Research article Special Issues

Model-free scheme using time delay estimation with fixed-time FSMC for the nonlinear robot dynamics

  • Received: 04 January 2024 Revised: 15 February 2024 Accepted: 04 March 2024 Published: 13 March 2024
  • MSC : 93C10, 93C40, 93D09, 93D21, 93B52

  • This paper presents a scheme of time-delay estimation (TDE) for unknown nonlinear robotic systems with uncertainty and external disturbances that utilizes fractional-order fixed-time sliding mode control (TDEFxFSMC). First, a detailed explanation and design concept of fractional-order fixed-time sliding mode control (FxFSMC) are provided. High performance tracking positions, non-chatter control inputs, and nonsingular fixed-time control are all realized with the FxSMC method. The proposed approach performs better and obtains superior performance when FxSMC is paired with fractional-order control. Furthermore, a TDE scheme is included in the suggested strategy to estimate the unknown nonlinear dynamics. Afterward, the suggested system's capacity to reach stability in fixed time is determined by using Lyapunov analyses. By showing the outcomes of the proposed technique applied to nonlinear robot dynamics, the efficacy of the recommended method is assessed, illustrated, and compared with the existing control scheme.

    Citation: Saim Ahmed, Ahmad Taher Azar, Ibraheem Kasim Ibraheem. Model-free scheme using time delay estimation with fixed-time FSMC for the nonlinear robot dynamics[J]. AIMS Mathematics, 2024, 9(4): 9989-10009. doi: 10.3934/math.2024489

    Related Papers:

    [1] Houssem Eddine Khochemane, Ali Rezaiguia, Hasan Nihal Zaidi . Exponential stability and numerical simulation of a Bresse-Timoshenko system subject to a neutral delay. AIMS Mathematics, 2023, 8(9): 20361-20379. doi: 10.3934/math.20231038
    [2] Abdelkader Moumen, Fares Yazid, Fatima Siham Djeradi, Moheddine Imsatfia, Tayeb Mahrouz, Keltoum Bouhali . The influence of damping on the asymptotic behavior of solution for laminated beam. AIMS Mathematics, 2024, 9(8): 22602-22626. doi: 10.3934/math.20241101
    [3] Khaled zennir, Djamel Ouchenane, Abdelbaki Choucha, Mohamad Biomy . Well-posedness and stability for Bresse-Timoshenko type systems with thermodiffusion effects and nonlinear damping. AIMS Mathematics, 2021, 6(3): 2704-2721. doi: 10.3934/math.2021164
    [4] Tae Gab Ha, Seyun Kim . Existence and energy decay rate of the solutions for the wave equation with a nonlinear distributed delay. AIMS Mathematics, 2023, 8(5): 10513-10528. doi: 10.3934/math.2023533
    [5] Abdelbaki Choucha, Sofian Abuelbacher Adam Saad, Rashid Jan, Salah Boulaaras . Decay rate of the solutions to the Lord Shulman thermoelastic Timoshenko model. AIMS Mathematics, 2023, 8(7): 17246-17258. doi: 10.3934/math.2023881
    [6] Abdelbaki Choucha, Salah Boulaaras, Asma Alharbi . Global existence and asymptotic behavior for a viscoelastic Kirchhoff equation with a logarithmic nonlinearity, distributed delay and Balakrishnan-Taylor damping terms. AIMS Mathematics, 2022, 7(3): 4517-4539. doi: 10.3934/math.2022252
    [7] Yudhveer Singh, Devendra Kumar, Kanak Modi, Vinod Gill . A new approach to solve Cattaneo-Hristov diffusion model and fractional diffusion equations with Hilfer-Prabhakar derivative. AIMS Mathematics, 2020, 5(2): 843-855. doi: 10.3934/math.2020057
    [8] Afraz Hussain Majeed, Sadia Irshad, Bagh Ali, Ahmed Kadhim Hussein, Nehad Ali Shah, Thongchai Botmart . Numerical investigations of nonlinear Maxwell fluid flow in the presence of non-Fourier heat flux theory: Keller box-based simulations. AIMS Mathematics, 2023, 8(5): 12559-12575. doi: 10.3934/math.2023631
    [9] Saima Rashid, Fahd Jarad, Hajid Alsubaie, Ayman A. Aly, Ahmed Alotaibi . A novel numerical dynamics of fractional derivatives involving singular and nonsingular kernels: designing a stochastic cholera epidemic model. AIMS Mathematics, 2023, 8(2): 3484-3522. doi: 10.3934/math.2023178
    [10] José Luis Díaz Palencia, Saeed ur Rahman, Antonio Naranjo Redondo . Analysis of travelling wave solutions for Eyring-Powell fluid formulated with a degenerate diffusivity and a Darcy-Forchheimer law. AIMS Mathematics, 2022, 7(8): 15212-15233. doi: 10.3934/math.2022834
  • This paper presents a scheme of time-delay estimation (TDE) for unknown nonlinear robotic systems with uncertainty and external disturbances that utilizes fractional-order fixed-time sliding mode control (TDEFxFSMC). First, a detailed explanation and design concept of fractional-order fixed-time sliding mode control (FxFSMC) are provided. High performance tracking positions, non-chatter control inputs, and nonsingular fixed-time control are all realized with the FxSMC method. The proposed approach performs better and obtains superior performance when FxSMC is paired with fractional-order control. Furthermore, a TDE scheme is included in the suggested strategy to estimate the unknown nonlinear dynamics. Afterward, the suggested system's capacity to reach stability in fixed time is determined by using Lyapunov analyses. By showing the outcomes of the proposed technique applied to nonlinear robot dynamics, the efficacy of the recommended method is assessed, illustrated, and compared with the existing control scheme.



    Filippov introduced a generalized Jacobi identity for n-ary skew-symmetric operation, which acts as a replacement for the classical Jacobi identity in the context of Lie algebras [5]. He also proposed the concept of n-Lie algebra, also known as Filippov n-algebra, with the corresponding generalized Jacobi identity referred to as the Filippov identity. Nambu and Takhtajan extended the concept of Poisson manifold to an n-ary generalization called Nambu-Poisson structure in order to study Hamiltonian mechanics more comprehensively [15,17]. It is worth noting that both the Nambu-Poisson structure and the n-Lie algebra share the same generalized Jacobi identity. Grabowski and Marmo introduced the concept of Filippov n-algebroids, an n-ary generalization of Lie algebroids, in order to determine the relationship between linear Nambu-Poisson structures and Filippov algebras [7]. Consequently, it is reasonable to anticipate that many tools used to study Lie algebroids could be enhanced or upgraded to the realm of Filippov algebroids. Therefore, we aim to address the absence of the concepts of connections and curvatures of Filippov algebroids in the literature and provide a primitive analysis of these topics from a geometric point of view.

    Recall that a Lie algebroid is a (real) vector bundle AM together with a bundle map ρ:ATM, called anchor, and a Lie bracket [,] on the section space Γ(A) of A, satisfying that ρ:Γ(A)Γ(TM) is a morphism of Lie algebras and the Leibniz rule

    [X,fY]=f[X,Y]+(ρ(X)f)Y,X,YΓ(A) and fC(M).

    By an easy smooth analysis, the bracket [,] can always be reformulated in the form

    [X,Y]=XYYX, (1.1)

    where :Γ(A)×Γ(A)Γ(A) satisfies the properties

    fXY=fXY and X(fY)=fXY+(ρ(X)f)Y.

    One calls a connection on the anchored bundle (A,ρ). (This is indeed a straightforward generalization of connections on vector bundles.)

    When the bracket [,] of a Lie algebroid A is expressed in the form (1.1), one says that the connection is torsion free. See [9] for the existence of torsion free connections on Lie algebroids. The curvature form RΓ(2AEnd(A)) of such a connection is defined in the standard manner:

    R(X,Y)(Z)=XYZYXZ[X,Y]Z,

    for all X,Y,ZΓ(A). Now, ρ being a morphism of Lie algebras is equivalent to the condition that R is a tensor in its third argument. Moreover, the Jacobi identity for [,] is transformed into the following Lie-Bianchi identity

    R(X,Y)(Z)+R(Y,Z)(X)+R(Z,X)(Y)=0.

    Therefore, Lie algebroids can be realized as anchored bundles equipped with special connections [16]. We wish to find an analogous characterization of the n-ary bracket of any Filippov algebroid. A significant difference between Lie algebroids and Filippov n-algebroids (for n3) is that the bracket and anchor of the latter are of more arguments (see Definition 2.2). So there is not an obvious way to extend Eq (1.1). We come up with a solution in Section 3. Below is a quick summary:

    ● First, we define (multi-input) connections compatible with a given (multi-input) n-anchor (see Definition 3.1). This is a quite straightforward extension of usual connections of Lie algebroids (when n=2).

    ● Second, we introduce the curvature form R stemming from a connection (see Eq (3.2)). We believe that this is a highly nontrivial invention of this note.

    ● Third, we prove in Theorem 3.1 that certain good connections, which we call Filippov connections, fully determines Filippov algebroid structures. This includes two points: (1) The n-ary bracket of any Filippov algebroid can be realized in a torsion free manner (see Eq (3.1)); (2) The generalized Jacobi identity is transformed to a constraint, called the Bianchi-Filippov identity (see Eq (3.3)) about the associated curvature R.

    We then illustrate a simple method via covariant differential operators to construct Filippov connections in Section 3.3.

    As vector bundles are fiber bundles with linear fibers, particular cases of homogeneity structures, linear geometrical structures on vector bundles are of particular interest (see also [6] on weighted structures for various geometric objects on manifolds with general homogeneity structures). We finally show that there exists a one-to-one correspondence between Filippov n-algebroid structures on a vector bundle A of rank n3 and linear Nambu-Poisson structures on its dual bundle A (see Theorem 4.1). As an interesting application of our result, one is able to construct linear Nambu-Poisson structures from Filippov connections (Corollary 4.4).

    In short summary, torsion-free connections subject to the Bianchi-Filippov identity are important geometric constraints for Filippov algebroids. It is well known that torsion free connections for Lie algebroids play a crucial role in various mathematical constructions, for example, in the construction of Poincaré-Birkhoff-Witt isomorphisms and Kapranov dg manifolds for Lie algebroid pairs [9]. Additionally, Bianchi identities are not only significant in Riemannian geometry, but also in Poisson geometry [3]. We believe that our approach to Filippov algebroids will be beneficial in this context.

    In this section, we recall the definition of Filippov algebroids from [7]. There is an alternative characterization of Filippov algebroids in terms of certain 1-derivations [13]. It is important to note that n2 is an integer, although the only interesting situation is when n3. Let us start with a notion of n-anchored bundles.

    Definition 2.1. An n-anchored vector bundle over a smooth manifold M is pair (A,ρ), where A is a vector bundle over M and ρ:n1ATM is a vector bundle morphism, called n-anchor of A.

    Definition 2.2. A Filippov n-algebroid over a smooth manifold M is an n-anchored bundle (A,ρ) over M together with an R-multilinear and skew-symmetric n-bracket on the section space Γ(A) of A:

    [,,]:Γ(A)××Γ(A)ncopiesΓ(A),

    satisfying the following compatibility conditions:

    (1) The n-anchor ρ intertwines the n-bracket and the standard Lie bracket [,]TM on Γ(TM):

    [ρ(X1Xn1),ρ(Y1Yn1)]TM=n1i=1ρ(Y1[X1,,Xn1,Yi]Yn1); (2.1)

    (2) The n-bracket is a derivation with respect to C(M)-multiplications:

    [X1,,Xn1,fY]=f[X1,,Xn1,Y]+ρ(X1Xn1)(f)Y; (2.2)

    (3) The following equation holds, to be called the (generalized) Jacobi identity (or Filippov identity):

    [X1,,Xn1,[Y1,,Yn]]=ni=1[Y1,,Yi1,[X1,,Xn1,Yi],Yi+1,,Yn], (2.3)

    for all Xi,YiΓ(A) and fC(M).

    Note that any Lie algebroid is a Filippov 2-algebroid. A Filippov n-algebra is a Filippov n-algebroid over the one-point base manifold. In fact, analogous to the Lie algebroid case (i.e., n=2 case), the condition (1) in the above definition follows from the conditions (2) and (3).

    The following examples (due to [7]) illustrate two Filippov n-algebroid structures on the trivial tangent bundle TRm of Rm for mn2.

    Example 2.3. Consider the trivial n-anchored bundle (TRm,ρ=0). For each Filippov n-algebra structure on Rm with structure constants {cji1,,in} and each smooth function gC(Rm), we have a Filippov n-algebroid (TRm,0) whose bracket is defined by

    [f1xi1,,fnxin]=gf1fnmj=1cji1,,inxj.

    Example 2.4. Equip TRm with the n-anchor map ρ defined by the tensor field

    dx1dxn1x1,

    where x1,,xn1,xn,,xm are coordinates of Rm. Then, the n-anchored bundle (TRm,ρ) together with the trivial n-bracket on generators xi produces a (nontrivial) Filippov n-algebroid over Rm.

    We emphasize a crucial but often overlooked point in the literature: the presence of a Filippov n-bracket on an n-anchored bundle (A,ρ) imposes a constraint on the rank of ρ for every integer n3.

    Proposition 2.5. Let (A,[,,],ρ) be a Filippov n-algebroid for n3. Then the rank of the image of ρ as a distribution on M can not exceed 1, i.e., rank(ρ(n1A))1.

    Proof. Suppose that the image of ρ at pM is not trivial. So we can find an open neighborhood U of p and some Y1Yn1Γ(n1A)|U such that ρ(Y1Yn1) is nowhere vanishing on U. The desired statement amounts to show that, if ρ(X1Xn1) is also nowhere vanishing on U, then there exists some cC(U) such that

    ρ(X1Xn1)=cρ(Y1Yn1).

    In fact, by the definition of Filippov n-algebroids, we obtain

    [ρ(fX1Xn1),ρ(Y1Yn1)]by Eq (2.1)=n1i=1ρ(Y1Yi1[fX1,X2,,Xn1,Yi]Yi+1Yn1)by Eq (2.2)=fn1i=1ρ(Y1Yi1[X1,X2,,Xn1,Yi]Yi+1Yn1)+n1i=1(1)n1ρ(X2Xn1Yi)(f)ρ(Y1Yi1X1Yi+1Yn1)by Eq (2.1)=f[ρ(X1Xn1),ρ(Y1Yn1)]+n1i=1(1)n1ρ(X2Xn1Yi)(f)ρ(Y1Yi1X1Yi+1Yn1).

    Moreover, since ρ is a morphism of vector bundles, we have

    [ρ(fX1Xn1),ρ(Y1Yn1)]=[fρ(X1Xn1),ρ(Y1Yn1)]=f[ρ(X1Xn1),ρ(Y1Yn1)]ρ(Y1Yn1)(f)ρ(X1Xn1).

    Setting Y1=X1 in the above two equations, we obtain

    ρ(X1Xn1)(f)ρ(X1Y2Yn1)=ρ(X1Y2Yn1)(f)ρ(X1Xn1). (2.4)

    Using Eq (2.4), we have

    ρ(X1Xn1)=g1ρ(X1Y2Yn1)=g1ρ(Y2X1Yn1)=g1g2ρ(Y2Y1Yn1)=g1g2ρ(Y1Yn1),

    for some g1,g2C(U).

    (1) If ρ(X1Y2Yn1) is nowhere vanishing on U, then the vector fields ρ(X1Xn1) and ρ(Y1Yn1) must be C(U)-linearly dependent.

    (2) If ρ(X1Y2Yn1)|p=0, then we let ˜X1=X1+Y1 and consider ρ(˜X1Y2Yn1), which is nowhere vanishing on U. By arguments in (1) as above, ρ(˜X1X2Xn1) and ρ(Y1Yn1) are C(U)-linearly dependent, and we obtain the desired statement as well.

    In this section, we characterize Filippov algebroids via connections on the underlying anchored bundles.

    Definition 3.1. A connection on an n-anchored bundle (A,ρ) is a bilinear map :Γ(n1A)×Γ(A)Γ(A) satisfying two conditions:

    fX1Xn1Xn=fX1Xn1Xn, and X1Xn1(fXn)=fX1Xn1Xn+ρ(X1Xn1)(f)Xn

    for all X1,,XnΓ(A) and fC(M).

    To see the existence of such a connection, one takes a TM-connection on A, say TM, and then define on the n-anchored bundle (A,ρ) as the pullback of TM:

    X1Xn1Xn:=TMρ(X1Xn1)Xn.

    The key point of this note is that any connection on (A,ρ) induces a skew-symmetric n-bracket on Γ(A) defined by

    [X1,,Xn]:=ni=1(1)n+iX1^XiXnXi=ni=1(1)(n1)iXi+1XnX1Xi1Xi. (3.1)

    For computational convenience, we denote the covariant derivative on Γ(A) along X1,,Xn1Γ(A) by

    X1n1:=[X1,,Xn1,]:Γ(A)Γ(A).

    It extends to all sections in A by

    X1n1(Y1Ym):=mi=1Y1Yi1X1n1(Yi)Yi+1Ym.

    We then introduce the curvature form of , an operation

    R(,)():Γ(A)××Γ(A)(n1)copies×Γ(n1A)×Γ(A)Γ(A),

    defined by

    R(X1,,Xn1,Y1Yn1)(Z):=[X1n1,Y1Yn1](Z)X1n1(Y1Yn1)Z:=X1n1Y1Yn1ZY1Yn1X1n1ZX1n1(Y1Yn1)Z, (3.2)

    for all X1,,Xn1,Y1,,Yn1,ZΓ(A) and n3. When n=3, it reads

    R(X1,X2,Y1Y2)(Z)=X12Y1Y2ZY1Y2X12ZX12(Y1Y2)Z=[X1,X2,Y1Y2Z]Y1Y2[X1,X2,Z] [X1,X2,Y1]Y2+Y1[X1,X2,Y2]Z.

    When n=4, the expression of R consists of twenty terms. As n gets larger, more terms are involved.

    It is easy to verify from the defining Eq (3.2) that the curvature R is C(M)-linear with respect to the argument Y1Yn1. However, R need not be tensorial in X1,,Xn1 although it is skew-symmetric in these arguments.

    Definition 3.2. A connection on an n-anchored bundle (A,ρ) is called a Filippov connection if the following two conditions are true:

    (1) The curvature R is C(M)-linear with respect to its last argument, i.e., for all fC(M) and all X1,,Xn1,Y1,,YnΓ(A), we have

    R(X1,,Xn1,Y1Yn1)(fYn)=fR(X1,,Xn1,Y1Yn1)(Yn);

    (2) The following equality holds, to be called the Bianchi-Filippov identity:

    0=n1i=0(1)(n1)iR(X1,,Xn1,Yi+1YnY1Yi1)Yi, (3.3)

    where Y0 means Yn.

    We are ready to state our main theorem, which characterizes Filippov algebroids fully by Filippov connections.

    Theorem 3.3. Let (A,ρ) be an n-anchored bundle. If is a Filippov connection on (A,ρ), then (A,ρ,[,,]) is a Filippov n-algebroid, where [,,] is the n-bracket given by Eq (3.1). Moreover, any Filippov n-algebroid structure on (A,ρ) arises from a Filippov connection in this way.

    The proof of Theorem 3.3 is divided, and will follow immediately from the three lemmas below.

    Lemma 3.4. Let be a connection on an n-anchored bundle (A,ρ). The curvature R satisfies the first condition of Definition 3.2 if and only if the anchor ρ intertwines the induced n-bracket [,,] and the Lie bracket [,]TM on Γ(TM), i.e.,

    [ρ(X1Xn1),ρ(Y1Yn1)]TM=n1i=1ρ(Y1[X1,,Xn1,Yi]Yn1),

    for all X1,,Xn1,Y1,,Yn1Γ(A).

    Proof. By the definition of curvature, we have

    R(X1,,Xn1,Y1Yn1)(fYn)by Eq (3.2)=[X1n1,Y1Yn1](fYn)X1n1(Y1Yn1)(fYn)=X1n1Y1Yn1(fYn)Y1Yn1X1n1(fYn)X1n1(Y1Yn1)(fYn)=fR(X1,,Xn1,Y1Yn1)(Yn)+ρ(X1Xn1)ρ(Y1Yn1)(f)Ynρ(Y1Yn1)ρ(X1Xn1)(f)Ynn1i=1ρ(Y1X1n1(Yi)Yn1)(f)Yn.

    Hence, the curvature R is C(M)-linear with respect to its last argument if and only if

    ρ(X1Xn1)ρ(Y1Yn1)ρ(Y1Yn1)ρ(X1Xn1)=n1i=1ρ(Y1X1n1(Yi)Yn1)=n1i=1ρ(Y1[X1,,Xn1,Yi]Yn1).

    Lemma 3.5. Let be a connection on an n-anchored bundle (A,ρ). The curvature R satisfies the second condition of Definition 3.2, i.e., the Bianchi-Filippov identity (3.3), if and only if the induced n-bracket [,,] satisfies the (generalized) Jacobi identity (2.3).

    Proof. The statement follows directly from the following lines of computation:

    n1i=0(1)(n1)iR(X1,,Xn1,Yi+1YnY1Yi1)Yiby Eq (3.2)=n1i=0(1)(n1)i([X1n1,Yi+1YnY1Yi1](Yi)X1n1(Yi+1YnY1Yi1)Yi)=[X1,,Xn1,n1i=0(1)(n1)iYi+1YnY1Yi1(Yi)]n1i=0(1)(n1)i(Yi+1YnY1Yi1X1n1(Yi)+jiYi+1X1n1(Yj)Yi1Yi)by Eq (3.1)=[X1,,Xn1,[Y1,,Yn]]n1i=0(1)(n1)iYi+1YnY1Yi1[X1,,Xn1,Yi]n1i=0ji(1)(n1)iYi+1[X1,,Xn1,Yj]Yi1Yiby Eq (3.1)=[X1,,Xn1,[Y1,,Yn]]ni=1[Y1,,[X1,,Xn1,Yi],,Yn].

    The next lemma shows that any Filippov algebroid can be realized by a Filippov connection.

    Lemma 3.6. Let (A,[,,],ρ) be a Filippov n-algebroid. Then there exists a Filippov connection on the underlying n-anchored bundle (A,ρ) such that [,,]=[,,] (the torsion-free property).

    Proof. Given a connection on (A,ρ), we are able to obtain an R-multilinear operation K(,,) on Γ(A) by

    K(X1,,Xn):=[X1,,Xn][X1,,Xn].

    Using axioms of Filippov algebroids, it is easy to see that K(,,) is indeed C(M)-multilinear. Then we define a new connection on (A,ρ) by

    X1Xn1Xn:=1nK(X1,,Xn)+X1Xn1Xn.

    It remains to check the desired equality:

    [X1,,Xn]=X1Xn1Xn+n1i=1(1)(n1)iXi+1XnX1Xi1Xi=K(X1,,Xn)+X1Xn1Xn+n1i=1(1)(n1)iXi+1XnX1Xi1Xi=[X1,,Xn].

    The following examples illustrate three Filippov connections on the trivial tangent bundle TRm of Rm for mn2.

    Example 3.7. Consider the trivial n-anchored vector bundle (TRm,ρ=0). Suppose that the vector space Rm is endowed with a Filippov n-algebra structure whose structure constants are {cji1in} with respect to the standard basis of Rm. Given a smooth function gC(Rm) and a set of constants {aji1in1;in} satisfying the equality:

    aji1in1;in+n1k=1(1)(n1)kajik+1ini1ik1;ik=cji1in, (3.4)

    we are able to obtain a connection on (TRm,ρ=0) generated by the only one nontrivial relation:

    xi1xin1xin:=gmj=1aji1in1;inxj.

    It follows from the recipe in Eqs (3.1) and (3.4) that

    [xi1,,xin]=gmi=1cji1inxj.

    So, what we recover is the Filippov structure on TRm as in Example 2.3. Hence, is indeed a Filippov connection.

    Example 3.8. Consider the n-anchor map ρ on the tangent bundle TRm defined by the tensor field dx1dxn1x1, where x1,,xn1,xn,,xm are coordinate functions of Rm. It is obvious that the (nontrivial) connection on (TRm,ρ) generated by the trivial relation:

    xi1xin1xin:=0,

    produces a nontrivial n-bracket which is compatible with ρ. Indeed, what we recover is the Filippov structure on TRm as in Example 2.4, and the said connection is a Filippov connection.

    Example 3.9. Continue to work with the anchored bundle (A,ρ) as in the previous example. We consider a different connection with the only nontrivial generating relations:

    Zxk={(1)σxk,if Z=xσ1xσn1,0,otherwise,

    where σ is a permutation {1,,n1}, for all xk{x1,,xm}. Then, the associated n-bracket is given by

    [xσ1,,xσn1,xk]={(1)σxk,ifk>n1,0,otherwise,

    where σ is a permutation {1,,n1}. By subtle analysis, one can find that the associated curvature R is just zero. Hence is truly a Filippov connection and the above bracket defines a Filippov algebroid structure on (A,ρ).

    Let AM be a vector bundle. Consider the bundle CDO(A) of covariant differential operators (cf. [10]*Ⅲ, [11], see also [8], where the notation D(A) is used instead of CDO(A)). An element D of Γ(CDO(A)), called a covariant differential operator, is an R-linear operator Γ(A)Γ(A) together with a vector field ˆDΓ(TM), called the symbol of D, satisfying

    D(fX)=fD(X)+ˆD(f)X,XΓ(A),fC(M).

    The operator D can be first extended by the Leibniz rule to an operator D:Γ(n1A)Γ(n1A). By taking dual we obtain an operator D:Γ(n1A)Γ(n1A) defined by

    X1Xn1|D(ˉη)=ˆDX1Xn1|ˉηn1i=1X1D(Xi)Xn1|ˉη, (3.5)

    for all X1,,Xn1Γ(A) and ˉηΓ(n1A).

    Given a pair (D,ˉξ), where DΓ(CDO(A)) and ˉξΓ(n1A), one is able to construct a map

    ρ(D,ˉξ):Γ(n1A)Γ(TM),X1Xn1X1Xn1|ˉξˆD.

    It is clear that ρ(D,ˉξ) makes A an n-anchored bundle, and the rank of the image of ρ(D,ˉξ) does not exceed 1.

    Define a connection on the n-anchored bundle (A,ρ(D,ˉξ)) by

    (D,ˉξ)X1Xn1Xn:=X1Xn1|ˉξD(Xn). (3.6)

    Proposition 3.10. If the pair (D,ˉξ) is subject to the relation

    D(ˉξ)=gˉξ,for some gC(M), (3.7)

    then (D,ˉξ) defined as in (3.6) is a Filippov connection on the n-anchored bundle (A,ρ(D,ˉξ)).

    Proof. We denote (D,ˉξ) by for simplicity below. It suffices to check the associated curvature R is C(M)-linear with respect to the last argument and satisfies the Bianchi-Filippov identity (3.3). In fact, we have

    R(X1,,Xn1,Y1Yn1)(fYn)by Eq (3.2)=X1n1Y1Yn1(fYn)Y1Yn1X1n1(fYn)X1n1(Y1Yn1)(fYn)by Eqs (3.1), (3.5), (3.6)=fR(X1,,Xn1,Y1Yn1)(Yn)+(X1Xn1|ˉξY1Yn1|D(ˉξ)Y1Yn1|ˉξX1Xn1|D(ˉξ))ˆD(f)Ynby Eq (3.7)=fR(X1,,Xn1,Y1Yn1)(Yn),

    and

    n1i=0(1)(n1)iR(X1,,Xn1,Yi+1YnY1Yi1)Yiby Eq (3.2)=n1i=0(1)(n1)i[X1n1,Yi+1YnY1Yi1](Yi)n1i=0(1)(n1)iX1n1(Yi+1YnY1Yi1)Yiby Eqs (3.1) and (3.6) =n1i=0(1)(n1)iYi+1YnY1Yi1|ˉξX1n1(D(Yi))n1i=0(1)(n1)iYi+1YnY1Yi1|ˉξD(X1n1(Yi))+n1i=0(1)(n1)iX1n1(Yi+1YnY1Yi1|ˉξ)D(Yi)n1i=0(1)(n1)ijiYi+1X1n1(Yj)Yi1|ˉξD(Yi)by Eqs (3.1), (3.5) and (3.6)=n1i=0(1)(n1)iX1Xn1|ˉξYi+1YnY1Yi1|D(ˉξ)n1i=0(1)(n1)iYi+1YnY1Yi1|ˉξX1Xn1|D(ˉξ)by Eq (3.7)=0.

    Hence, (D,ˉξ) defined as in (3.6) is indeed a Filippov connection.

    As a consequence of Theorem 3.3, a pair (D,ˉξ) subject to condition (3.7) produces a Filippov n-algebroid structure on A. Its n-bracket reads:

    [X1,,Xn](D,ˉξ)=X1Xn1|ˉξD(Xn)+(1)n1X2Xn|ˉξD(X1)+ +Xn1XnX1Xn3|ˉξD(Xn2)+(1)n1XnX1Xn2|ˉξD(Xn1).

    In this section, we unravel under certain conditions a relation between linear Nambu-Poisson structures and Filippov connections.

    Definition 4.1. [14,17] A Nambu-Poisson structure of order n on a smooth manifold P is an R-multilinear and skew-symmetric n-bracket on the smooth function space C(P):

    {,,}:C(P)××C(P)ncopiesC(P),

    satisfying the following two conditions:

    (1) The n-bracket is a derivation with respect to C(P)-multiplications:

    {f1,,fn1,g1g2}=g1{f1,,fn1,g2}+{f1,,fn1,g1}g2;

    (2) The (generalized) Jacobi identity (also known as the fundamental identity):

    {f1,,fn1,{g1,,gn}}=ni=1{g1,,{f1,,fn1,gi},,gn},

    holds for all fi and gjC(P).

    The pair (P,{,,}) is called a Nambu-Poisson manifold.

    Alternatively, one could express the said bracket via an n-vector field π on P such that

    {f1,,fn}=π(df1,,dfn),f1,,fnC(P). (4.1)

    Given a smooth vector bundle p:AM, the section space Γ(A) are identified as the space Clin(A) of fiberwise linear functions on A, the dual vector bundle of A; while elements in p(C(M)) are called basic functions on A. To fix the notations, for any section XΓ(A), let ϕXClin(A) be the corresponding linear function on A.

    Definition 4.2. [1] A Nambu-Poisson structure of order n on the vector bundle AM is said to be linear, if it satisfies the following three conditions:

    (1) The bracket of n linear functions is again a linear function;

    (2) The bracket of (n1) linear functions and a basic function is a basic function;

    (3) The bracket of n functions is zero if there are more than one basic functions among the arguments.

    In fact, the second and the third condition in the above definition can be derived from the first condition.

    Note that any Poisson manifold is a Nambu-Poisson manifold of order 2. A well-known fact is the following: A Lie algebroid A over M gives rise to a linear Poisson manifold A, and vice versa. It is pointed out in [1]*Theorem 4.4 that a linear Nambu-Poisson structure of order n on A corresponds to a Filippov n-algebroid structure on A (see also [2]). However, the reverse process is generally not valid for the cases of n3, mainly because the condition of a Nambu-Poisson structure is very strong (cf. [12]). Nevertheless, in this paper, we will require that A be a rank n vector bundle and establish the one-to-one correspondence between Filippov n-algebroid structures on A and linear Nambu-Poisson structures on A. In specific, under the said condition, our main theorem below serves as a complement to [1]*Theorem 4.4.

    Theorem 4.3. Let (A,ρ,[,,]) be a Filippov n-algebroid over a smooth manifold M, where AM is a vector bundle of rank n3. Then there exists a unique linear Nambu-Poisson structure on the dual bundle AM such that for all sections X1,,XnΓ(A),

    {ϕX1,,ϕXn}=ϕ[X1,,Xn]. (4.2)

    Note that, if Eq (4.2) holds, then it is easy to deduce that the linear Nambu-Poisson structure on A and the anchor map ρ are also related:

    {ϕX1,,ϕXn1,pf}=p(ρ(X1Xn1)(f)), (4.3)

    for all fC(M) and X1,,Xn1Γ(A).

    As a direct application of Theorems 3.3 and 4.3, one can construct linear Nambu-Poisson structures out of Filippov connections:

    Corollary 4.4. If is a Filippov connection on an n-anchored bundle (A,ρ), where AM is a vector bundle of rank n3, then the dual bundle A admits a unique linear Nambu-Poisson structure of order n defined by

    {ϕX1,,ϕXn}=ϕ[X1,,Xn],

    for all X1,,XnΓ(A).

    The proof is divided into three steps.

    Step 1. Since functions of type ϕX (for XΓ(A)) and pf (for fC(M)) generate C(A), there exists a unique R-multilinear n-bracket {,,} on C(A) satisfying Eqs (4.2) and (4.3). We wish to write the corresponding n-vector field π on A explicitly.

    To this end, we work locally and consider the trivialization A|UU×Rn over an open subset UM with coordinates x1,,xm; let {X1,,Xn} be a local basis of Γ(A|U). Then

    {y1=ϕX1,,yn=ϕXn,px1,,pxm}

    forms a chart on A|U. For convenience, pxi is denoted by xi.

    Suppose further that the Filippov algebroid A|U is described by the structure functions ck and f1ˆlnC(U) such that

    [X1,,Xn]=nk=1ckXk,
    and ρ(X1^XlXn)=f1ˆlnx1.

    Here we have utilized Proposition 2.5. Then one is able to find the expression of the n-vector field π on A|U:

    π=nk=1ckyky1yn+nl=1f1ˆlny1^ylynx1, (4.4)

    which corresponds to the n-bracket {} on A|U.

    Step 2. We need to set up a preparatory lemma.

    Lemma 4.5. There exists a local basis {X1,,Xn} of Γ(A|U) such that the corresponding structure functions ck and f1ˆln satisfy the following relations: for all ij (in {1,,n}),

    f1ˆinf1ˆjnx1=f1ˆjnf1ˆinx1; (4.5)
    (1)if1ˆincj=(1)jf1ˆjnci. (4.6)

    Proof. Consider the map ρ:n1A|UTU. By Proposition 2.5, for any point pU, we have rank(ρ(n1A)p)1, and thus dim(ker(ρp))(n1). Note that the subset VU where dim(ker(ρp)) is locally constant is open and dense. By a continuity argument if necessary, we may assume that dim(ker(ρp)) is locally constant on U. Thus, we are able to find a local basis {Z1,,Zn} of Γ(n1A|U) such that ρ(Z2), , ρ(Zn) are trivial.

    Take an arbitrary ΩΓ(nA|U) which is nowhere vanishing on U. Consider

    Ω:A|Un1A|U,    Ω(α):=iαΩ,

    which is an isomorphism of vector bundles. Then we obtain a basis {α1, , αn} of Γ(A|U) by setting αi:=(Ω)1(Zi). Let {X1,,Xn} be the dual basis of Γ(A|U) corresponding to {α1,,αn}. There exists a nowhere vanishing smooth function gC(U) such that Ω=gX1Xn, and hence Zi=iαiΩ=gX1^XiXn.

    Since we have ρ(Zi)=0 (i{2,,n}), we also have

    ρ(X1^XiXn)=0,i{2,,n}. (4.7)

    Using the axiom of a Filippov algebroid, we have a relation

    f1ˆinf1ˆjnx1x1f1ˆjnf1ˆinx1x1=[f1ˆinx1,f1ˆjnx1]TM=[ρ(X1^XiXn),ρ(X1^XjXn)]TMby Eq. (2.1)=nk=1,k<jρ(X1[X1,,^Xi,,Xn,Xk]^XjXn)+nk=1,k>jρ(X1^Xj[X1,,^Xi,,Xn,Xk]Xn)=(1)(ni)cif1ˆjnx1(1)(nj)cjf1ˆinx1.

    According to the previous fact (4.7), all the lines above must be trivial, and thus the desired two equalities (4.5) and (4.6) are proved.

    Step 3. We wish to show that the n-bracket {,,} given in Step 1, or the n-vector field π locally given in Eq (4.4), is a linear Nambu-Poisson structure on A.

    We need the following proposition due to Dufour and Zung [4].

    Proposition 4.6. [4] Let Ω be a volume form on an l-dimensional manifold P, and π an n-vector filed on P, where l>n3. Consider the (ln)-form ω:=ιπΩ on P. Then π defines a Nambu-Poisson structure (via Eq (4.1)) if and only if ω satisfies the following two conditions:

    (ιKω)ω=0, (4.8)
    (ιKω)dω=0, (4.9)

    for any (ln1)-vector field K on P.

    Consider the volume form Ω=dy1dyndx1dxm on A|U, where U, yi, and xj are as earlier, and we suppose that such a coordinate system stems from {X1,,Xn} fulfills Lemma 4.5. According to Proposition 4.6, we need to examine the m-form defined by:

    ω:=ιπΩ=nk=1ckykdx1dxm+nj=1(1)nj+1f1ˆjndyjdx2dxm.

    We can easily check that ω satisfies Eq (4.8) and hence it remains to check Eq (4.9). One first finds that

    dω=nk=1ykdckdx1dxm+nk=1ckdykdx1dxm +nj=1(1)nj+1df1ˆjndyjdx2dxm=nk=1ckdykdx1dxm+nj=1(1)nj+1f1ˆjnx1dx1dyjdx2dxm.

    Consider the following special type of (m1)-vector field on A|U: K=x2xm. Then one computes:

    (ιKω)dω=(1)m1(nk=1ckykdx1+nj=1(1)nj+1f1ˆjndyj)dω=nj=1(1)m+njf1ˆjndyjni=1cidyidx1dxm+nj=1(1)m+njf1ˆjndyjni=1(1)ni+1f1ˆinx1dx1dyidx2dxm=(1)mnj=1ni=1,ij((1)(ni)cif1ˆjn(1)(nj)cjf1ˆin)dyjdyi+nj=1ni=1,ij(1)m+i+j(f1ˆjnf1ˆinx1f1ˆinf1ˆjnx1)dyjdyiby Eqs. (4.5) and (4.6)=0.

    This justifies Eq (4.9) for this particular K. For other types of K, it is easy to verify Eq (4.9) as well. This completes the proof of Theorem 4.3.

    In order to understand the geometry of Filippov n-algebroids, we introduced a kind of multi-input connections on n-anchored vector bundles. Filippov n-brackets could be reconstructed from such connections. Moreover, all compatibility conditions in the definition of Filippov n-algebroids correspond to some natural conditions on connections. These connections are called Filippov connections. We also provided with concrete constructions on Filippov connections, which led to many examples of Filippov n-algebroids. As an application, we obtain a one-to-one correspondence between Filippov n-algebroid structures on a vector bundle A of rank n and linear Nambu-Poisson structures on its dual bundle A.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    We are grateful to the anonymous referees for constructive suggestions to improve the presentation of the manuscript. This work is supported by the National Natural Science Foundation of China (NSFC) grants 11961049, 11901221 and 12071241, and by the Key Project of Jiangxi Natural Science Foundation grant 20232ACB201004.

    The authors declare no conflicts of interest in this paper.



    [1] A. T. Azar, Q. Zhu, A. Khamis, D. Zhao, Control design approaches for parallel robot manipulators: a review, Int. J. Model. Identif. Control, 28 (2017), 199–211. https://doi.org/10.1504/IJMIC.2017.086563 doi: 10.1504/IJMIC.2017.086563
    [2] K. K. Ayten, M. H. Çiplak, A. Dumlu, Implementation a fractional-order adaptive model-based PID-type sliding mode speed control for wheeled mobile robot, Proceedings of the Institution of Mechanical Engineers, Part Ⅰ: Journal of Systems and Control Engineering, 233 (2019), 1067–1084. https://doi.org/10.1177/0959651819847395 doi: 10.1177/0959651819847395
    [3] M. S. Zanjani, S. Mobayen, Event-triggered global sliding mode controller design for anti-sway control of offshore container cranes, Ocean Eng., 268 (2023), 113472. https://doi.org/10.1016/j.oceaneng.2022.113472 doi: 10.1016/j.oceaneng.2022.113472
    [4] M. Bakouri, A. Alqarni, S. Alanazi, A. Alassaf, I. AlMohimeed, M. A. Aboamer, et al., Robust dynamic control algorithm for uncertain powered wheelchairs based on sliding neural network approach, AIMS Math., 8 (2023), 26821–26839. https://doi.org/10.3934/math.20231373 doi: 10.3934/math.20231373
    [5] A. Almasoud, Jamming-aware optimization for UAV trajectory design and internet of things devices clustering, Complex Intell. Syst., 9 (2023), 4571–4590. https://doi.org/10.1007/s40747-023-00970-3 doi: 10.1007/s40747-023-00970-3
    [6] S. Ahmed, A. T. Azar, Adaptive fractional tracking control of robotic manipulator using fixed-time method, Complex Intell. Syst., 9 (2023), 369–382. https://doi.org/10.1007/s40747-023-01164-7 doi: 10.1007/s40747-023-01164-7
    [7] S. Ahmed, A. T. Azar, M. Tounsi, I. K. Ibraheem, Adaptive control design for Euler-Lagrange systems using fixed-time fractional integral sliding mode scheme, Fractal Fract., 7 (2023), 712. https://doi.org/10.3390/fractalfract7100712 doi: 10.3390/fractalfract7100712
    [8] S. J. Gambhire, D. R. Kishore, P. S. Londhe, S. N. Pawar, Review of sliding mode based control techniques for control system applications, Int. J. Dyn. Control, 9 (2021), 363–378. https://doi.org/10.1007/s40435-020-00638-7 doi: 10.1007/s40435-020-00638-7
    [9] H. Yin, B. Meng, Z. Wang, Disturbance observer-based adaptive sliding mode synchronization control for uncertain chaotic systems, AIMS Math., 8 (2023), 23655–23673. https://doi.org/10.3934/math.20231203 doi: 10.3934/math.20231203
    [10] D. Zhao, S. Li, F. Gao, A new terminal sliding mode control for robotic manipulators, Int. J. Control, 82 (2009), 1804–1813. https://doi.org/10.1080/00207170902769928 doi: 10.1080/00207170902769928
    [11] Y. Feng, X. Yu, Z. Man, Non-singular terminal sliding mode control of rigid manipulators, Automatica, 38 (2002), 2159–2167. https://doi.org/10.1016/S0005-1098(02)00147-4 doi: 10.1016/S0005-1098(02)00147-4
    [12] L. Yang, J. Yang, Nonsingular fast terminal sliding-mode control for nonlinear dynamical systems, Int. J. Robust Nonlinear Control, 21 (2011), 1865–1879. https://doi.org/10.1002/rnc.1666 doi: 10.1002/rnc.1666
    [13] C. Ton, C. Petersen, Continuous fixed-time sliding mode control for spacecraft with flexible appendages, IFAC-PapersOnLine, 51 (2018), 1–5. https://doi.org/10.1016/j.ifacol.2018.07.079 doi: 10.1016/j.ifacol.2018.07.079
    [14] H. Khan, S. Ahmed, J. Alzabut, A. T. Azar, J. F. Gómez-Aguilar, Nonlinear variable order system of multi-point boundary conditions with adaptive finite-time fractional-order sliding mode control, Int. J. Dyn. Control, 2024, 1–17. https://doi.org/10.1007/s40435-023-01369-1
    [15] Y. Su, C. Zheng, P. Mercorelli, Robust approximate fixed-time tracking control for uncertain robot manipulators, Mech. Syst. Signal Pr., 135 (2020), 106379. https://doi.org/10.1016/j.ymssp.2019.106379 doi: 10.1016/j.ymssp.2019.106379
    [16] S. Ahmed, A. T. Azar, I. K. Ibraheem, Nonlinear system controlled using novel adaptive fixed-time SMC, AIMS Math., 9 (2024), 7895–7916. https://doi.org/10.3934/math.2024384 doi: 10.3934/math.2024384
    [17] Z. Zhu, Z. Duan, H. Qin, Y. Xue, Adaptive neural network fixed-time sliding mode control for trajectory tracking of underwater vehicle, Ocean Eng., 287 (2023), 115864. https://doi.org/10.1016/j.oceaneng.2023.115864 doi: 10.1016/j.oceaneng.2023.115864
    [18] Q. D. Nguyen, D. D. Vu, S. C. Huang, V. N. Giap, Fixed-time supper twisting disturbance observer and sliding mode control for a secure communication of fractional-order chaotic systems, J. Vib. Control, 2023. https://doi.org/10.1177/10775463231180947
    [19] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.
    [20] A. Ali, K. Shah, T. Abdeljawad, H. Khan, A. Khan, Study of fractional order pantograph type impulsive antiperiodic boundary value problem, Adv. Differ. Equ., 2020 (2020), 572. https://doi.org/10.1186/s13662-020-03032-x doi: 10.1186/s13662-020-03032-x
    [21] S. Ahmad, A. Ullah, Q. M. Al-Mdallal, H. Khan, K. Shah, A. Khan, Fractional order mathematical modeling of COVID-19 transmission, Chaos Soliton. Fract., 139 (2020), 110256. https://doi.org/10.1016/j.chaos.2020.110256 doi: 10.1016/j.chaos.2020.110256
    [22] K. Shah, Z. A. Khan, A. Ali, R. Amin, H. Khan, A. Khan, Haar wavelet collocation approach for the solution of fractional order COVID-19 model using Caputo derivative, Alexandria Eng. J., 59 (2020), 3221–3231. https://doi.org/10.1016/j.aej.2020.08.028 doi: 10.1016/j.aej.2020.08.028
    [23] A. I. Ahmed, M. S. Al-Sharif, M. S. Salim, T. A. Al-Ahmary, Numerical solution of fractional variational and optimal control problems via fractional-order Chelyshkov functions, AIMS Math., 7 (2022), 17418–17443. https://doi.org/10.3934/math.2022960 doi: 10.3934/math.2022960
    [24] R. Ayad, W. Nouibat, M. Zareb, Y. B. Sebanne, Full control of quadrotor aerial robot using fractional-order FOPID, Iran. J. Sci. Technol. Trans. Electr. Eng., 43 (2019), 349–360. https://doi.org/10.1007/s40998-018-0155-4 doi: 10.1007/s40998-018-0155-4
    [25] H. Khan, S. Ahmed, J. Alzabut, A. T. Azar, A generalized coupled system of fractional differential equations with application to finite time sliding mode control for Leukemia therapy, Chaos Soliton. Fract., 174 (2023), 113901. https://doi.org/10.1016/j.chaos.2023.113901 doi: 10.1016/j.chaos.2023.113901
    [26] T. T. Nguyen, Fractional-order sliding mode controller for the two-link robot arm, Int. J. Electr. Comput. Eng., 10 (2020), 5579–5585. https://doi.org/10.11591/ijece.v10i6.pp5579-5585 doi: 10.11591/ijece.v10i6.pp5579-5585
    [27] S. Huang, L. Xiong, J. Wang, P. Li, Z. Wang, M. Ma, Fixed-time fractional-order sliding mode controller for multimachine power systems, IEEE Trans. Power Syst., 36 (2020), 2866–2876. https://doi.org/10.1109/TPWRS.2020.3043891 doi: 10.1109/TPWRS.2020.3043891
    [28] B. D. H. Phuc, V. D. Phung, S. S. You, T. D. Do, Fractional-order sliding mode control synthesis of supercavitating underwater vehicles, J. Vib. Control, 26 (2020), 1909–1919. https://doi.org/10.1177/1077546320908412 doi: 10.1177/1077546320908412
    [29] X. Zhang, F. Wu, M. Liu, X. Chen, Fractional-order robust fixed-time sliding mode control for deployment of tethered satellite, Acta Astronaut., 209 (2023), 172–178. https://doi.org/10.1016/j.actaastro.2023.04.041 doi: 10.1016/j.actaastro.2023.04.041
    [30] T. C. Lin, T. Y. Lee, V. E. Balas, Adaptive fuzzy sliding mode control for synchronization of uncertain fractional order chaotic systems, Chaos Soliton. Fract., 44 (2011), 791–801. https://doi.org/10.1016/j.chaos.2011.04.005 doi: 10.1016/j.chaos.2011.04.005
    [31] S. Huang, J. Wang, C. Huang, L. Zhou, L. Xiong, J. Liu, et al., A fixed-time fractional-order sliding mode control strategy for power quality enhancement of PMSG wind turbine, Int. J. Electr. Power Energy Syst., 134 (2022), 107354. https://doi.org/10.1016/j.ijepes.2021.107354 doi: 10.1016/j.ijepes.2021.107354
    [32] S. Han, H. Wang, Y. Tian, N. Christov, Time-delay estimation based computed torque control with robust adaptive RBF neural network compensator for a rehabilitation exoskeleton, ISA Trans., 97 (2020), 171–181. https://doi.org/10.1016/j.isatra.2019.07.030 doi: 10.1016/j.isatra.2019.07.030
    [33] A. T. Azar, H. H. Ammar, M. Y. Beb, S. R. Garces, A. Boubakari, Optimal design of PID controller for 2-DOF drawing robot using bat-inspired algorithm, In: A. Hassanien, K. Shaalan, M. Tolba, Proceedings of the International Conference on Advanced Intelligent Systems and Informatics 2019, Springer, 1058 (2019), 175–186. https://doi.org/10.1007/978-3-030-31129-2_17
    [34] M. Van, S. S. Ge, H. Ren, Finite time fault tolerant control for robot manipulators using time delay estimation and continuous nonsingular fast terminal sliding mode control, IEEE Trans. Cybernetics, 47 (2016), 1681–1693. https://doi.org/10.1109/TCYB.2016.2555307 doi: 10.1109/TCYB.2016.2555307
    [35] K. Y. Toumi, O. Ito, A time delay controller for systems with unknown dynamics, J. Dyn. Sys., Meas., Control, 112 (1990), 133–142. https://doi.org/10.1115/1.2894130 doi: 10.1115/1.2894130
    [36] T. C. Hsia, L. S. Gao, Robot manipulator control using decentralized linear time-invariant time-delayed joint controllers, Proceedings., IEEE International Conference on Robotics and Automation, IEEE, 1990, 2070–2075. https://doi.org/10.1109/ROBOT.1990.126310
    [37] S. Ahmed, I. Ghous, F. Mumtaz, TDE based model-free control for rigid robotic manipulators under nonlinear friction, Sci. Iran., 2022. https://doi.org/10.24200/sci.2022.57252.5141
    [38] Y. Wu, H. Fang, T. Xu, F. Wan, Adaptive neural fixed-time sliding mode control of uncertain robotic manipulators with input saturation and prescribed constraints, Neural Process. Lett., 54 (2022), 3829–3849. https://doi.org/10.1007/s11063-022-10788-8 doi: 10.1007/s11063-022-10788-8
    [39] A. Polyakov, Fixed-time stabilization via second order sliding mode control, IFAC Proc. Vol., 45 (2012), 254–258. https://doi.org/10.3182/20120606-3-NL-3011.00109 doi: 10.3182/20120606-3-NL-3011.00109
    [40] J. Zhai, Z. Li, Fast-exponential sliding mode control of robotic manipulator with super-twisting method, IEEE Trans. Circuits Syst. II, 69 (2021), 489–493. https://doi.org/10.1109/TCSII.2021.3081147 doi: 10.1109/TCSII.2021.3081147
    [41] S. Ahmed, A. T. Azar, M. Tounsi, Adaptive fault tolerant non-singular sliding mode control for robotic manipulators based on fixed-time control law, Actuators, 11 (2022), 353. https://doi.org/10.3390/act11120353 doi: 10.3390/act11120353
    [42] S. Ahmed, H. Wang, Y. Tian, Fault tolerant control using fractional-order terminal sliding mode control for robotic manipulators, Stud. Inform. Control, 27 (2018), 55–64. https://doi.org/10.24846/v27i1y201806 doi: 10.24846/v27i1y201806
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1227) PDF downloads(69) Cited by(11)

Figures and Tables

Figures(16)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog