Peng Han, Weidong Zhang, Simon X. Yang, and Hongtian Chen
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[28].22.2 Models of Heterogeneous Systems with FaultsConsidering the heterogeneous cooperative unmannedsystem described above, the kinematic equations of motionfor USV and UAV are given as:.ps = Rsvs,.Rs = RsS(ωs),.pa = Rava,.Ra = RaS(ωa)(1)where ps, pa ∈ R3represent the positions within thegeodetic coordinate system FI; vs, va ∈ R3, respectively,denote the linear velocities within body coordinate systemsFs and Fa; ωs, ωa ∈ R3, respectively, indicate theangular velocities within Fs and Fa; the rotation matricesRs ≡FIFsR, Ra ≡FIFaR ∈ R3×3facilitate the transformationbetween the respective coordinate systems; the operatorS(·) : R3→ R3×3is designated such that for any x, y ∈ R3,S(x)y = x × y.The dynamic equations of USV and UAV areformulated as:Ms.vs = −S(ωs)Msvs − Dsvvs + e1Ts + fsv,Js.ωs = −dsωωs + e3τs + e3fsω,ma.va = −S(ωa)mava + magRa e3 − e3Ta + fav,Ja.ωa = −S(ωa)Jaωa + τa + faω(2)In these dynamic equations, Ms = diag(ms−msx, ms−msy, 1) represents USV’s mass matrix, where ms is USV’smass, msx and msy are USV’s added masses, and madenotes UAV’s mass; Js and Ja = diag(Jax, Jay, Jaz) rep-resent the inertial moments; Dsv = diag(dsvx, dsvy, 0), dsware the damping coefficients; Ts, τs denote USV’s thrustand torque inputs; the thrust input for UAV, representingthe collective force of all motors in the direction of gravity,is denoted by Ta; τa represents UAV’s torque input; unitvectors are defined as e1 ≡ [1, 0, 0] , e2 = [0, 1, 0] , ande3 = [0, 0, 1] . Under the instantaneous fault detectionassumption, the time-varying actuator faults fsv, fsω, fav,and faω are bounded with positive constants fsv, fsω, fav,and faω, and are detected and isolated without delay forcompensation.A virtual leader is introduced to limit the impact of asingle agent’s faults on the overall formation. The virtualleader’s path is denoted as pl(γd) ∈ R3, where γd∈ R isa continuous parameter. A tangent frame Fp is attachedto the formation. Assuming pl(γi) = 0, without loss ofgenerality, the tangent vector of the path pl(γi) is given byrT (γi) = pl(γi)/ pl(γi) .Let rd be a unit vector unparalleled to rT (γi). Therotation matrix IT R(γi) is constructed as IT R(γi) =[r1(γi), r2(γi), rd] , where:r1(γi) =rT (γi)−(rT (γi)rd)rdrT (γi)−(rT (γi)rd)rd,r2(γi) = rd × r1 (γi) . The formation is describedby a distance matrix D(γi) = [d1(γi), . . . , dn(γi)],where di(γi) ∈ R3represents the time-varying distancevector from the virtual leader to the ith followerin the tangent frame Tp. Then the individualpath for each agent pdi(γi) can be generatedas pdi(γi) = pl(γi) +IT R(γi)di(γi).2.3 Objective and Design IssuesThe topological structure of a multi-agent systemcomprising n agents can be represented by a graph G,characterised by its Laplacian matrix L, and orientedincidence matrix N.The formation path coordination error zcp is definedas:zcp = N γ (3)where γ = [γ1, . . . , γn]. The formation speed coordinationerror zcv is defined as:zcv =.γ − 1n×1.γd (4)For the ith follower in the formation, the kinematicand dynamic models are defined by (1) and (2). Itsdesired path pdi(γd) is belongs to class C4. Since thecontroller is computed at the current time t, withcorresponding valuables γi, the valuables pdi(γi) and pi(t)are, respectively, abbreviated as pdi and pifor notationalsimplicity, where pi and pdi are the actual and desiredpositions of the ith followers. The formation followingerrors zpi are defined as:zpi = pi − pdi, i = 1, . . . , n (5)The objectives and design issues of the heterogeneouscooperative fault-tolerant formation system are split asfollows:1. Design cooperative timing laws to achieve the conver-gence of formation path and speed coordination errorsfor any given bounded initial states, which is formulatedas limt→∞zcp = 0, limt→∞zcv = 0, and the constraintshall be satisfied that the path coordination parametersγ is continuous and differentiable, described as γi ∈ C4for i = 1, . . . , n.2. Design formation control laws, including the thrust andtorque forces, to ensure the convergence of formationfollowing errors despite the presence of time-varyingactuator faults, which can be formulated as limt→∞zpi =0, i = 1, . . . , n.3. The Proposed MethodologyIn this section, the formation coordination method isfirst introduced. Then, the fault-tolerant formation controlmethods are given.3.1 Formation Coordination MethodFollowing the definitions of the path and speed coordi-nation errors, the Lyapunov function is established asVc = 12 zcpzcp + 12 zcvΛ−1c zcv, where Λc is a positive definitediagonal matrix. The time derivative of the Lyapunov3function Vc is given by:˙Vc = γ L zcv + 1n×1.γd + zcvΛ−1c..γ − 1n×1¨γd= zcv Lγ + Λ−1c..γ − 1n×1¨γd (6)where L1n×1 = 0n×1. By adding and subtracting a non-negative term zcvKczcv, where Kc is a positive definitediagonal matrix, the derivative is expressed as:˙Vc = −zcvKczcv+ zcv Lγ + Λ−1c..γ − 1n×1¨γd + Kczcv (7)To construct a negative definite function ˙Vc, thecooperative timing law is chosen as:..γ = 1n×1¨γd − Λc (Lγ + Kczcv) (8)Following the timing law, the time derivative of theLyapunov function ˙Vc is given as ˙Vc = −zcvKczcv, whichis non-positive and equals to zero if and only if zcv = 0n×1.With the implementation of the cooperative timing law(8), the cooperative path pdi for each agent is effectivelydetermined.3.2 Fault-Tolerant Formation Control MethodsFor notational clarity, vehicle-specific subscripts areemployed: s for USV agents and a for UAV agents,replacing the generic agent subscript i throughout ourformulation.Based on the definition of the formation following error(8), the position errors zs1 and za1 are defined as:zs1 = Rs zps = Rs (ps − pds) ,za1 = Ra zpa = Ra (pa − pda) (9)The first Lyapunov functions are established asVs1 = 12 zs1zs1 and Va1 = 12 za1za1. The deriva-tives of the first Lyapunov functions are given as˙Vs1 = −Ws1 (zs1) + zs1 vs − Rs.pds + ks1M−1s zs1 and˙Va1 = −Wa1 (za1) + za1 va − Ra.pda + ka1m−1a za1 , whereWs1(zs1) = ks1zs1M−1s zs1, Wa1(za1) = ka1za1m−1a za1 andks1, ka1 are positive constants.To ensure that the derivatives ˙Vs1 and ˙Va1 are negativedefinite, linear velocity errors are defined as:zs2 = vs − Rs.pds + ks1M−1s zs1 − δs,za2 = va − Ra.pda + ka1m−1a za1 − δa (10)where δs = (δs1, δs2, δs3) and δa = (δa1, δa2, δa3) arepositive constant vectors. With the velocity errors, thesecond Lyapunov functions are established as Vs2 = Vs1 +12 zs2M2s zs2 and Va2 = Va1 + 12 za2m2aza2.The time derivatives of the second Lyapunov functionsare given as:˙Vs2 = −Ws2(zs1, zs2) + zs1δs + zs2Ms (αs (ωs)+e1Ts + ηs + fsv) ,˙Va2 = −Wa2(za1, za2) + za1δa + za2ma (αa (ωa)−e3Ta + ηa + fav)(11)where:Ws2(zs1, zs2) = ks1zs1M−1s zs1 + ks2zs2zs2,Wa2(za1, za2) = ka1za1m−1a za1 + ka2za2za2,αs (ωs) = MsS (ωs) Rs.pds − S (ωs) MsRs.pds−S (ωs) Msδs,ηs = −Dsvvs − MsRs..pds + ks1 vs − Rs.pds+M−1s zs1 + k2M−1s zs2, αa (ωa) = −S (ωa) maδa,ηa = magRa e3 − maRa..pda + ka1 va − Ra.pda+m−1a za1 + ka2m−1a za2(12)and ks2, ka2 are positive constants.In (11), the faults fsv and fav are unknown, thuscannot be directly incorporated into the control laws. Thefirst estimates of these faults are denoted as fsv1 andfav1. The corresponding estimation errors are respectivelyrepresented as fsv1 and fav1. Based on the first estimationerrors, the third Lyapunov functions are defined asVs3 = Vs2 + 12 fsv1Λ−1s1 fsv1 and Va3 = Va2 + 12 fav1Λ−1a1 fav1,where Λs1 and Λa1 are positive-definite diagonal parametermatrices. The time derivatives of the third Lyapunovfunctions are given as:˙Vs3 = −Ws2(zs1, zs2) + zs1δs + zs2Ms(αs (ωs)+e1Ts + ηsf ) + fsv1 Mszs2 − Λ−1s1.fsv1˙Va3 = −Wa2(za1, za2) + za1δa + za2ma(αa (ωa)−e3Ta + ηaf ) + fav1 maza2 − Λ−1a1.fav1(13)where ηsf = ηs + fsv1, ηaf = ηa + fav1.To ensure the unknown estimation terms are renderednegative definite, a projection operator P is defined as:P (x, y) =x, if p(y) ≤ 0;x, if p(y) > 0 and p (y) x ≤ 0;q(y)x, otherwise;(14)where p(y) = yTy−y22+2 y , q(y) = E3×3− p(y)p (y)p (y)p (y) p (y). Inthe above definitions, is an arbitrary positive constant,y denotes the maximum of y , and E3×3denotes theidentity matrix of order 3.The updating laws for the first estimates fsv1 and fav1are set as:.fsv1 = Λs1P Mszs2, fsv1 ,.fav1 = Λa1P maza2, fav1 (15)By utilising the properties of the projection [29],inequalities can be obtained that fsv1 Mszs2 − Λ−1s1.fsv1≤ 0 and fav1 maza2 − Λ−1a1.fav1 ≤ 0.4To guarantee that the time derivatives ˙Vs3 and ˙Va3remain negative definite, the thrust forces are chosen as:Ts = −e1 (αs (ωs) + ηsf ) , Ta = e3 (αa (ωa) + ηaf ) (16)The angular velocity error for USV is given as:zs3 = Π23 (αs (ωs) + ηsf ) (17)where Π23 = e2e2 + e3e3 . The fourth Lyapunov functionfor USV is defined as Vs4 = Vs3 + 12 zs3zs3.The time derivative of the fourth Lyapunov function˙Vs4 is formulated as:˙Vs4 ≤ −Ws3(zs1, zs2, zs3) + zs1δs + zs3e2(ψsJ−1s τs− ψsJ−1s dsωe3 ωs + ψsJ−1s fsω + ˙ψse3 ωs+ e2.ηsf + e2 Mszs2 + ks3e2 zs3)(18)where ψs = e1 (msx − msy) Rs.pds − (ms − msx) δs, Ws3(zs1, zs2, zs3) = Ws2(zs1, zs2) + ks3zs3zs3, and ks3 is apositive constant.In (19), the time-varying faults exist in both ψsJ−1s fsωand e2.ηsf , which cannot be directly incorporated into thecontrol laws. The fault fsω is estimated as fsω, with thecorresponding estimation error denoted as fsω. The secondestimation for the fault fsv is represented as fsv2, with theassociated estimation error denoted as fsv2.The term.ηsf can be decomposed as.ηsf =.ηsf + ∂ηsf∂vsfsv2,where.ηsf represents the estimate of.ηsf based on the secondestimation fsv2. The time derivative of the fifth Lyapunovfunction Vs5 is given as:˙Vs5 = ˙Vs4 + fsv2Λ−1s2.fsv2 + λ−1s3 fsω˙fsω≤ −Ws3(zs1, zs2, zs3) + zs1δs+ zs3e2(ψsJ−1s τs − ψsJ−1s dsωe3 ωs+ ψsJ−1s fsω + ˙ψse3 ωs + e2.ηsf + e2 Mszs2+ ks3e2 zs3) + fsv2∂ηsf∂vse2e2 zs3 − Λ−1s2.fsv2+ fsω zs3e2ψsJ−1s − λ−1s3˙fsω(19)To guarantee the time derivative ˙Vs5 negative definite,the torque input for USV is chosen as:τs = ψ−1s Js(ψsJ−1s dsωe3 ωs − ψsJ−1s fsω − ˙ψse3 ωs− e2.ηsf − e2 Mszs2 − ks3e2 zs3) (20)The parameter δs is selected such that |δs1| >|(msx − msy) vds/(ms − msx)| to avoid the singularity inthe term ψ−1s , where vds ≥.pds is the upper bound of thedesired velocity.The updating laws for fsv2 and fsω are chosen as:.fsv2 = Λs2P∂ηsf∂vse2e2 zs3, fsv2 ,˙fsω = λs3P zs3e2ψsJ−1s , fsω (21)Figure 2. Experimental control architecture and inter-communication topology of the agents, where dottedlines represent the existence of communication links andinformation exchanges.By applying the estimation updating laws, the inequal-ities can be derived that fsv2∂ηsf∂vse2e2 zs3 − Λ−1s2.fsv2≤ 0 and fsω zs3e2ψsJ−1s − λ−1s3˙fsω ≤ 0.By utilising the control laws (16) and (20), alongwith the fault estimation updating laws (15) and (21),the time derivative of the fifth Lyapunov function satisfies˙Vs5 ≤ −Ws3(zs1, zs2, zs3) + zs1δs.As a result, the USV’s tracking errors zs1, zs2,and zs3, along with the fault estimation errors fsv1,fsv2, and fsω, will converge to a region around theorigin. The radius of the region is determined by12 k−1s1 Msδs + Msδs δs/(4ksmin), where ksmin representsthe minimum value among ks1/(ms−msx), ks1/(ms−msy),ks2/(ms − msx)2, ks2/(ms − msy)2, and ks3.The virtual desired angular velocity for UAV is definedas ωda = m−1a Ψa Ψaηaf , where:Ψa =0 δa3 −δa2−δa3 0 δa10 0 0,Ψa =1δa δaδ2a3δ2a1 + δ2a3 δa1δa2 0δa1δa2 δ2a2 + δ2a3 00 0 0(22)To prevent the singularity in the term Ψa, theparameter δa is selected such that δa3 > 0.The angular velocity error for UAV is given as:za3 = ωa − ωda (23)The fourth Lyapunov function for UAV is defined asVa4 = Va3 + 12 za3za3. The time derivative of the fourthLyapunov function Va4 is formulated as:˙Va4 ≤ −Wa3(za1, za2, za3) + za1δa+za3(−maΨa za2 − J−1a S(ωa)Jaωa + J−1a τa+J−1a faω − m−1a Ψa Ψa.ηaf + ka3za3)(24)5where Wa3(za1, za2, za3) = Wa2(za1, za2) + ka3za3za3, andka3 is a positive constant.In (24), time-varying faults exist in the items J−1a faωand m−1a Ψa Ψa.ηaf , which cannot be directly included inthe control laws. The fault faω is estimated as faω, withthe corresponding estimation error denoted as faω. Thesecond estimation for the fault fav is represented as fav2,with the associated error denoted as fav2. The item.ηaf canbe decomposed as.ηaf =.ηaf + ∂ηaf∂vafav2, where.ηaf is theestimation of.ηaf with the estimation fav2.Based on the estimate errors fav2 and faω, the fifthLyapunov function for UAV is established as Va5 =Va4 + 12 fav2Λ−1a2 fav2 + 12 faωΛ−1a3 faω, where Λa2 and Λa3 arepositive-definite diagonal parameter matrices. The timederivative of the fifth Lyapunov function Va5 is given as:˙Va5 = ˙Va4 + fav2Λ−1a2.fav2 + faωΛ−1a3.faω≤ −Wa3 (za1, za2, za3) + za1δaf+za3(−maΨa za2 − J−1a S(ωa)Jaωa + J−1a τa+J−1a faω − m−1a Ψa Ψa.ηaf + ka3za3)+fav2 m−1a∂ηsf∂vaΨa Ψaza3 − Λ−1a2.fav2+faω J−1a za3 − Λ−1a3.faω(25)To guarantee that the time derivative of the fifthLyapunov functions ˙Va5 remains negative definite, thetorque force τa is chosen as:τa = maJaΨa za2 + S(ωa)Jaωa− faω + m−1a JaΨa Ψa.ηaf − ka3Jaza3 (26)The updating laws for fav2 and faω are chosen as:.fav2 = Λa2P m−1a∂ηaf∂vaΨa Ψaza3, fav2 ,.faω = Λa3P J−1a za3, faω (27)Using the estimation updating laws, the inequali-ties can be obtained that fav2 m−1a∂ηaf∂vaΨa Ψaza3−Λ−1a2.fav2 ≤ 0 and faω J−1a za3 − Λ−1a3.faω ≤ 0.By applying the control laws (16) and (26), alongwith the fault estimation updating laws (15) and (27),the time derivative of the fifth Lyapunov function satisfies˙Va5 ≤ −Wa3 (za1, za2, za3) + za1δa.As a result, the UAV’s tracking errors za1, za2, andza3, together with the fault estimation errors fav1, fav2,and faω, will converge to a ball centered at the origin. Theradius is given by 12 k−1a1 maδa + maδa δa/(4kamin), wherekamin = minka1/ma,ka2,ka3.4. Experimental ResultsThe effectiveness and validity of the proposed methodsare demonstrated through a series of experiments. Thissection provides a thorough overview of the experimentalplatform and details the results.4.1 Experimental SetupAs illustrated in Fig. 2, this paper employs a heterogeneousunmanned system consisting of a rear-propelled, twinhull-based USV and four quadcopters with a generic X-geometry frame to validate the effectiveness of the proposedmethods. USV is equipped with two rear propellers, abattery, and a relay signal converter. Each quadcoptercomprises four independent brushless motors, a battery, aflight control unit, and a relay signal converter. Due toconstraints at the experimental site, USV and UAVs weretested sequentially, as illustrated in Fig. 3(a) and (b). Allparameters used in the proposed methods are detailed inTable 1.The thrust Ts for USV is generated by the combinedforce of the propellers, denoted as Tss and Tsp. The torqueinput τs for USV is produced by the difference betweenthese two propellers, expressed as τs = (Tss − Tsp) /l. Thequadcopters are controlled through thrust and angularvelocity commands. To implement the proposed controller,the quadrotors are augmented with an artificial statedescribed by Ja.ωa = −S(ωa)Jaωa + τa + faω, simulated inMATLAB.The shells of both USV and quadcopters aremanufactured by 3D printing technology, which offers alow-cost and highly adaptable solution for test platforms.Due to their compact sizes, USV and quadcopters lackthe payload capacity for onboard sensors and computingprocess power. As a result, the vehicles’ states aremeasured using external sensors. In the experiments, aNOKOV motion capture system is employed to capturethe poses and velocities with sub-millimeter accuracy.A serial of reflective markers attached to the vehiclesare tracked by 12 high-frequency cameras operatingat 100Hz.The controller computations are carried out on abase station computer running MATLAB. Subsequently,the controller outputs are transmitted to the vehiclesvia a 2.4Ghz Wifi network. Each vehicle is equippedwith a relay signal converter to minimise communicationload and delay. For USV, a Raspberry Pi 4B moduleserves as the signal converter, which also generates pulse-width modulation signals that are sent to the electronicspeed controllers. For each quadcopter, an Orange PiZero module receives signals and transmits them to aPX4 flight control unit via the Mavlink communicationprotocol. The overall control architecture is illustratedin Fig. 2.In the experiments, the USV is designated as the firstfollower in the formation, with the quadcopters labeledsequentially. The intercommunication topology is shown inFig. 2, with the oriented incidence matrix N and Laplacian6Table 1Parameters in the ExperimentsQuantity Symbol Values (Unit)USV’s mass ms 1748 (g)USV’s add masses msx, msy -87.4, 2179 (g)UAV’s mass ma 315 (g)USV’s inertial moment Js 0.036 (kg/m2)UAV’s inertial moments Ja diag(6.6, 6.6, 13)×10−4(kg/m2)USV’s linear damping coef dsvx, dsvy 0.83, 36.3 (N s/m)USV’s angular damping coef dsω 0.0968 (N s/m)Coordination parametersΛc E6×6· 15Kc E6×6· 2/3USV’s control parametersks1, ks2, ks3 2, 1.3, 10δs [1.5, 0, 0]Λs1 diag(0.07, 0.01, 0)Λs2 diag(0.04, 0.005, 0)λs3 5 × 10−5UAV’s control parameterska1, ka2, ka3 0.2, 0.3, 0.4δa [0, 0, 0.6]Λa1 diag(1, 1, 5)Λa2 diag(0.1, 0.1, 0.2)Λa3 diag(1, 1, 0)×8.7 × 10−11USV’s hull length - 0.32 (m)USV’s propeller distance - 0.15 (m)USV’s draught - 0.068 (m)UAV’s base axis length - 0.15 (m)matrix L defined as:N =1 0 0 0 0 0−1 1 1 0 0 00 −1 0 1 0 00 0 0 −1 1 00 0 0 0 −1 10 0 −1 0 0 −1,L =1 −1 0 0 0 0−1 3 −1 0 0 −10 −1 2 −1 0 00 0 −1 2 −1 00 0 0 −1 2 −10 −1 0 0 −1 2(28)The desired path for the virtual leader pl(γi) isconstructed with several waypoints. Smooth transitionsbetween these waypoints are generated using the B-splinefunctions of degree 5 to generate the connecting segments,guaranteeing that the paths are continuously derivable toorder 4. The horizontal projection of the formation forms aregular pentagon. The distance matrix D(γi) is defined as:D(γi) =0 0 0df 0 0cos(0.4π)df sin(0.4π)df dh (γi)−cos(0.2π)df sin(0.2π)df dh (γi)−cos(0.2π)df −sin(0.2π)df dh (γi)cos(0.4π)df −sin(0.4π)df dh (γi)(29)where df represents the side length of the formation, anddh (γi) is the flight altitude. rd is chosen as e3. The initial7Figure 3. The experiment sites: (a) The experimental site featuring a pool for USV and (b) The experimental site designatedfor UAVs.Figure 4. (a) The 3D trajectories of the heterogeneousformation system in the experiment, with the blue dottedlines representing the connections between agents in theformation at a specific moment and (b) The x, y, andz positions of the heterogeneous formation system in theexperiment.values of the path parameters γl and γi, i = 1 . . . 5 are setas [0, 1, 2, 3, 4, 5].Actuator faults are artificially induced after thecalculation of the controller output is completed andmodeled as a combination of different constant terms andzero-mean white bounded Gaussian noises to simulatefaults fsv, fsω, fav, and faω in the dynamic model.Remark 1. The waypoints used in the desired pathsimulation reflect a realistic application scenario. Inpractical missions, desired paths are typically defined by asequence of waypoints, from which the connecting segments’algebraic expressions are derived. In the experiment, thesewaypoints are classified into distinct categories: initial,destination, mission start, and transit points.4.2 Data AnalysisThe trajectories of the heterogeneous unmanned systemduring the experiment are shown in Fig. 4(a), while theindividual positions of USV and UAVs are depicted inFig. 4(b). UAVs ascend from a low altitude to their cruisinglevel, perform a circular maneuver, and then descend.The control outputs for USV and UAVs are respectivelypresented in Fig. 5(a) and (b). Since UAVs exhibit isotropicbehaviour in the horizontal plane, the formation systemdoes not impose any orientation requirements for UAVs.Consequently, UAVs’ torque outputs τa along the z-axisare zero and are omitted from Fig. 5(b). Due to the varyingamplitudes of the faults, the thrust outputs of the UAVsdiffer accordingly.Figure 6(a) illustrates the formation coordinationerrors zcp and speed errors zcv, defined in (6) and(7). The path parameter γ serves as a control variablefor regulating the path speed within the heterogeneousformation. Due to initial differences in the path parametersfor each agent, their initial positions and speeds differ.After approximately 2.5 s, the formation coordinationand velocity errors associated with the path parametersconverge, enabling the heterogeneous system to achieveand maintain formation successfully.The path following errors of the proposed fault-tolerant formation controller are presented in Fig. 6(b).The formation following errors zp have the same normas the Lyapunov position errors zs1 and za1, respectivelydefined in (16). As proven in Section 3, the path followingerrors will converge to a bounded region centered at theorigin. The norms of the controller gain factors determinethe radius of this region. Due to the underactuated natureof USV, a large control gain parameter can result insignificant overshoot and oscillation. To ensure a smoothpath following behaviour, the control gains for USV are8Figure 5. (a) The control outputs Ts and τs for USV and(b) The control outputs Ta and τa for UAVs, where τaalong the z-axis are zero and are omitted.set to slightly lower values. Consequently, UAVs achievegreater tracking accuracy due to their higher mobility.Figure 7 illustrates the norms of fault estimation errorsfv1 , fv2 , and fω . The first fault estimation fv1 isderived from deviations in position and velocity statesand affect only the thrust. In contrast, the second faultestimation fv2 is primarily based on deviations in velocityand acceleration states, influencing the torque alongsidethe third estimation fω. Due to their different generationsources, the first fault estimation fv1 converges moreslowly but is more stable than fv2. This behaviour isevident during parameter adjustment. The controller gainfactors for fv1 and fv2, provided in Table 1, indirectlyconfirm this observation. The traces of Λs1 and Λa1are an order of magnitude larger than those of Λs2and Λa2, respectively. However, f2 demonstrates a fasterconvergence rate compared to fv1.After the convergence of the heterogeneous formation,the mean formation error norms are 0.0661, 0.0356, 0.0345,0.0298, and 0.031 m. The mean first linear fault estimationerror norms are 0.0123, 0.0092, 0.0131, 0.0291, and 0.0079N, while the mean second linear fault estimation errornorms are 0.0017, 0.0241, 0.0115, 0.0057, and 0.0093 N.Additionally, the mean angular fault estimation errorFigure 6. (a) The path and speed coordination errorszcp, zcv of the formation system, defined in (6) and (7),which converge at t = 2s and (b)The following errorsz1, z2, z3 of the heterogeneous formation system, defined in(9), (10), (17), and (23).norms are 0.0339, 0.0421, 0.0391, 0.0131, and 0.0412N. Experimental errors can stem from various sources,including model uncertainties, calibration inaccuracies inmodel parameters, asynchronous sensor data reading andprocessing times, environmental disturbances in confinedspaces, and agent interference effects.Table 2 provides a quantitative comparison conductedunder identical experimental conditions, including plat-form specifications and fault injection protocol, with allmethods evaluated using the mean formation error normNp , where p = [pl, p1, p2, p3, p4, p5] . The PID controllerperforms reasonably in UAV path following but suffers fromlarge USV tracking errors due to sluggish fault-affecteddynamics, while the learning-based backstepping controller(LBBC) [30] exhibits slower recovery and reduced accuracy9Figure 7. The norms of fault estimation errors fv1 , fv2 ,and fω , defined in (15), (21), and (27). Due to dynamicmodels’ lag characteristic, USV’s convergence speed islower than that of UAVs.Table 2Comparative Performance AnalysisMethod Np (m) Fault Recovery Time (s)Proposed Method 0.039 0.2PID 0.21 N/ALBBC [30] 0.065 0.35DBSMC 0.085 N/A (passive)from its unconstrained integrator gain. The bioinspiredsliding mode controller (DBSMC) mitigates chatteringbut lacks fault identification, yielding higher formationerrors. Our method outperforms these approaches inboth formation accuracy and fault recovery speed,demonstrating robust fault tolerance in heterogeneousmulti-agent systems.5. ConclusionIn this paper, cooperative fault-tolerant formation methodswere developed for a heterogeneous unmanned system.First, the coordination and formation errors were describedbased on vehicle models with dynamic the presence offaults. A coordination method was then introduced tosynchronise the agents and mitigate the impact of faultson the overall system. Fault-tolerant formation controllerswere designed for both the USV and UAV to ensureaccurate path following, as dictated by the proposedcoordination method. Experimental results validated theeffectiveness and performance of the coordination andformation control approaches. However, our frameworkrequires reliable inter-agent communication, a connectedformation topology with spanning tree, bounded modeluncertainties, and fourth-order differentiable referencepaths to maintain guaranteed performance. Our methodscales effectively through its decentralised architecture,universal control laws for any fleet size, and localinteraction protocols that maintain formation stability.Future work could explore the integration of metaheuristicoptimisation algorithms for real-time trajectory planningin fault-tolerant formations, leveraging their proven effi-ciency in robotic visual servo tasks. Future work should alsoexplore delay-compensation control techniques to addresscommunication delays in disturbed environments, ensuringrobust system performance under real-world conditions.AcknowledgementThis work is partly supported by the National Sci-ence and Technology Major Project (2022ZD0119900),the National Natural Science Foundation of China(62303308, 62303305, U2141234), Shanghai Pujiang Pro-gram (23PJ1404700), Joint Research Fund of ShanghaiAcademy of Spaceflight Technology (USCAST2023-22),Shanghai Science and Technology Program 22015810300,and Hainan Province Science and Technology Special Fund(ZDYF2024GXJS003).References[1] W. Bi, M. Zhang, H. Chen, and A. Zhang, Cooperativetask allocation method for air-sea heterogeneous unmannedsystem with an application to ocean environment informationmonitoring, Ocean Engineering, 309, 2024, 118496.[2] J. Li, G. Zhang, C. Jiang, and W. 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