Fault-tolerant dynamic formation control of the heterogeneous multi-agent system for cooperative wildfire tracking

This article addresses the collaborative wildfire tracking problem in air-ground heterogeneous multiple agent systems (HMAS) with faulty actuators during fire tracking missions. A fault-tolerant dynamicformationcontroltechniqueisproposedtoenableHMAStodynamicallytrackwildfireprop-agationusingthefractional-ordernonsingularterminalslidingmode(FO-NFTSM)controlapproach. Additionally,ahigher-orderslidingmodeobserver(HOSMO)isdesignedtoestimateunavailablelinearvelocitiesandmulti-sourcedisturbancesarisingfromexternaldisturbances,systemuncertain-ties,andactuatorfaults.Theresultsdemonstratetheeffectivenessoftheproposedcontroltechnique in guiding all agents for accurate wildfire tracking, leading to the convergence of collaborative tracking errors within a finite time. Comparative numerical simulations further validate the efficiency of the proposed approach.


Introduction
Forest fires present a significant threat to forested landscapes, wildlife, properties, and the courageous firefighters battling them.The resulting socio-economic challenges and irreversible environmental losses have prompted the development of effective techniques to combat these wildfires (Barmpoutis et al., 2020;Hossain et al., 2020;Mohapatra & Trinh, 2022).While conventional approaches, still in use today, often rely on extensive human resources and visual observations by fire suppression experts, this dependence on human visual assessment can lead to disastrous consequences due to factors like exhaustion, smoke obstruction, and inaccuracies in evaluating critical information (Ngombane et al., 2022).
Regrettably, the lack of timely and accurate information about the spread and state of wildfires has led to fatal accidents in firefighting operations.For instance, in August 2021, multiple wildfires erupted in Algeria's northern region, overwhelming emergency services and civilian volunteers.Despite efforts by deployed soldiers, factors like high winds, scorching heatwaves, and limited information about the dynamic spread of wildfires resulted in a devastating outcome, claiming the lives of 90 individuals, including 33 soldiers and 57 civilians (Africanews, 2021;Wikipedia, 2021).
CONTACT Joewell T. Mawanza joelmawanza@gmail.comTo mitigate the catastrophic effects of forest fires, continuous tracking of their dynamic evolution emerges as a potential solution.By continuously monitoring wildfires, critical information about their current status and progression can be gathered and provided to fire scene experts (Hu et al., 2022;Tariq et al., 2021).This detailed information empowers these experts to design more effective firefighting plans, determine optimal resource deployment locations, and identify safer escape routes for people, livestock, and valuable properties away from dangerous areas.
In this context, uncrewed ground (UG) vehicles and uncrewed aerial (UA) vehicles, commonly referred to as drones, have proven invaluable in enhancing operational safety and reducing human risks in various domains.These autonomous platforms have been successfully deployed in chemical spill inspections, landmine clearance, and structural damage assessments.UA vehicles offer aerial perspectives of forest fires, monitor their spread, and enable the deployment of water or fireextinguishing bombs to suppress the flames (Bouguettaya et al., 2022;Seraj et al., 2022;Sudhakar et al., 2020).Conversely, UG vehicles provide precise ground monitoring, possess higher payload capacity, and offer extended operation times (Radmanesh et al., 2021).Consequently, both UG and UA vehicles have become highly soughtafter platforms for forest fire monitoring and firefighting operations.
It is worth noting that relying on a single uncrewed vehicle (UV) would be insufficient to gather crucial forest fire parameters due to limitations in communication range and sensor capability.Moreover, a single UV lacks robustness and parallelism to effectively handle various tasks during forest fire monitoring.Therefore, the use of multiple UVs is necessary for collaborative forest fire monitoring, enhancing effectiveness and obtaining fire information from multiple locations along the fire spread perimeter simultaneously.Authors in Akhloufi et al. (2018) employed a UG-UA vehicle framework to collaboratively monitor and strategically suppress wildfires.Each UG and UA vehicle in the framework is equipped with a multimodal stereo-vision system, enabling precise fire detection and segmentation using different spectra.A time-varying formation control framework for a multi-fixed-wing UAV with actuator faults and external disturbances is proposed in Yu et al. (2021c).This control scheme consists of a sliding mode differentiator to estimate faults and perturbations in the system, as well as sliding mode control.To autonomously and actively monitor the progress of a wildfire, Pham et al. (2018) designed a decentralized control scheme for a group of UA vehicles.This control scheme allows UA vehicles to provide complementary views of the fire, increasing situational awareness by improving image resolution at a safe distance from the fire perimeter.
Authors in Harikumar et al. (2018) developed a control algorithm that enables a multiple UA vehicle system to effectively search for and dynamically track a forest fire.A reconfigurable fault-tolerant (F-T) collaborative control scheme capable of adapting the geometric shape of a group of UA vehicles based on different forest fire tasks is presented in Ghamry and Zhang (2016).In Kumar et al. (2011), the authors explored the development of cooperative control algorithms for a group of UA vehicles used in combating wildfires, focussing on monitoring and suppression objectives.They assigned a candidate utility function for optimization to each goal.These studies demonstrate the potential of using multiple UA vehicles for real-time forest fire detection and monitoring.
However, the collaborative dynamic monitoring problem, especially when an air-ground heterogeneous multiagent system (HMAS) used in monitoring missions is affected by unexpected actuator faults and external perturbations simultaneously, has received limited research attention.Failure to address actuator faults and external perturbations in a timely manner can significantly reduce the monitoring performance of an HMAS.In an HMAS composed of both UA and UG vehicles, actuator faults can lead to catastrophic mishaps, such as collisions, if not promptly addressed.Furthermore, a fault occurring in one agent can propagate to the rest, resulting in performance deterioration or even instability of the entire system.Therefore, it is crucial to mitigate the detrimental effects of actuator faults in an HMAS and improve performance and operational safety.
To address these challenges, several F-T control techniques have been developed in recent years.These techniques aim to mitigate the effects of actuator faults in an HMAS, enhancing performance and ensuring operational safety.Authors in Liu et al. (2020) proposed a distributed F-T formation control framework with an adaptive mechanism for a fleet of quadrotor UA vehicles, considering unexpected actuator faults.The control protocol combines the boundary layer theory and adaptive update technique to ensure that a group of UA vehicles form and maintain a specific geometric shape while tracking a desired trajectory, even in the presence of actuator malfunctions.An adaptive decentralized control framework to ensure the desired geometric shape of both UA and UG vehicle collaborative systems, even in the presence of actuator faults is proposed in Gong et al. (2022).In Yu, Zhang, et al. (2020) a decentralized control framework is proposed for fixed UA vehicles to maintain a predefined geometric pattern, despite wind perturbations and actuator malfunctions.A distributed adaptive F-T control scheme for a mixed multi-agent system, allowing it to maintain a time-varying formation and track desired trajectories, even in the presence of actuator malfunctions and external disturbances is presented in Ren et al. (2022).
Fractional-order (FO) calculus, an extension of integerorder (IO) calculus, has been extensively studied in various domains, including signal processing, bioengineering, control theory, dynamic systems, and system modelling (Birs et al., 2019;Monje et al., 2010;Sheng et al., 2011;Sun et al., 2018).In control theory, FO calculus is integrated with different control schemes to develop FO-control methods for both IO and FO systems.FO-control schemes offer advantages such as improved convergence rate, robustness, greater flexibility, and reduced energy consumption compared to IOcontrol schemes (Yu et al., 2021a).Recent advancements include FO-backstepping, FO-proportional resonant, FOfuzzy logic, FO-sliding mode, and FO-optimal control techniques for various systems (Ahmed et al., 2019;Yu, Badihi, et al., 2020;Yu et al., 2021b).However, there is a lack of research on integrating FO calculus into F-T dynamic formation control for collaborative surveillance of wildfires using an air-ground HMAS.
Based on the analysis mentioned above, this article aims to develop an F-T control framework for HMAS to collaboratively track wildfires, considering circular and elliptical propagation information.A higher-order sliding mode observer (HOSMO) is used to estimate the multi-source perturbation caused by uncertainties, external disturbances, and unexpected actuator faults.Leveraging FO calculus, a novel FO-NFTSM control framework is proposed to achieve collaborative dynamic monitoring of wildfires.The key contributions of this article are as follows: • In contrast to the dynamic wildfire tracking approach, which utilizes only one type of uncrewed vehicle in the system (Hossain et al., 2020;Radmanesh et al., 2021;Sudhakar et al., 2020), this article further investigates cooperative dynamic tracking of wildfire using an airground HMAS comprising of UG and quadrotor UA vehicles.Air-ground HMAS overcomes the limitations of both UG and quadrotor UA vehicles by offering several advantages, including increased localization capabilities, configuration capabilities, and payload capabilities.
• Different from the wildfire tracking approaches formulated in Yu et al. (2021c), Pham et al. (2018), Ghamry and Zhang (2016) and Harikumar et al. (2018), a novel F-T dynamic control scheme for an air-ground HMAS tasked with monitoring the propagation of wildfires in an elliptical and circular pattern is developed.This control scheme considers aerodynamic drag, actuator faults, external perturbations, and system uncertainties in the system.It ensures fast finite-time convergence of cooperative tracking errors to zero.• In contrast to the well-studied IO control frameworks in Seraj et al. (2022), Pham et al. (2018), Harikumar et al. (2018), Ghamry and Zhang (2016) and Mawanza et al. (2022), this article proposes a novel F-T dynamic formation control algorithm for air-ground HMAS to track wildfire propagation dynamically.The control algorithm comprises a HOSMO, which estimates both multi-source perturbation and unknown linear velocities of agents, an improved NFTSM control, which has an improved nonsingular fast terminal sliding surface integrated with an exponential terminal and FO calculus.
The remainder of this article is organized as follows.Preliminaries, dynamic models and problem description are presented in Section 2. The control scheme design and theoretical stability analysis are presented in Section 3. Numerical results are presented in Section 4 to illustrate the effectiveness of the proposed control scheme.Lastly, the conclusion is presented in Section 5.

Graph theory
In this article, G = (V, E, A) describes a weighted digraph is used characterize the exchange of information between N agents in the air-ground HMAS where In this article, all self-loops are non-existent, i.e. [a ii ] = 0.
A graph is considered to include a spanning tree if there exists a root agent with a directed link to the rest of the agents in the HMAS.We define the interaction of the virtual leader agent and follower agents using the diagonal matrix Q = diag{q 1 , q 2 , . . ., q N } where q i >0 if the virtual leader agent's information is accessible to the ith follower agent, otherwise q i = 0.

UG vehicle dynamic model
Using a 2-DOF point model, we can describe the dynamical model of the ith UG vehicle as follows (Xiao & Dong, 2020) where x i (t) and y i (t) represent the position of the ith UG vehicle, v i (t) denote the linear velocity, ω i (t) signifies the angular velocity and ϕ i (t) denotes orientation of the ith robot.F i and τ i represent the ith UG vehicle's force input and input torque.M i and J r i represent the mass, and moment of inertia of the ith UG vehicle, respectively.The control input of the ith UG vehicle can be expressed as: If the ith UG vehicle's centre point is directly included in the formation, this may lead to the non-holonomic constraint issue described below.
As a result of the non-holonomic constraint, the determinant of the matrix representing the system state is zero.
To cope with this problem, it is considered that the front point of the ith UG vehicle must be in the formation, and the front point is formulated as Ding et al. (2021): where P r is the distance between the reference point and the midpoint of the two wheels of the ith UG vehicle.x h i and y h i represent the positions of the front point.These are shown in Figure 1.The following position dynamics of the ith UG vehicle are obtained by differentiating Equation ( 4) and substituting it into Equation (3): where The ith UG vehicle's nonlinear term f i ( * , t) and its new control input U i (t) may be reformulated as follows:

Quadrotor UA vehicle dynamic model
The body-fixed frame denoted by where s * , t * and c * are the acronyms for the sin( * ), tan( * ) and cos( * ), respectively.The Euler angles θ i , φ i , ψ i denote the pitch, roll, and yaw for the ith quadrotor UA vehicle, respectively.It is worth noting that the Euler angles are constrained to (− π 2 , π 2 ), to avoid singularity problems.
Based on the work in Mawanza et al. (2022) and Zhang et al. (2019), the translational dynamic model for the ith quadrotor UA vehicle is formulated as: The rotational dynamic model for the ith quadrotor UA vehicle is formulated as: where signifies the position of the ith quadrotor UA vehicle in the Earth fixed-frame, while i = [φ i , θ i , ψ i ] T ∈ R 3 represents Euler angles that describe the posture of the aircraft.δ i,x , δ i,y , δ i,z , δ i,φ , δ i,θ and δ i,ψ are the bounded external perturbation in position and attitude dynamics, caused by wind gusts.k i , i = 1, . . ., 6 denotes the coefficients of aerodynamic drag forces and J x , J y and J z are the inertia moments of the aircraft.F i,T and U i,1 , U i,2 and U i,3 denote the total thrust of all rotors and control torque generated by four rotors, respectively.The relationship of are the control inputs F i,T and U i,1 , U i,2 and U i,3 with the angular velocities of the four rotors is as follows: where r b , k c and ω i i = 1, . . ., 4 are the aerodynamic lift force coefficient, torque coefficient and rotor angular speed, respectively.
It is worth noting the translational dynamics of the ith quadrotor UA vehicle can be rewritten in compact  form as: where T represents the nonlinear terms of the translational dynamics of the ith quadrotor UA vehicle.
In addition, the rotational dynamics of the ith quadrotor UA vehicle can be written in compact form as: where represents the nonlinear terms of the attitude dynamics of the ith quadrotor UA vehicle.The virtual controls of the quadrotor UA vehicle and the abbreviations of the used in Equations ( 12) and ( 13) are shown as follow: According to Equation ( 14), the desired attitude for the ith quadrotor UA vehicle, represented by id = [φ id , θ id , ψ id ] T , is produced from the inputs of the virtual control U i = [u i,x , u i,y , u i,z ] T .More precisely, we obtain where F i,t represent the total thrust of all rotors, φ id denotes the desired pitch angle, θ i,d signifies the desired roll angle, while ψ i,d denotes the yaw angle.It is worth noting that ψ i,d is a free variable that can take any value depending on the designer.

Fire propagation model
As shown in Mawanza et al. (2022) and Yu et al. (2021c) the fire propagation can be simulated by using an empirical model, which yields a reliable estimate of the fire front.In this article, fire is assumed to ignite at (x r0 , y r0 ) and propagates following both the elliptical and circular shape depending on fire propagation rate along X and Y axes.The fire spread model is shown in Figure 3, and has been mathematically formulated as: x ri = wt r + nt r cos α i + x r0 y ri = bt r cos α i + y r0 (18) where 0 ≤ α i ≤ 2π is the fire-boundary-associated included angle at point i.The propagation time of the fire is represented by t r , while the ignition point is denoted by (x r0 , y r0 ).n, b represent the rates of fire propagation along the X and Y axes, respectively.c is the speed at which the fire propagation centre moves in the direction of the wind.For the sake of simplicity, the direction of the wind is considered to be orthogonal to the Y-axis.It is worth noting the fire propagation model considered in this work can be either elliptical or circular depending on the changes in spread rates n and b.That is if: n = b : the fire propagation shape is circular n>b : the fire propagation shape is elliptical In the case of cooperative forest fire surveillance using an air-ground HMAS, the ith agent corresponds to the location i and the P 0 point is associated with a centre of the fire, which provides overall situation awareness.The points i and the P 0 are illustrated in Figure 3.The agents need to dynamically track the centre given as P 0 = [x r0 + wt r , y r0 , z 0 ] T ∈ R 3 , which has time-varying displacement defined as T ∈ R 3 is the ith agent's safe distance vector from the fire boundary.Hence, the ith agent's desired position signal is formulated as h i p = P 0 + T i r ∈ R 3 .

Factional order calculus
Here FO calculus preliminaries are presented, which will be utilized in the design of F-T dynamic formation control law for an air-ground HMAS employed in the cooperative forest fire monitoring application.It is worth noting that a wide variety of fractional definitions exist, and the Grünwald-Letnikov (G-L) is the most often utilized and implemented in different practical systems.The G-L definition of fractional differentiation is given as follows (Mainardi, 2018): where (t − a)/h is the integer value, the bounds of the operator are a and t, while α>0.α k denotes the binomial variable and is given as: )dt is the Gamma function.More detail may be found in Mainardi ( 2018)'s research.

Problem description
Dynamic formation control of air-ground HMAS is complicated compared to MAS due to the different dynamic characteristics and dimensions of the state space varying across the agents.For this control problem to be resolved, the dynamic model of quadrotor UA vehicles and UG vehicles must be merged.Thus, the dynamic model of the ith agent in the air-ground HMAS is as follows: where x i (t) ∈ R n represent the position, v i (t) ∈ R n denote the velocity and u F i (t) ∈ R n is the control input vector with faulty actuators and δ i (t) ∈ R n denotes the external perturbation affecting the ith agent.f i ( * , t) signified the system uncertainties emanating from internal friction.Taking into account that agents are prone to encounter faults when performing different tasks.To improve the practical application of the proposed control scheme, the modes of actuator faults are defined below as: where i (t) = diag{ i,1 (t), i,2 (t), . . ., in (t)}, 0< i,j (t) ≤ 1 denotes the unknown actuator efficiency factor and i = diag{ i,1 , i,2 , . . ., i,n } denotes bias fault.
Substituting Equations ( 22) into (23) yields By accounting for the actuator faults, external perturbations, and parameter uncertainties Equation ( 24) can be rewritten as where Assumption 2.1: The external perturbation δ i is bounded.i.e. |δ i | ≤ δi where δi is the upper bound of the perturbations.
Remark 2.1: The direct control of air-ground HMAS is challenging because the second-order dynamic model of both the quadrotor UA vehicles and UG vehicles are 3-D and 2-D, respectively.Thus, UG vehicles undergo dimension upgrade, which entails zeroing off the Z-axis positions, velocities and control inputs.

Higher-order sliding mode observer (HOSMO)
In this article, it is assumed that only x i the state variable is accessible due to uncertainties and external perturbations.A HOSMO is utilized to determine unavailable states and multi-source perturbations.First, we define Then, the HOSMO is designed as follows: where x1,i , x2,i and x 3,i are the estimation of x 1,i , x 2,i and x 3,i for the ith UV.While x1,i is the estimation error and is defined as x1,i = x 1,i − x1,i .λ 1,i , λ 2,i and λ 3,i are the positive observer gain.
Assumption 2.2: The multi-source perturbation ζ i is a Lipschitz and satisfy | ζi | ≤ i , where i is a positive constant.
Other estimation errors are defined as Then, the error dynamics of the HOSMO is defined as Utilizing Assumption 2.2, the estimation error dynamics are proven to be finite-time stable (Chalanga et al., 2016;Moreno, 2012).By appropriately designing the observer gains, the observer errors x1,i , x2,i and x3,i will converge to zero.Consequently, the state information x 1,i = x1,i , x 2,i = x2,i and x 3,i = x3,i can be obtained within a finite time.

Control objective
The control objective of this article is to develop an F-T control scheme that enables N air-ground HMAS to collaboratively monitor a dynamic wildfire with either circular or elliptical spread.The control scheme ensures a fast finite-time convergence rate even when a subset of agents experiences actuator faults and all follower agents are subjected to both nonlinear uncertainties and external disturbances.

Useful Lemmas
Lemma 2.1 (Ali et al., 2020): Given a nonlinear system ẏ = g(y(t)) with g(0) = 0 that satisfies the inequality V ≤ −σ V b 1 − λV where σ , λ>0 and b 1 ∈ (0, 1].Define V 0 as the starting value of V, then the system's state y(t) converges to the equilibrium point in finite-time from any initial point y 0 .
The settling time is constrained as T ≤ Mawanza et al., 2022):

Main results
The position dynamics of the virtual leader vehicle are defined as: In this article, the ith follower vehicle's position and velocity tracking errors are denoted as e i,x and e i,v , respectively.They are defined as: Suppose that only a fraction of follower agents in the HMAS have access to the virtual leader's information and the consistency between neighbouring follower agents is limited.The cooperative position and velocity control errors of the ith follower are expressed as: It is worth mentioning that the lack of velocity information precludes the direct utilization of the errors E i,v (i = 1, 2, . . ., N) to the formulation of the control scheme.As a result, an adjunct error Êi,v variable is introduced as: 31) and ( 32) may be rewritten as The non-singular fast terminal sliding surface for the ith vehicle is defined as: where sig κ 2 ( Êi,v ) = | Êi,v | κ 2 sign( Êi,v ) and 0<γ <1 is the FO operator.η 1 , η 2 and η 3 are positive constants.While, κ 1 >0, κ 2 >0 and they satisfy κ 1 >κ 2 .To improve the convergence speed, the reaching law is defined as: where k 4 and k 5 are positive integers.

Stability analysis
Theorem 3.1: Consider an air-ground HMAS, comprised of N agents, with it's dynamics described in Equation (25) tasked with monitoring a wildfire which is simulated by the mathematical model described in Equation (18).The collaborative tracking errors will converge to zero in fast finitetime with the proposed sliding surface in Equation (34) the control scheme in matrix form is developed as follows: Proof: . ., q N } and L represent the Laplace matrix.usw = diag{usw 1 , usw 2 , . . ., usw N } and sign(x 1 ) = diag{sign(x i,1 ), sign(x i,2 ), . . ., sign(x i,N )}.
Let S = [s 1 , s 2 , . . ., s N ] T and sig The sliding surface defined in Equation ( 34) can be represented in combat form as: Differentiating Equation (37) once, we obtain Substituting Equations ( 36) into (38) yields Based on the HOSMO's attributes, E v − Êv will converge to zero in finite time; hence Ṡ is rewritten as: The Lyapunov function is defined as The time derivative of Equation ( 41) is given as The terms 1 and 2 are positive constants, and they follow the following inequalities; ϒ 1 > 1 , ϒ 2 > 2 .Thus, the following deduction can be made below Based on Lemma 2.1, it can be concluded that the sliding mode will be achieved within a finite time if the condition Êv = 0 is satisfied.It is important to note that according to Xiong et al. (2017), when Êv = 0, it is not an attractor during the reaching phase, ensuring the guaranteed finite-time reachability of the sliding surface defined in Equation ( 34).Additionally, the tracking error E for neighbourhood formation will converge to zero along the sliding surface within a finite time.

Numerical simulation
Here, numerical simulations are provided to validate and demonstrate the efficacy of the proposed control scheme presented in the preceding sections.Consider an airground HMAS consisting of one virtual leader and six followers composed of two UG and four quadrotor UA vehicles.The unified position dynamics of these vehicles are described in Equation ( 25).The interaction topology of the HMAS is depicted in Figure 4.The Laplacian matrix L and interaction weight between the leader and follower matrix Q is defined as 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 (1, 0, 0, 0, 0, 0)   For the simulation, it is assumed that the forest fire has been spreading for 6 min.At this point, the follower agents are positioned along the perimeter to collaboratively monitor the wildfire and the virtual leader will be overhead monitoring the situation.The parameters of the fire propagation model are assumed as w = 0, b = 0.9 m/s, x r0 = 0 m, y r0 = 0 m and n = (1.43 − 0.43t/50) if t ≤ 50 1 e l s e The distance vector that ensures the safety of the airground HMAS is defined as o = [0.5, 0.5].Then, the desired position vector h i (t) is defined as The initial position states are set as x 1 = −9, y 1 = 1, x 2 = −3, y 2 = 7, x 3 = 5.57, y 3 = 5.73,z 3 = 0, x 4 = 9, y 4 = −1, z 4 = 0 x 5 = 3, y 5 = −7, z 5 = 0 x 6 = −5.578,y 6 = −5.73 and z 6 = 0.The simulation considers the impact of actuator faults to validate the proposed controller's robustness and fault tolerance.Additionally, the effect of unknown external disturbances is regarded throughout the simulation of the system.According to the fault model in Equation ( 23), when all agents are healthy, one has   Additionally, Figure 7 presents the estimation tracking error.Figure 8 showcases the multi-source disturbances, including those caused by external factors, uncertainties, and actuator faults, along with their corresponding estimates.Figures 9 and 10 illustrate the change from the elliptical to circular pattern of the fire as well as the trajectories of the agents in the system.Finally, the formation tracking errors are depicted in Figure 11.
The simulation results demonstrate that the proposed control scheme enables an air-ground HMAS to collaboratively monitor a dynamic wildfire.This is achieved even when all agents are affected by external disturbances and uncertainties, while a subset of agents may also experience actuator faults.

Conclusion
This article presented a finite-time cooperative wildfire monitoring approach utilizing a dynamic formation control scheme for air-ground HMASs, even in the presence of actuator malfunctions and multi-source perturbations.To address disparities between distinct agent models, a comprehensive dynamical model was developed to capture the motion characteristics of different agents effectively.The integration of a HOSMO enabled the estimation of the HMAS's multi-source perturbations and unknown states.Furthermore, an FO-NFTSM control scheme for HMAS was devised by incorporating the FO calculus into the dynamic wildfire tracking method.The theoretical analysis demonstrated that the proposed control algorithm ensured all agents in the HMAS monitored the wildfire perimeter at safe distances, resulting in the convergence of cooperative tracking errors to zero within a finite time.Comparative numerical results further validated the efficiency of the proposed strategy.

Disclosure statement
No potential conflict of interest was reported by the author(s).
Figure 2 are utilized to develop the dynamic model of a quadrotor UA vehicle.The transformation matrix of the ith quadrotor UA vehicle in the HMAS from O B to O E is defined as

Figure 4 .
Figure 4. Interaction topology of the HMAS.

Figure 6 .
Figure 6.2D trajectories of the air-ground HMAS.

Figure 7 .
Figure 7. Position and estimation tracking of the agents in the ground-air HMAS.

Figure 9 .
Figure 9. Snapshot at 7 s and interaction among agents.

Figure 10 .
Figure 10.Snapshot at 100 s and interaction among agent.