehat{\eta },p,\dot{p},t)& +\parallel \lambda \parallel \mathrm{\Pi }(\eta ,p,\dot{p},t)\le \parallel \lambda \parallel {\displaystyle \frac{\partial \mathrm{\Pi }(\widehat{\eta },p,\dot{p},t)}{\partial \eta }}(\eta -\widehat{\eta }).\hfill \end{array}$$

(44)

By replacing the second term on the right-hand side of Equation (25) into the adaptive law in Equation (19), we obtain

$$\begin{array}{c}2{\varrho }_1^{-1}(1+\rho ){(\widehat{\eta }-\eta )}^{\mathrm{T}}\dot{\widehat{\eta }}=2{\varrho }_1^{-1}(1+\rho ){(\widehat{\eta }-\eta )}^{\mathrm{T}}\left({\varrho }_1{\displaystyle \frac{\partial {\mathrm{\Pi }}^{\mathrm{T}}}{\partial \eta }}(\widehat{\eta },p,\dot{p},t)\parallel \lambda \parallel -{\varrho }_2\widehat{\eta }\right)\hfill \\ ⩽2(1+\rho ){(\widehat{\eta }-\eta )}^{\mathrm{T}}{\displaystyle \frac{\partial {\mathrm{\Pi }}^{\mathrm{T}}}{\partial \eta }}(\widehat{\eta },p,\dot{p},t)\parallel \lambda \parallel -2{\varrho }_1^{-1}{\varrho }_2(1+\rho )\parallel \widehat{\eta }{-\eta \parallel }^2+2{\varrho }_1^{-1}{\varrho }_2(1+\rho )\parallel \widehat{\eta }-\eta \parallel \parallel \eta \parallel .\hfill \end{array}$$

(45)

By substituting Equations (44) and (45) into Equation (24) $({\parallel \varpi \parallel }^2={\parallel \lambda \parallel }^2+$$\parallel \widehat{\eta }{-\eta \parallel }^2\parallel \widehat{\eta }-\eta \parallel \le \parallel \varpi \parallel )$, we have

$$\begin{array}{cc}\hfill \dot{V}⩽& -{2v\parallel \lambda \parallel }^2-2{\varrho }_1^{-1}{\varrho }_2(1+\rho ){\parallel \widehat{\eta }-\eta \parallel }^2\hfill \\ \hfill & +2{\varrho }_1^{-1}{\varrho }_2(1+\rho )\parallel \widehat{\eta }-\eta \parallel \parallel \eta \parallel +(1+\rho ){\displaystyle \frac{\xi }{2}}\hfill \\ \hfill ⩽& -{\kappa }_1{\parallel \varpi \parallel }^2+{\kappa }_2\parallel \varpi \parallel +{\kappa }_3.\hfill \end{array}$$

(46)

Here, ${\kappa }_1=min\left\{2v,2{\varrho }_1^{-1}{\varrho }_2(1+\rho )\right\},{\kappa }_2=2{\varrho }_1^{-1}$${\varrho }_2(1+\rho )\parallel \eta \parallel ,{\kappa }_3=(1+\rho )\xi /2$.

It then follows that the closed-loop system is uniformly bounded. Specifically,

(a) Uniform boundedness:

$$d\left(r\right)=\left\{\begin{array}{cc}\sqrt{{\displaystyle \frac{{\chi }_2}{{\chi }_1}}R}\hfill & r\le R\hfill \\ \sqrt{{\displaystyle \frac{{\chi }_2}{{\chi }_1}}}r\hfill & r>R\hfill \end{array}\right.,$$

(47)

$$R={\displaystyle \frac{1}{2{\kappa }_1}}\left({\kappa }_2+\sqrt{{\kappa }_{2^2}+4{\kappa }_2{\kappa }_3}\right).$$

(48)

where ${\chi }_1=min\left\{{\chi }_{min}\left(h\right),{\varrho }_1^{-1}(1+\rho )\right\},{\chi }_2=$$max\left\{{\chi }_{max}\left(h\right),{\varrho }_1^{-1}(1+\rho )\right\}$.

(b) Uniform ultimate boundedness:

$$\overline{d}>\underline{d}=\sqrt{{\displaystyle \frac{{\chi }_2}{{\chi }_1}}}R$$

(49)

$$L(\overline{d},r)=\left\{\begin{array}{cc}0\hfill & r\le \overline{d}\sqrt{{\displaystyle \frac{{\chi }_1}{{\chi }_2}}}\hfill \\ {\displaystyle \frac{{\chi }_2{r}^2-\left({\chi }_1^2/{\chi }_2\right){\overline{d}}^2}{{\kappa }_1{\overline{d}}^2\left({\chi }_1/{\chi }_2\right)-{\kappa }_2\overline{d}{\left({\chi }_1/{\chi }_2\right)}^{\frac{1}{2}}-{\kappa }_3}}\hfill & r>\overline{d}\sqrt{{\displaystyle \frac{{\chi }_1}{{\chi }_2}}}\hfill \end{array}\right..$$

(50)

These expressions indicate that, by appropriately selecting the design parameters v, ${\varrho }_1$, ${\varrho }_2$, and $\xi$, the ultimate bound $\overline{d}$ can be made sufficiently small. □

Remark 3.

This section develops an adaptive robust controller for the VRUAV under water-spraying disturbances, based on the nonlinear dynamic model (1) and the constraint-following formulation introduced in Section 3.1.

4. Control Framework

The overall control framework of the proposed ARCFC method for the VRUAV is illustrated in Figure 4. First, the nonlinear dynamics of the VRUAV are established with the explicit inclusion of water-jet-induced disturbances and parametric uncertainties. The desired spraying-state regulation objective is then converted into servo-constraints, from which the constraint-following error is derived. Based on this formulation, the generalized control input u is constructed as the sum of a nominal realization term $P_1$, a stabilizing feedback term $P_2$, and an adaptive robust compensation term $P_3$. The resulting generalized command is subsequently delivered to the lower-level control allocation module to generate feasible rotor thrusts and tilting actions. Through this upper-level/lower-level architecture, the VRUAV achieves bounded state regulation under strong time-varying disturbances.

5. Simulation and Results

This section evaluates the ability of the proposed controller to suppress reaction forces induced by water spraying, which constitute the dominant source of uncertainty in the considered firefighting scenario. The ARCFC method is assessed through two representative disturbance cases and benchmarked against a sliding mode control (SMC) regulator. In addition, the structural effect of the VRUAV configuration is illustrated through comparison with the QUAV under the same disturbance conditions.

5.1. System Model and Parameters

To ensure fairness and reproducibility in the comparative simulations, the VRUAV and the QUAV are assigned identical mass, geometric dimensions, and inertia parameters. Specifically, both platforms have a mass of $\overline{m}=50\mathrm{kg}$; moments of inertia ${\overline{I}}_{xx}=2.4\mathrm{kg}·{\mathrm{m}}^2$, ${\overline{I}}_{yy}=2.4\mathrm{kg}·{\mathrm{m}}^2$, and ${\overline{I}}_{zz}=3.0\mathrm{kg}·{\mathrm{m}}^2$; a wheelbase of $d=1.1\mathrm{m}$; and gravitational acceleration $g=9.8\mathrm{N}/\mathrm{kg}$. The detailed parameters are listed in

Table 1. Simulation parameters.

Parameter	Description	Value
$\overline{m}$	Mass of UAV	50 kg
d	Wheelbase of UAV	1.1 m
${\overline{I}}_{xx}$	Roll inertia of UAV	2.4 kg·m2
${\overline{I}}_{yy}$	Pitch inertia of UAV	2.4 kg·m2
${\overline{I}}_{zz}$	Yaw inertia of UAV	3.0 kg·m2
g	Gravitational acceleration	9.8 N/kg

.

The selected mass represents a heavy-duty UAV configuration for firefighting scenarios. For reference, industrial heavy-lift UAV platforms such as the DJI Agras T40 have a total weight of approximately 50 kg (including battery) and a maximum take-off weight of up to 90 kg. Although such platforms are primarily designed for spraying applications, they demonstrate that the selected mass falls within a realistic range for heavy-load multirotor UAV systems. The inertia parameters are selected consistently with the system mass.

Under high-intensity time-varying water-jet conditions, the uncertainties acting on the UAV mainly arise from the internal parametric uncertainties of the UAV itself and the external disturbances induced by the water jet. In this study, the mass and moments of inertia are treated as uncertain quantities in practical flight operations. Accordingly, these parameters are decomposed into nominal values and time-varying perturbations, namely

$$m=\overline{m}+\Delta m\left(t\right),I=\overline{I}+\Delta I\left(t\right),$$

where $\Delta m\left(t\right)$ and $\Delta I\left(t\right)$ denote time-varying uncertainties in the mass and inertia, respectively. In the simulations, these uncertainty terms are specified as

$$\Delta m\left(t\right)=0.001\overline{m}sin\left(t\right),\Delta I\left(t\right)=0.001\overline{I}sin\left(t\right),$$

so as to emulate small but time-varying parameter perturbations encountered in practical flight conditions.

It should be noted that in practical firefighting applications, the time-varying weight and the dynamic tension of the water hose are not explicitly modeled as coupled flexible dynamics; instead, they are reasonably incorporated into the lumped parametric uncertainties and external disturbances, which will be actively compensated for by the proposed ARCFC scheme.

5.2. Control Performance Under High-Intensity Fast Time-Varying Water-Jet Disturbances

To validate the effectiveness of the proposed controller under physically meaningful disturbance conditions, the water-jet-induced disturbances are modeled using two representative scenarios in this section. A benchmark SMC regulator is implemented for comparison with the proposed ARCFC method. SMC has been widely used in UAV control applications [39] and is therefore adopted here as a representative baseline. Based on the original system (1), the corresponding SMC dynamics are expressed as

$$\overline{M}(\dot{s}-\iota \dot{q}+{\ddot{p}}_d)+\overline{C}\dot{p}+\overline{G}+\overline{K}=u,$$

(51)

where $q=p-p_d$, the sliding surface is defined as $s=\iota q+\dot{q}$, and the reaching law is given by

$$\dot{s}=-\mathrm{\Lambda }\mathrm{sign}\left(s\right)-\mathrm{\Theta }s.$$

5.2.1. High-Intensity Transient Disturbance Caused by Water Hammer

The water-hammer effect is equivalently modeled as an external transient disturbance acting on the UAV. This disturbance is suddenly introduced at a prescribed instant with bounded but unknown magnitude to characterize the impact caused by the rapid opening or closing of the nozzle. It then gradually decays toward the steady-state disturbance corresponding to normal water-spraying operation, as illustrated in Figure 5. Its mathematical expression is given by

$$K_1\left(t\right)=K_{end}+\left(K_{start}-K_{end}\right)N\left(t\right),$$

(52)

where $K_{start}$ denotes the impulsive disturbance generated by the water-hammer effect, and $K_{end}$ denotes the steady disturbance associated with normal water spraying. The continuously differentiable transition function $N\left(t\right)$ is defined as

$$N\left(t\right)=\left\{\begin{array}{cc}0,\hfill & t<t_0,\hfill \\ {sin}^2\left({\displaystyle \frac{\pi (t-t_0)}{2T_{r+f}}}\right),\hfill & t_0\le t\le t_0+T_{r+f},\hfill \\ 1,\hfill & t>t_0+T_{r+f}.\hfill \end{array}\right.$$

(53)

Here, $t_0$ denotes the starting time of the water jet, and $T_{r+f}$ denotes the duration of the transient transition process.

At the beginning of the simulation, the UAV is assumed to have already reached the firefighting target, and the initial state is set as

$$p_1={\left[\begin{array}{cccccc}0& 0& 10& 0& 0& 0\end{array}\right]}^T.$$

The water jet is directed approximately along the positive x-axis of the body-fixed frame. In the experimental setup, the nozzle is mounted 20 cm below the center of mass of the VRUAV. At $t=1\mathrm{s}$, the nozzle generates a water-hammer-induced transient disturbance. The ARCFC controller is first evaluated on both the QUAV and VRUAV platforms. For the QUAV system, the adaptive-law parameters are chosen as ${\varrho }_{11}=0.1$ and ${\varrho }_{12}=0.1$. For the VRUAV system, the adaptive-law parameters are chosen as ${\varrho }_{21}=0.1$ and ${\varrho }_{22}=0.1$. The corresponding results are shown below.

Figure 6 shows the pitch-angle responses of the QUAV and VRUAV under the transient water-jet disturbance. The QUAV converges to a nonzero pitch angle, whereas the VRUAV maintains its pitch angle in the vicinity of zero, with only negligible fluctuations throughout the disturbance process. Although both platforms can partially compensate for the disturbance, their compensation mechanisms are fundamentally different. The QUAV relies on body tilting to generate a balancing horizontal force, which inevitably changes the spraying direction in practical operation. In contrast, the VRUAV can resist the disturbance while maintaining an approximately horizontal attitude, which is more favorable for aerial firefighting tasks requiring stable jet orientation.

To further evaluate the proposed ARCFC method, it is compared with the SMC controller on the VRUAV platform. For the SMC law (51) under the water-hammer disturbance, the controller parameters are selected as

$$\begin{array}{cc}\hfill {\iota }_1& =diag\left(\right[10,10,5,5,5,5\left]\right),\hfill \\ \hfill {\mathrm{\Lambda }}_1& =diag\left(\right[1,1,1.2,1.2,1.2,1.2\left]\right),\hfill \\ \hfill {\mathrm{\Theta }}_1& =diag\left(\right[1,1,1,1,1,1\left]\right).\hfill \end{array}$$

As shown in Figure 7a, under the transient water-hammer disturbance, the VRUAV controlled by SMC exhibits slower position recovery and a larger residual deviation along the x-axis. By contrast, the ARCFC controller confines the position error to a smaller range and drives the system back to the equilibrium state more rapidly. This indicates that the proposed controller achieves stronger disturbance rejection and better recovery performance under severe transient uncertainty.

Figure 7b compares the attitude responses of ARCFC and SMC. It can be observed that the SMC controller is more sensitive to the transient high-intensity disturbance and produces more pronounced oscillations in the tilt angle. In contrast, ARCFC suppresses the disturbance-induced attitude oscillation more effectively and yields a smoother attitude response.

Figure 7c presents the rotor control input generated by the two control strategies. The SMC controller exhibits more pronounced fluctuations throughout the simulation, indicating stronger sensitivity to the disturbance. By comparison, the ARCFC controller produces a smoother control input with a smaller overall magnitude. This suggests that the proposed controller not only preserves control performance but also reduces control effort.

As shown in Figure 8, the adaptive parameters gradually decrease and converge to a bounded range. This behavior is due to the introduced leakage term, which prevents parameter drift once the state error becomes small and the disturbance is effectively compensated. The result is consistent with the boundedness analysis of the adaptive estimation process and closed-loop system.

5.2.2. Bubble-Induced Rapidly Varying Disturbances

The second scenario considers local flow instabilities caused by bubble generation and collapse, which make the reaction force acting on the UAV exhibit rapid and time-varying fluctuations. This effect is equivalently modeled as a time-varying periodic disturbance. Sinusoidal components are used to capture the dominant rapidly varying characteristics of the water-spraying disturbance, leading to the following model:

$$K_2\left(t\right)=K_s+A_{1}cos(2\pi f_{1}t+l_1)+A_{2}cos(2\pi f_{2}t+l_2)+W_r,$$

(54)

where $K_s$ denotes the steady disturbance associated with normal water spraying; $A_1$ and $A_2$ are the amplitudes of the disturbance components; $f_1$ and $f_2$ are their corresponding frequencies; and $l_1$ and $l_2$ denote the phase shifts. $W_r$ is defined as a zero-mean, unit-variance Gaussian stochastic process with a 100 Hz sampling frequency. It is postulated that all uncertainties and disturbances are bounded; however, the precise magnitudes of these bounds remain unknown a priori to the controller. The resulting disturbance profile is shown in Figure 9.

In this case, the UAV is assumed to have passed the initial transient stage of spraying and entered a stable spraying condition. The initial state is set as

$$p_2={\left[\begin{array}{cccccc}0& 0& 10& 0& 0& 0\end{array}\right]}^T.$$

The water jet is generated by a nozzle mounted 20 cm below the center of mass of the VRUAV, and its direction remains aligned with the positive x-axis of the body-fixed frame. The bubble-induced rapidly varying disturbance is then applied to represent the oscillatory reaction force during sustained spraying. For the QUAV system, the adaptive-law parameters are selected as ${\varrho }_{31}=0.5$ and ${\varrho }_{32}=0.5$. For the VRUAV system, the adaptive-law parameters are selected as ${\varrho }_{41}=1$ and ${\varrho }_{42}=1$. The simulation results are shown below.

Figure 10 shows the pitch-angle responses of the QUAV and VRUAV under bubble-induced rapidly varying disturbance. The QUAV continuously adjusts its pitch angle to compensate for the oscillatory reaction force, which causes coupled fluctuations in the spraying direction. In contrast, the VRUAV experiences only minor pitch-angle deviations during the disturbance process, indicating its capability to suppress disturbance effects without requiring significant body-attitude adjustment.

To further assess disturbance-rejection performance, the proposed ARCFC method is again compared with SMC on the VRUAV platform. In this scenario, the SMC parameters in (51) are selected as

$$\begin{array}{cc}\hfill {\iota }_2& =diag\left(\right[15,15,3,3,3,3\left]\right),\hfill \\ \hfill {\mathrm{\Lambda }}_2& =diag\left(\right[0.1,0.1,0.5,0.5,0.5,0.5\left]\right),\hfill \\ \hfill {\mathrm{\Theta }}_2& =diag\left(\right[1,1,5,5,5,5\left]\right).\hfill \end{array}$$

As shown in Figure 11a, under the rapidly varying disturbance, the SMC-controlled VRUAV exhibits larger deviation and weaker recovery performance along the x-axis. By comparison, the proposed ARCFC controller effectively confines the position error within a smaller range and restores the system state more quickly. These results indicate that ARCFC retains strong disturbance rejection and state-regulation capability even under fast time-varying uncertainty.

Figure 11b compares the tilt-angle responses of ARCFC and SMC under the rapidly varying disturbance. The SMC controller exhibits more pronounced oscillatory behavior, whereas ARCFC maintains a noticeably smoother attitude response. This result further shows that the proposed method is less sensitive to rapidly varying disturbance components.

Figure 11c shows the rotor control inputs produced by the two controllers. The SMC controller exhibits larger fluctuations and stronger sensitivity to high-frequency disturbance components. In contrast, the control input generated by ARCFC is smoother and has a smaller overall amplitude, indicating that the proposed controller can reduce control effort while maintaining satisfactory control performance.

Finally, Figure 12 shows that the adaptive parameters remain bounded and converge into a stable range over time. Again, this behavior is attributed to the leakage term in the adaptive law, which suppresses parameter drift after the disturbance has been effectively compensated. This result is consistent with the theoretical boundedness properties established in the previous section.

6. Conclusions

This paper investigated the state-regulation problem of a VRUAV operating under high-intensity time-varying water-jet disturbances in high-rise firefighting scenarios. To address the strong nonlinearity of VRUAV dynamics and the presence of rapidly varying uncertainties with unknown bounds, an ARCFC strategy was proposed. By explicitly incorporating the generalized disturbance induced by water-jet reaction forces, the VRUAV’s dynamics under water-spraying conditions were established, and the regulation problem was reformulated as an approximate constraint-following problem through the introduction of the constraint-following error. Based on this framework, an adaptive robust controller was developed to compensate for uncertainties with unknown bounds, and Lyapunov-based analysis was carried out to establish uniform boundedness and uniform ultimate boundedness of the closed-loop error system. Simulation studies under two representative disturbance scenarios, namely water-hammer-induced transient disturbance and bubble-induced rapidly varying disturbance, showed that the proposed method enables the VRUAV to maintain the spraying direction more effectively than the QUAV configuration, while also achieving improved position regulation performance, smoother attitude responses, and lower control effort than the benchmark SMC regulator.

Future work will focus on hardware-in-the-loop and real-flight experimental validation, tighter integration between upper-level control and lower-level control allocation, and the extension of the proposed method to more realistic coupled aerodynamic–fluid disturbance environments.