【Paper】2023_Distributed adaptive fixed-time formation control for UAV-USV heterogeneous multi-agent

程序开发 2023-09-04 20:36:04

Liu H, Weng P, Tian X, et al. Distributed adaptive fixed-time formation control for UAV-USV heterogeneous multi-agent systems[J]. Ocean Engineering, 2023, 267: 113240.

文章目录

3. Main results

Ref

1. Introduction

2. Preliminaries and problem statement

2.1. Graph theory

2.2. Problem formulation

Consider one quadrotor as the UAV system. The dynamics model of the UAV is as follows:

x ¨ a = ( cos ⁡ φ sin ⁡ θ cos ⁡ ξ + sin ⁡ φ sin ⁡ ξ ) u 1 / m a − ρ x x ˙ a / m a y ¨ a = ( cos ⁡ φ sin ⁡ θ sin ⁡ ξ − sin ⁡ φ cos ⁡ ξ ) u 1 / m a − ρ y y ˙ a / m a z ¨ a = ( cos ⁡ φ sin ⁡ θ ) u 1 / m a − ρ z z ˙ a / m a − g φ ¨ = θ ˙ ξ ˙ ( I y − I z ) / I x − θ ˙ Λ I r / I x + u 2 / I x − ρ φ φ ˙ / I x θ ¨ = φ ˙ ξ ˙ ( I z − I x ) / I y − φ ˙ Λ I r / I y + u 3 / I y − ρ θ θ ˙ / I y ξ ¨ = φ ˙ θ ˙ ( I x − I y ) / I z + u 4 / I z − ρ ξ ξ ˙ / I z (1) begin{aligned} ddot{x}_a &= (cosvarphi sintheta cosxi + sinvarphi sinxi) u_1 / m_a - rho_x dot{x}_a / m_a \ ddot{y}_a &= (cosvarphi sintheta sinxi - sinvarphi cosxi) u_1 / m_a - rho_y dot{y}_a / m_a \ ddot{z}_a &= (cosvarphi sintheta) u_1 / m_a - rho_z dot{z}_a / m_a - g \ ddot{varphi} &= dot{theta} dot{xi} (I_y - I_z) / I_x - dot{theta} varLambda I_r / I_x + u_2 / I_x - rho_varphi dot{varphi} / I_x \ ddot{theta} &= dot{varphi} dot{xi} (I_z - I_x) / I_y - dot{varphi} varLambda I_r / I_y + u_3 / I_y - rho_theta dot{theta} / I_y \ ddot{xi} &= dot{varphi} dot{theta} (I_x - I_y) / I_z + u_4 / I_z - rho_xi dot{xi} / I_z end{aligned} tag{1} x¨ay¨az¨aφ¨θ¨ξ¨=(cosφsinθcosξ+sinφsinξ)u1/ma−ρxx˙a/ma=(cosφsinθsinξ−sinφcosξ)u1/ma−ρyy˙a/ma=(cosφsinθ)u1/ma−ρzz˙a/ma−g=θ˙ξ˙(Iy−Iz)/Ix−θ˙ΛIr/Ix+u2/Ix−ρφφ˙/Ix=φ˙ξ˙(Iz−Ix)/Iy−φ˙ΛIr/Iy+u3/Iy−ρθθ˙/Iy=φ˙θ˙(Ix−Iy)/Iz+u4/Iz−ρξξ˙/Iz(1)

where
χ a = [ x a , y a , z a ] T red{chi_a} = [red{x_a}, red{y_a}, red{z_a}]^text{T} χa=[xa,ya,za]T denote the position variable,
ω a = [ φ , θ , ξ ] T red{omega_a} = [red{varphi}, red{theta}, red{xi}]^text{T} ωa=[φ,θ,ξ]T denote angle variable,
u 1 , u 2 , u 3 , u 4 blue{u_1, u_2, u_3, u_4} u1,u2,u3,u4 denote control thrust, three control torques,
I x , I y , I z red{I_x, I_y, I_z} Ix,Iy,Iz denote the moments of inertia,
ρ x , ρ y , ρ z , ρ φ , ρ θ , ρ ξ red{rho_x, rho_y, rho_z, rho_varphi, rho_theta, rho_xi} ρx,ρy,ρz,ρφ,ρθ,ρξ represent the aerodynamic damping coefficients,
m a red{m_a} ma is the mass,
g red{g} g is the acceleration of gravity,
I r red{I_r} Ir is the rotor blade of inertia and
Λ red{varLambda} Λ is the rotor angular.

For the leader quadrotor, the attitude system is stable in (Tayebi and McGilvray, 2006). Thus, the researched UAV control problem in this paper concentrates primarily on position systems. Considering external disturbances, parametric uncertainties and input saturation, the UAV three degrees of freedom model can be redefined as follows:

χ ¨ a ( t ) = A sat ( u a ) + f ( χ a ) + Δ a f ( χ a ) + d ( χ a , t ) (2) begin{aligned} ddot{chi}_a (t) &= A text{sat}(u_a) + f(chi_a) + Delta_a f(chi_a) + d(chi_a, t) \ end{aligned} tag{2} χ¨a(t)=Asat(ua)+f(χa)+Δaf(χa)+d(χa,t)(2)

where
sat ( u a ) = [ sat ( u x a ) , sat ( u y a ) , sat ( u z a ) ] T ∈ R 3 red{text{sat}(u_a)} = [red{text{sat}(u_{xa})}, red{text{sat}(u_{ya})}, red{text{sat}(u_{za})}]^text{T} in R^3 sat(ua)=[sat(uxa),sat(uya),sat(uza)]T∈R3 denote the actual control input vector,
u a ( t ) = [ u x a ( t ) , u y a ( t ) , u z a ( t ) ] T ∈ R 3 red{u_a(t)} = [red{u_{xa}(t)}, red{u_{ya}(t)}, red{u_{za}(t)}]^text{T} in R^3 ua(t)=[uxa(t),uya(t),uza(t)]T∈R3
Δ a f ( χ a ) = [ Δ a f x , Δ a f y , Δ a f z ] T ∈ R 3 red{Delta_a f(chi_a)} = [red{Delta_a f_x}, red{Delta_a f_y}, red{Delta_a f_z}]^text{T} in R^3 Δaf(χa)=[Δafx,Δafy,Δafz]T∈R3 denote the model uncertainties,
d ( ⋅ ) = [ d x ( t ) , d y ( t ) , d z ( t ) ] T ∈ R 3 red{d (cdot)} = [red{d_x(t)}, red{d_y(t)}, red{d_z(t)}]^text{T} in R^3 d(⋅)=[dx(t),dy(t),dz(t)]T∈R3 denote the external disturbances.

{ u x a = ( cos ⁡ φ sin ⁡ θ cos ⁡ ξ + sin ⁡ φ sin ⁡ ξ ) u 1 u y a = ( cos ⁡ φ sin ⁡ θ sin ⁡ ξ − sin ⁡ φ cos ⁡ ξ ) u 1 u z a = ( cos ⁡ φ cos ⁡ θ ) u 1 (3) left{begin{aligned} u_{xa} &= (cos varphi sintheta cosxi + sinvarphi sinxi) u_1 \ u_{ya} &= (cos varphi sintheta sinxi - sinvarphi cosxi) u_1 \ u_{za} &= (cos varphi costheta) u_1 end{aligned}right. tag{3} ⎩ ⎨ ⎧uxauyauza=(cosφsinθcosξ+sinφsinξ)u1=(cosφsinθsinξ−sinφcosξ)u1=(cosφcosθ)u1(3)

Let
{ χ l , 1 = χ a χ l , 2 = χ ˙ a (5) left{begin{aligned} chi_{l,1} &= chi_a \ chi_{l,2} &= dot{chi}_a end{aligned}right. tag{5} {χl,1χl,2=χa=χ˙a(5)

Then leader UAV dynamic model can be rewritten as:
Let
{ χ ˙ l , 1 = χ l , 2 χ ˙ l , 2 = A sat ( u a ) + f ( χ l , 1 ) + F a ( χ l , 1 , t ) (6) left{begin{aligned} dot{chi}_{l,1} &= chi_{l,2} \ dot{chi}_{l,2} &= A text{sat}(u_a) + f(chi_{l,1}) + F_a(chi_{l,1}, t) end{aligned}right. tag{6} {χ˙l,1χ˙l,2=χl,2=Asat(ua)+f(χl,1)+Fa(χl,1,t)(6)

Consider N N N surface vessels as the USV systems. The dynamics of the k k kth USV are described below:

{ η ˙ s , k = R ( Ψ s , k ) v s , k M ˉ k v ˙ s , k + C ˉ k v s , k + D ˉ k v s , k = sat ( u s , k ) + R T ( Ψ s , k ) δ k ( t ) (7) left{begin{aligned} dot{eta}_{s,k} &= R(varPsi_{s,k}) v_{s,k} \ bar{M}_k dot{v}_{s,k} + bar{C}_k v_{s,k} + bar{D}_k v_{s,k} &= text{sat}(u_{s,k}) + R^{text{T}}(varPsi_{s,k}) delta_k(t) end{aligned}right. tag{7} {η˙s,kMˉkv˙s,k+Cˉkvs,k+Dˉkvs,k=R(Ψs,k)vs,k=sat(us,k)+RT(Ψs,k)δk(t)(7)

where
η s , k = [ x s , k , y s , k , Ψ s , k ] T red{eta_{s,k}} = [red{x_{s,k}}, red{y_{s,k}}, red{varPsi_{s,k}}]^text{T} ηs,k=[xs,k,ys,k,Ψs,k]T denote the position variable,
v s , k = [ v x s , k , v y s , k , v Ψ s , k ] T red{v_{s,k}} = [red{v_{xs,k}}, red{v_{ys,k}}, red{v_{varPsi s,k}}]^text{T} vs,k=[vxs,k,vys,k,vΨs,k]T denote the velocity variable,
sat ( u s k ) = [ sat ( u x s k ) , sat ( u y s k ) , sat ( u Ψ s k ) ] T blue{text{sat}(u_{sk})} = [blue{text{sat}(u_{xsk})}, blue{text{sat}(u_{ysk})}, blue{text{sat}(u_{varPsi sk})}]^text{T} sat(usk)=[sat(uxsk),sat(uysk),sat(uΨsk)]T represent the actual control input vector,
u s k ( t ) = [ u x s k , u y s k , u Ψ s k ] T blue{u_{sk}(t)} = [blue{u_{xsk}}, blue{u_{ysk}}, blue{u_{varPsi sk}}]^text{T} usk(t)=[uxsk,uysk,uΨsk]T represent the desired control inputs vector of the k k kth USV,
δ k ( t ) = [ δ x k , δ y k , δ Ψ k ] T red{delta_{k}(t)} = [red{delta_{xk}}, red{delta_{yk}}, red{delta_{varPsi k}}]^text{T} δk(t)=[δxk,δyk,δΨk]T denote external disturbances of the k k kth USV,
R ( Ψ s , k ) red{R ( varPsi_{s,k})} R(Ψs,k) is a rotation matrix
M ˉ k red{bar{M}_k} Mˉk denotes the inertia matrix,
D ˉ k red{bar{D}_k} Dˉk denotes the damping matrix,
C ˉ k red{bar{C}_k} Cˉk denotes the Coriolis and centripetal matrix.

Let
{ χ s k , 1 = η s , k χ s k , 2 = η ˙ s , k (9) left{begin{aligned} chi_{sk,1} &= eta_{s,k} \ chi_{sk,2} &= dot{eta}_{s,k} end{aligned}right. tag{9} {χsk,1χsk,2=ηs,k=η˙s,k(9)

Then leader UAV dynamic model can be rewritten as:
Let
{ χ ˙ s , 1 = χ s , 2 χ ˙ s , 2 = M s − 1 ( R s sat ( u s ) − C s χ s , 2 − D s χ s , 2 ) + F s (10) left{begin{aligned} dot{chi}_{s,1} &= chi_{s,2} \ dot{chi}_{s,2} &= M_s^{-1} ( R_s text{sat}(u_s) - C_s chi_{s,2} - D_s chi_{s,2} ) + F_s end{aligned}right. tag{10} {χ˙s,1χ˙s,2=χs,2=Ms−1(Rssat(us)−Csχs,2−Dsχs,2)+Fs(10)

{ [ χ ˙ s 1 , 1 χ ˙ s 2 , 1 χ ˙ s 3 , 1 χ ˙ s 4 , 1 ] = [ χ s 1 , 2 χ s 2 , 2 χ s 3 , 2 χ s 4 , 2 ] [ χ ˙ s 1 , 2 χ ˙ s 2 , 2 χ ˙ s 3 , 2 χ ˙ s 4 , 2 ] = M s − 1 ( R s sat ( u s ) − C s [ χ s 1 , 2 χ s 2 , 2 χ s 3 , 2 χ s 4 , 2 ] − D s [ χ s 1 , 2 χ s 2 , 2 χ s 3 , 2 χ s 4 , 2 ] ) + F s (10) left{begin{aligned} left[begin{matrix} dot{chi}_{s1,1} \ dot{chi}_{s2,1} \ dot{chi}_{s3,1} \ dot{chi}_{s4,1} \ end{matrix}right] &= left[begin{matrix} {chi}_{s1,2} \ {chi}_{s2,2} \ {chi}_{s3,2} \ {chi}_{s4,2} \ end{matrix}right] \ left[begin{matrix} dot{chi}_{s1,2} \ dot{chi}_{s2,2} \ dot{chi}_{s3,2} \ dot{chi}_{s4,2} \ end{matrix}right] &= M_s^{-1} ( R_s text{sat}(u_s) - C_s left[begin{matrix} chi_{s1,2} \ chi_{s2,2} \ chi_{s3,2} \ chi_{s4,2} \ end{matrix}right] - D_s left[begin{matrix} chi_{s1,2} \ chi_{s2,2} \ chi_{s3,2} \ chi_{s4,2} \ end{matrix}right] ) + F_s end{aligned}right. tag{10} ⎩ ⎨ ⎧ χ˙s1,1χ˙s2,1χ˙s3,1χ˙s4,1 χ˙s1,2χ˙s2,2χ˙s3,2χ˙s4,2 = χs1,2χs2,2χs3,2χs4,2 =Ms−1(Rssat(us)−Cs χs1,2χs2,2χs3,2χs4,2 −Ds χs1,2χs2,2χs3,2χs4,2 )+Fs(10)

2.3. Radial basis function neural networks

Θ ( L ) = w ∗ T S ( L ) + ε ( L ) (11) varTheta(L) = w^{*text{T}} S(L) + varepsilon(L) tag{11} Θ(L)=w∗TS(L)+ε(L)(11)

S ( L ) = exp ⁡ ( − ∥ L − μ ∥ 2 2 h 2 ) (12) S(L) = exp (-frac{|L-mu|^2}{2h^2}) tag{12} S(L)=exp(−2h2∥L−μ∥2)(12)

3. Main results

3.1. Fixed-time control for the leader UAV

The trajectory tracking error of the UAV can be given as follows:
{ e l , 1 = χ l , 1 − χ a d e l , 2 = χ l , 2 − χ ˙ a d (13) left{begin{aligned} e_{l,1} &= chi_{l,1} - chi_{ad} \ e_{l,2} &= chi_{l,2} - dot{chi}_{ad} end{aligned}right. tag{13} {el,1el,2=χl,1−χad=χl,2−χ˙ad(13)

{ [ e x a , 1 e y a , 1 e z a , 1 ] = [ χ x a , 1 χ y a , 1 χ z a , 1 ] − χ a d [ e x a , 2 e y a , 2 e z a , 2 ] = [ χ x a , 2 χ y a , 2 χ z a , 2 ] − χ ˙ a d (13) left{begin{aligned} left[begin{matrix} e_{xa,1} \ e_{ya,1} \ e_{za,1} \ end{matrix}right] &= left[begin{matrix} chi_{xa,1} \ chi_{ya,1} \ chi_{za,1} \ end{matrix}right] - chi_{ad} \ left[begin{matrix} e_{xa,2} \ e_{ya,2} \ e_{za,2} \ end{matrix}right] &= left[begin{matrix} chi_{xa,2} \ chi_{ya,2} \ chi_{za,2} \ end{matrix}right] - dot{chi}_{ad} end{aligned}right. tag{13} ⎩ ⎨ ⎧ exa,1eya,1eza,1 exa,2eya,2eza,2 = χxa,1χya,1χza,1 −χad= χxa,2χya,2χza,2 −χ˙ad(13)

where
χ a d red{chi_{ad}} χad is the UAV desired position,
χ ˙ a d red{dot{chi}_{ad}} χ˙ad is the UAV desired velocity,
χ l , 1 red{chi_{l,1}} χl,1 is the UAV position,
χ ˙ l , 2 red{dot{chi}_{l,2}} χ˙l,2 is the UAV velocity.
e l , 1 = [ e x a , 1 , e y a , 1 , e z a , 1 ] T red{e_{l,1}} = [e_{xa,1}, e_{ya,1}, e_{za,1}]^text{T} el,1=[exa,1,eya,1,eza,1]T,
e l , 2 = [ e x a , 2 , e y a , 2 , e z a , 2 ] T red{e_{l,2}} = [e_{xa,2}, e_{ya,2}, e_{za,2}]^text{T} el,2=[exa,2,eya,2,eza,2]T.

Now, a fixed-time nonsingular fast terminal sliding mode surface of the leader UAV is defined as follows:

s l = e l , 2 + g 1 H l λ 1 ( e l , 1 ) + g 2 Sig λ 2 ( e l , 1 ) (14) s_l = e_{l,2} + g_1 H^{lambda_1}_l (e_{l,1}) + g_2 text{Sig}^{lambda_2}(e_{l,1}) tag{14} sl=el,2+g1Hlλ1(el,1)+g2Sigλ2(el,1)(14)

s l = [ e x a , 2 e y a , 2 e z a , 2 ] + [ g 11 g 12 g 13 ] [ h l λ 1 ( e x a , 1 ) h l λ 1 ( e y a , 1 ) h l λ 1 ( e z a , 1 ) ] + [ g 21 g 22 g 23 ] [ Sig λ 2 ( e x a , 1 ) Sig λ 2 ( e y a , 1 ) Sig λ 2 ( e z a , 1 ) ] (14) s_l = left[begin{matrix} e_{xa,2} \ e_{ya,2} \ e_{za,2} \ end{matrix}right] + left[begin{matrix} g_{11} \ & g_{12} \ && g_{13} \ end{matrix}right] left[begin{matrix} h^{lambda_1}_l(e_{xa,1}) \ h^{lambda_1}_l(e_{ya,1}) \ h^{lambda_1}_l(e_{za,1}) \ end{matrix}right] + left[begin{matrix} g_{21} \ & g_{22} \ && g_{23} \ end{matrix}right] left[begin{matrix} text{Sig}^{lambda_2}(e_{xa,1}) \ text{Sig}^{lambda_2}(e_{ya,1}) \ text{Sig}^{lambda_2}(e_{za,1}) \ end{matrix}right] tag{14} sl= exa,2eya,2eza,2 + g11g12g13 hlλ1(exa,1)hlλ1(eya,1)hlλ1(eza,1) + g21g22g23 Sigλ2(exa,1)Sigλ2(eya,1)Sigλ2(eza,1) (14)

where
H l λ 1 ( e l , 1 ) = [ h l λ 1 ( e x a , 1 ) , h l λ 1 ( e y a , 1 ) , h l λ 1 ( e z a , 1 ) ] T red{H^{lambda_1}_l(e_{l,1})} = [h^{lambda_1}_l(e_{xa,1}), h^{lambda_1}_l(e_{ya,1}), h^{lambda_1}_l(e_{za,1})]^text{T} Hlλ1(el,1)=[hlλ1(exa,1),hlλ1(eya,1),hlλ1(eza,1)]T,
Sig λ 2 ( e l , 1 ) = [ Sig λ 2 ( e x a , 1 ) , Sig λ 2 ( e y a , 1 ) , Sig λ 2 ( e z a , 1 ) ] T red{text{Sig}^{lambda_2}(e_{l,1})} = [text{Sig}^{lambda_2}(e_{xa,1}), text{Sig}^{lambda_2}(e_{ya,1}), text{Sig}^{lambda_2}(e_{za,1})]^text{T} Sigλ2(el,1)=[Sigλ2(exa,1),Sigλ2(eya,1),Sigλ2(eza,1)]T,
g 1 = diag { g 11 , g 12 , g 13 } red{g_1} = text{diag} {g_{11}, g_{12}, g_{13}} g1=diag{g11,g12,g13},
g 2 = diag { g 21 , g 22 , g 23 } red{g_2} = text{diag} {g_{21}, g_{22}, g_{23}} g2=diag{g21,g22,g23},
p i a , i = x , y , z red{p_{ia}, i=x,y,z} pia,i=x,y,z is a designed positive,
λ 1 , λ 2 red{lambda_1, lambda_2} λ1,λ2 are positive constants.

in which:
h l λ 1 ( e i a , 1 ) = { ∣ e i a , 1 ∣ λ 1 sign ( e i a , 1 ) , ∣ e i a , 1 ∣ ≥ p i a p i a λ 1 − 1 e i a , 1 , ∣ e i a , 1 ∣ < p i a Sig λ 2 ( e i a , 1 ) = ∣ e i a , 1 ∣ λ 2 sign ( e i a , 1 ) , i = x , y , z (15) begin{aligned} red{h^{lambda_1}_l(e_{ia,1})} &= left{begin{aligned} |e_{ia,1}|^{lambda_1} text{sign} (e_{ia,1}), quad |e_{ia,1}| ge p_{ia} \ p^{lambda_1 - 1}_{ia} e_{ia,1}, quad |e_{ia,1}| < p_{ia} end{aligned}right. \ red{text{Sig}^{lambda_2}(e_{ia,1})} &= |e_{ia,1}|^{lambda_2} text{sign} (e_{ia,1}), quad i=x,y,z end{aligned} tag{15} hlλ1(eia,1)Sigλ2(eia,1)={∣eia,1∣λ1sign(eia,1),∣eia,1∣≥piapiaλ1−1eia,1,∣eia,1∣

u a = A − 1 [ − f ( χ l 1 ) + χ ¨ a d − g 1 ϑ 1 e l , 2 − g 2 ϑ 2 e l , 2 − ( k 1 + σ ) sign ( s l ) − k 2 s l s l T s l ] (20) u_a = A^{-1} left[ -f(chi_{l1}) + ddot{chi}_{ad} - g_1 vartheta_1 e_{l,2} - g_2 vartheta_2 e_{l,2} - (k_1 + sigma) text{sign}(s_l) - k_2 s_l s^text{T}_l s_l right] tag{20} ua=A−1[−f(χl1)+χ¨ad−g1ϑ1el,2−g2ϑ2el,2−(k1+σ)sign(sl)−k2slslTsl](20)

3.2. Fixed-time control for the follower USVs

In view of assumption 5 and the USV model Equation (27), the follower USVs control law can be given as follows:

u s = R s − 1 [ C s χ s , 2 + D s χ s , 2 − M s ( − β ˙ s , 2 − χ ˙ l , 2 + ( ( L + B ) − 1 ⊗ I 3 ) ( b 1 Φ 1 e s , 2 + b 2 Φ 2 e s , 2 + ( r 1 s + τ ) sign ( s p ) + r 2 s s p s p T s p ) ) ] (29) begin{aligned} u_s &= R^{-1}_s [ C_s chi_{s,2} + D_s chi_{s,2} - M_s ( -dot{beta}_{s,2} - dot{chi}_{l,2} + ((L+B)^{-1}otimes I_3) \ &(b_1 varPhi_1 e_{s,2} + b_2 varPhi_2 e_{s,2} + (r_{1s}+tau) text{sign}(s_p) + r_{2s} s_p s_p^text{T} s_p) ) ] \ end{aligned} tag{29} us=Rs−1[Csχs,2+Dsχs,2−Ms(−β˙s,2−χ˙l,2+((L+B)−1⊗I3)(b1Φ1es,2+b2Φ2es,2+(r1s+τ)sign(sp)+r2sspspTsp))](29)

{ χ ˙ s , 1 = χ s , 2 χ ˙ s , 2 = M s − 1 ( R s sat ( u s ) − C s χ s , 2 − D s χ s , 2 ) + F s = M s − 1 ( R s u s − C s χ s , 2 − D s χ s , 2 ) + F s = M s − 1 ( C s χ s , 2 + D s χ s , 2 − M s ( − β ˙ s , 2 − χ ˙ l , 2 + ( ( L + B ) − 1 ⊗ I 3 ) ( b 1 Φ 1 e s , 2 + b 2 Φ 2 e s , 2 + ( r 1 s + τ ) sign ( s p ) + r 2 s s p s p T s p ) ) − C s χ s , 2 − D s χ s , 2 ) + F s = M s − 1 ( − M s ( − β ˙ s , 2 − χ ˙ l , 2 + ( ( L + B ) − 1 ⊗ I 3 ) ( b 1 Φ 1 e s , 2 + b 2 Φ 2 e s , 2 + ( r 1 s + τ ) sign ( s p ) + r 2 s s p s p T s p ) ) ) + F s = ( − ( − β ˙ s , 2 − χ ˙ l , 2 + ( ( L + B ) − 1 ⊗ I 3 ) ( b 1 Φ 1 e s , 2 + b 2 Φ 2 e s , 2 + ( r 1 s + τ ) sign ( s p ) + r 2 s s p s p T s p ) ) ) + F s = ( β ˙ s , 2 + χ ˙ l , 2 − ( ( L + B ) − 1 ⊗ I 3 ) ( b 1 Φ 1 e s , 2 + b 2 Φ 2 e s , 2 + ( r 1 s + τ ) sign ( s p ) + r 2 s s p s p T s p ) ) + F s (10) left{begin{aligned} dot{chi}_{s,1} &= chi_{s,2} \ dot{chi}_{s,2} &= M_s^{-1} ( R_s text{sat}(u_s) - C_s chi_{s,2} - D_s chi_{s,2} ) + F_s \ &= M_s^{-1} ( R_s u_s - C_s chi_{s,2} - D_s chi_{s,2} ) + F_s \ &= M_s^{-1} ( C_s chi_{s,2} + D_s chi_{s,2} - M_s ( -dot{beta}_{s,2} - dot{chi}_{l,2} + ((L+B)^{-1}otimes I_3) (b_1 varPhi_1 e_{s,2} + b_2 varPhi_2 e_{s,2} + (r_{1s}+tau) text{sign}(s_p) + r_{2s} s_p s_p^text{T} s_p) ) - C_s chi_{s,2} - D_s chi_{s,2} ) + F_s \ &= M_s^{-1} ( - M_s ( -dot{beta}_{s,2} - dot{chi}_{l,2} + ((L+B)^{-1}otimes I_3) (b_1 varPhi_1 e_{s,2} + b_2 varPhi_2 e_{s,2} + (r_{1s}+tau) text{sign}(s_p) + r_{2s} s_p s_p^text{T} s_p) ) ) + F_s \ &= ( - ( -dot{beta}_{s,2} - dot{chi}_{l,2} + ((L+B)^{-1}otimes I_3) (b_1 varPhi_1 e_{s,2} + b_2 varPhi_2 e_{s,2} + (r_{1s}+tau) text{sign}(s_p) + r_{2s} s_p s_p^text{T} s_p) ) ) + F_s \ &= ( dot{beta}_{s,2} + dot{chi}_{l,2} - ((L+B)^{-1}otimes I_3) (b_1 varPhi_1 e_{s,2} + b_2 varPhi_2 e_{s,2} + (r_{1s}+tau) text{sign}(s_p) + r_{2s} s_p s_p^text{T} s_p) ) + F_s end{aligned}right. tag{10} ⎩ ⎨ ⎧χ˙s,1χ˙s,2=χs,2=Ms−1(Rssat(us)−Csχs,2−Dsχs,2)+Fs=Ms−1(Rsus−Csχs,2−Dsχs,2)+Fs=Ms−1(Csχs,2+Dsχs,2−Ms(−β˙s,2−χ˙l,2+((L+B)−1⊗I3)(b1Φ1es,2+b2Φ2es,2+(r1s+τ)sign(sp)+r2sspspTsp))−Csχs,2−Dsχs,2)+Fs=Ms−1(−Ms(−β˙s,2−χ˙l,2+((L+B)−1⊗I3)(b1Φ1es,2+b2Φ2es,2+(r1s+τ)sign(sp)+r2sspspTsp)))+Fs=(−(−β˙s,2−χ˙l,2+((L+B)−1⊗I3)(b1Φ1es,2+b2Φ2es,2+(r1s+τ)sign(sp)+r2sspspTsp)))+Fs=(β˙s,2+χ˙l,2−((L+B)−1⊗I3)(b1Φ1es,2+b2Φ2es,2+(r1s+τ)sign(sp)+r2sspspTsp))+Fs(10)