Processing math: 100%
Review

Immersive virtual reality application for intelligent manufacturing: Applications and art design

  • Intelligent manufacturing (IM), sometimes referred to as smart manufacturing (SM), is the use of real-time data analysis, machine learning, and artificial intelligence (AI) in the production process to achieve the aforementioned efficiencies. Human-machine interaction technology has recently been a hot issue in smart manufacturing. The unique interactivity of virtual reality (VR) innovations makes it possible to create a virtual world and allow users to communicate with that environment, providing users with an interface to be immersed in the digital world of the smart factory. And virtual reality technology aims to stimulate the imagination and creativity of creators to the maximum extent possible for reconstructing the natural world in a virtual environment, generating new emotions, and transcending time and space in the familiar and unfamiliar virtual world. Recent years have seen a great leap in the development of intelligent manufacturing and virtual reality technologies, yet little research has been done to combine the two popular trends. To fill this gap, this paper specifically employs Preferred Reporting Items for Systematic Reviews and Meta-analysis (PRISMA) guidelines to conduct a systematic review of the applications of virtual reality in smart manufacturing. Moreover, the practical challenges and the possible future direction will also be covered.

    Citation: Yu Lei, Zhi Su, Xiaotong He, Chao Cheng. Immersive virtual reality application for intelligent manufacturing: Applications and art design[J]. Mathematical Biosciences and Engineering, 2023, 20(3): 4353-4387. doi: 10.3934/mbe.2023202

    Related Papers:

    [1] Ting Liu, Guo-Bao Zhang . Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, 2021, 29(4): 2599-2618. doi: 10.3934/era.2021003
    [2] Cui-Ping Cheng, Ruo-Fan An . Global stability of traveling wave fronts in a two-dimensional lattice dynamical system with global interaction. Electronic Research Archive, 2021, 29(5): 3535-3550. doi: 10.3934/era.2021051
    [3] Bin Wang . Random periodic sequence of globally mean-square exponentially stable discrete-time stochastic genetic regulatory networks with discrete spatial diffusions. Electronic Research Archive, 2023, 31(6): 3097-3122. doi: 10.3934/era.2023157
    [4] Yijun Chen, Yaning Xie . A kernel-free boundary integral method for reaction-diffusion equations. Electronic Research Archive, 2025, 33(2): 556-581. doi: 10.3934/era.2025026
    [5] Yongwei Yang, Yang Yu, Chunyun Xu, Chengye Zou . Passivity analysis of discrete-time genetic regulatory networks with reaction-diffusion coupling and delay-dependent stability criteria. Electronic Research Archive, 2025, 33(5): 3111-3134. doi: 10.3934/era.2025136
    [6] Meng Wang, Naiwei Liu . Qualitative analysis and traveling wave solutions of a predator-prey model with time delay and stage structure. Electronic Research Archive, 2024, 32(4): 2665-2698. doi: 10.3934/era.2024121
    [7] Minzhi Wei . Existence of traveling waves in a delayed convecting shallow water fluid model. Electronic Research Archive, 2023, 31(11): 6803-6819. doi: 10.3934/era.2023343
    [8] Hongquan Wang, Yancai Liu, Xiujun Cheng . An energy-preserving exponential scheme with scalar auxiliary variable approach for the nonlinear Dirac equations. Electronic Research Archive, 2025, 33(1): 263-276. doi: 10.3934/era.2025014
    [9] Cheng Wang . Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations. Electronic Research Archive, 2021, 29(5): 2915-2944. doi: 10.3934/era.2021019
    [10] Shao-Xia Qiao, Li-Jun Du . Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, 2021, 29(3): 2269-2291. doi: 10.3934/era.2020116
  • Intelligent manufacturing (IM), sometimes referred to as smart manufacturing (SM), is the use of real-time data analysis, machine learning, and artificial intelligence (AI) in the production process to achieve the aforementioned efficiencies. Human-machine interaction technology has recently been a hot issue in smart manufacturing. The unique interactivity of virtual reality (VR) innovations makes it possible to create a virtual world and allow users to communicate with that environment, providing users with an interface to be immersed in the digital world of the smart factory. And virtual reality technology aims to stimulate the imagination and creativity of creators to the maximum extent possible for reconstructing the natural world in a virtual environment, generating new emotions, and transcending time and space in the familiar and unfamiliar virtual world. Recent years have seen a great leap in the development of intelligent manufacturing and virtual reality technologies, yet little research has been done to combine the two popular trends. To fill this gap, this paper specifically employs Preferred Reporting Items for Systematic Reviews and Meta-analysis (PRISMA) guidelines to conduct a systematic review of the applications of virtual reality in smart manufacturing. Moreover, the practical challenges and the possible future direction will also be covered.



    In this article, we consider the following spatially discrete diffusion system with time delay

    {tv1(x,t)=d1D[v1](x,t)αv1(x,t)+h(v2(x,tτ1)),tv2(x,t)=d2D[v2](x,t)βv2(x,t)+g(v1(x,tτ2)) (1)

    with the initial data

    vi(x,s)=vi0(x,s), xR, s[τi,0], i=1,2, (2)

    where t>0, xR, di0 and

    D[vi](x,t)=vi(x+1,t)2vi(x,t)+vi(x1,t), i=1,2.

    Here v1(x,t) and v2(x,t) biologically stand for the spatial density of the bacterial population and the infective human population at point xR and time t0, respectively. Both bacteria and humans are assumed to diffuse, d1 and d2 are diffusion coefficients; the term αv1 is the natural death rate of the bacterial population and the nonlinearity h(v2) is the contribution of the infective humans to the growth rate of the bacterial; βv2 is the natural diminishing rate of the infective population due to the finite mean duration of the infectious population and the nonlinearity g(v1) is the infection rate of the human population under the assumption that the total susceptible human population is constant during the evolution of the epidemic, and τ1, τ2 are time delays. The nonlinearities g and h satisfy the following hypothesis:

    (H1) gC2([0,K1],R), g(0)=h(0)=0, K2=g(K1)/β>0, hC2([0,K2],R), h(g(K1)/β)=αK1, h(g(v)/β)>αv for v(0,K1), where K1 is a positive constant.

    According to (H1), the spatially homogeneous system of (1) admits two constant equilibria

    (v1,v2)=0:=(0,0)and(v1+,v2+)=K:=(K1,K2).

    It is clear that (H1) is a basic assumption to ensure that system (1) is monostable on [0,K]. When g(u)0 for u[0,K1] and h(v)0 for v[0,K2], system (1) is a quasi-monotone system. Otherwise, if g(u)0 for u[0,K1] or h(v)0 for v[0,K2] does not hold, system (1) is a non-quasi-monotone system. In this article, we are interested in the existence and stability of traveling wave solutions of (1) connecting two constant equilibria (0,0) and (K1,K2). A traveling wave solution (in short, traveling wave) of (1) is a special translation invariant solution of the form (v1(x,t),v2(x,t))=(ϕ1(x+ct),ϕ2(x+ct)), where c>0 is the wave speed. If ϕ1 and ϕ2 are monotone, then (ϕ1,ϕ2) is called a traveling wavefront. Substituting (ϕ1(x+ct),ϕ2(x+ct)) into (1), we obtain the following wave profile system with the boundary conditions

    {cϕ1(ξ)=d1D[ϕ1](ξ)αϕ1(ξ)+h(ϕ2(ξcτ1)),cϕ2(ξ)=d2D[ϕ2](ξ)βϕ2(ξ)+g(ϕ1(ξcτ2)),(ϕ1,ϕ2)()=(v1,v2),(ϕ1,ϕ2)(+)=(v1+,v2+), (3)

    where ξ=x+ct, =ddξ, D[ϕi](ξ)=ϕi(ξ+1)2ϕi(ξ)+ϕi(ξ1), i=1,2.

    System (1) is a discrete version of classical epidemic model

    {tv1(x,t)=d1xxv1(x,t)a1v1(x,t)+h(v2(x,tτ1)),tv2(x,t)=d2xxv2(x,t)a2v2(x,t)+g(v1(x,tτ2)). (4)

    The existence and stability of traveling waves of (4) have been extensively studied, see [7,19,21,24] and references therein. Note that system (1) is also a delay version of the following system

    {tv1(x,t)=d1D[v1](x,t)a1v1(x,t)+h(v2(x,t)),tv2(x,t)=d2D[v2](x,t)a2v2(x,t)+g(v1(x,t)). (5)

    When system (5) is a quasi-monotone system, Yu, Wan and Hsu [27] established the existence and stability of traveling waves of (5). To the best of our knowledge, when systems (1) and (5) are non-quasi-monotone systems, no result on the existence and stability of traveling waves has been reported. We should point out that the existence of traveling waves of (1) can be easily obtained. Hence, the main purpose of the current paper is to establish the stability of traveling waves of (1).

    The stability of traveling waves for the classical reaction-diffusion equations with and without time delay has been extensively investigated, see e.g., [4,9,10,12,13,14,16,22,24]. Compared to the rich results for the classical reaction-diffusion equations, limited results exist for the spatial discrete diffusion equations. Chen and Guo [1] took the squeezing technique to prove the asymptotic stability of traveling waves for discrete quasilinear monostable equations without time delay. Guo and Zimmer [5] proved the global stability of traveling wavefronts for spatially discrete equations with nonlocal delay effects by using a combination of the weighted energy method and the Green function technique. Tian and Zhang [19] investigated the global stability of traveling wavefronts for a discrete diffusive Lotka-Volterra competition system with two species by the weighted energy method together with the comparison principle. Later on, Chen, Wu and Hsu [2] employed the similar method to show the global stability of traveling wavefronts for a discrete diffusive Lotka-Volterra competition system with three species. We should point out that the methods for the above stability results heavily depend on the monotonicity of equations and the comparison principle. However, the most interesting cases are the equations without monotonicity. It is known that when the evolution equations are non-monotone, the comparison principle is not applicable. Thus, the methods, such as the squeezing technique, the weighted energy method combining with the comparison principle are not valid for the stability of traveling waves of the spatial discrete diffusion equations without monotonicity.

    Recently, the technical weighted energy method without the comparison principle has been used to prove the stability of traveling waves of nonmonotone equations, see Chern et al. [3], Lin et al. [10], Wu et al. [22], Yang et al. [24]. In particular, Tian et al. [20] and Yang et al. [26], respectively, applied this method to prove the local stability of traveling waves for nonmonotone traveling waves for spatially discrete reaction-diffusion equations with time delay. Later, Yang and Zhang [25] established the stability of non-monotone traveling waves for a discrete diffusion equation with monostable convolution type nonlinearity. Unfortunately, the local stability (the initial perturbation around the traveling wave is properly small in a weighted norm) of traveling waves can only be obtained. Very recently, Mei et al. [15] developed a new method to prove the global stability of the oscillatory traveling waves of local Nicholson's blowflies equations. This method is based on some key observations for the structure of the govern equations and the anti-weighted energy method together with the Fourier transform. Later on, Zhang [28] and Xu et al. [23], respectively, applied this method successfully to a nonlocal dispersal equation with time delay and obtained the global stability of traveling waves. More recently, Su and Zhang [17] further studied a discrete diffusion equation with a monostable convolution type nonlinearity and established the global stability of traveling waves with large speed. Motivated by the works [15,28,23,17,18], in this paper, we shall extend this method to study the global stability of traveling waves of spatial discrete diffusion system (1) without quasi-monotonicity.

    The rest of this paper is organized as follows. In Section 2, we present some preliminaries and summarize our main results. Section 3 is dedicated to the global stability of traveling waves of (1) by the Fourier transform and the weighted energy method, when h(u) and g(u) are not monotone.

    In this section, we first give the equivalent integral form of the initial value problem of (1) with (2), then recall the existence of traveling waves of (1), and finally state the main result on the global stability of traveling waves of (1). Throughout this paper, we assume τ1=τ2=τ.

    First of all, we consider the initial value problem (1) with (2), i.e.,

    {tv1(x,t)=d1D[v1](x,t)αv1(x,t)+h(v2(x,tτ)),tv2(x,t)=d2D[v2](x,t)βv2(x,t)+g(v1(x,tτ)),vi(x,s)=vi0(x,s), xR, s[τ,0], i=1,2. (6)

    According to [8], with aid of modified Bessel functions, the solution to the initial value problem

    {tu(x,t)=d[u(x+1,t)2u(x,t)+u(x1,t)], xR, t>0,u(x,0)=u0(x), xR,

    can be expressed by

    u(x,t)=(S(t)u0)(x)=e2dtm=Im(2dt)u0(xm),

    where u0()L(R), Im(), m0 are defined as

    Im(t)=k=0(t/2)m+2kk!(m+k)!,

    and Im(t)=Im(t) for m<0. Moreover,

    Im(t)=12[Im+1(t)+Im1(t)], t>0,mZ, (7)

    and Im(0)=0 for m0 while I0=1, and Im(t)0 for any t0. In addition, one has

    etm=Im(t)=et[I0(t)+2I1(t)+2I2(t)+I3(t)+]=1. (8)

    Thus, the solution (v1(x,t),v2(x,t)) of (6) can be expressed as

    {v1(x,t)=e(2d1+α)tm=Im(2d1t)v10(xm,0)+m=t0e(2d1+α)(ts)Im(2d1(ts))(h(v2(xm,sτ)))ds,v2(x,t)=e(2d2+β)tm=Im(2d2t)v20(xm,0)+m=t0e(2d2+β)(ts)Im(2d2(ts))(g(v1(xm,sτ)))ds. (9)

    In fact, by [8,Lemma 2.1], we can differentiate the series on t variable in (9). Using the recurrence relation (7), we obtain

    tv1(x,t)=(2d1+α)e(2d1+α)tm=Im(2d1t)v10(xm,0)   +e(2d1+α)tm=2d1Im(2d1t)v10(xm,0)   +m=Im(0)(h(v2(xm,tτ)))   (2d1+α)m=t0e(2d1+α)(ts)Im(2d1(ts))(h(v2(xm,sτ)))ds   +m=t0e(2d1+α)(ts)2d1Im(2d1(ts))(h(v2(xm,sτ)))ds=d1[v1(x+1,t)2v1(x,t)+v1(x1,t)]αv1(x,t)+h(v2(x,tτ))

    and

    tv2(x,t)=(2d2+β)e(2d2+β)tm=Im(2d2t)v20(xm,0)   +e(2d2+β)tm=2d2Im(2d2t)v20(xm,0)   +m=Im(0)(g(v1(xm,tτ)))   (2d2+β)m=t0e(2d2+β)(ts)Im(2d2(ts))(g(v1(xm,sτ)))ds   +m=t0e(2d2+β)(ts)2d2Im(2d2(ts))(g(v1(xm,sτ)))ds=d2[v2(x+1,t)2v2(x,t)+v2(x1,t)]βv2(x,t)+g(v1(x,tτ)).

    Next we investigate the characteristic roots of the linearized system for the wave profile system (3) at the trivial equilibrium 0. Clearly, the characteristic function of (3) at 0 is

    P1(c,λ):=f1(c,λ)f2(c,λ)

    for c0 and λC, where

    f1(c,λ):=Δ1(c,λ)Δ2(c,λ),f2(c,λ):=h(0)g(0)e2cλτ,

    with

    Δ1(c,λ)=d1(eλ+eλ2)cλα,Δ2(c,λ)=d2(eλ+eλ2)cλβ.

    It is easy to see that Δ1(c,λ)=0 admits two roots λ1<0<λ+1, and Δ2(c,λ)=0 has two roots λ2<0<λ+2. We denote λ+m=min{λ+1,λ+2}.

    Similar to [27,Lemma 3.1], we can obtain the following result.

    Lemma 2.1. There exists a positive constant c such that if c>c, then P1(c,λ)=0 has two distinct positive real roots λ1:=λ1(c) and λ2:=λ2(c) with λ1(c)<λ2(c)<λ+m, i.e. P1(c,λ1)=P1(c,λ2)=0, and P(c,λ)>0 for λ(λ1(c),λ2(c)). In addition, limccλ1(c)=limccλ2(c)=λ>0, i.e., P1(c,λ)=0.

    Furthermore, we show the existence of traveling wave of (1). When system (1) is a quasi-monotone system, the existence of traveling wavefronts follows from [6,Theorem 1.1]. When system (1) is a non-quasi-monotone system, the existence of traveling waves can also be obtained by using auxiliary equations and Schauder's fixed point theorem [21,24], if we assume the following assumptions:

    (H2) There exist K±=(K±1,K±2)0 with K<K<K+ and four continuous and twice piecewise continuous differentiable functions g±:[0,K+1]R and h±:[0,K+2]R such that

    (i) K±2=g±(K±1)/β, h±(1βg±(K±1))=αK±1, and h±(1βg±(v))>αv for v(0,K±1);

    (ii) g±(u) and h±(v) are non-decreasing on [0,K+1] and [0,K+2], respectively;

    (iii) (g±)(0)=g(0), (h±)(0)=h(0) and

    0<g(u)g(u)g+(u)g(0)u for u[0,K+1],0<h(v)h(v)h+(v)h(0)v for v[0,K+2].

    Proposition 1. Assume that (H1) and (H2) hold, τ0, and let c be defined as in Lemma 2.1. Then for every c>c, system (1) has a traveling wave (ϕ1(ξ),ϕ2(ξ)) satisfying (ϕ1(),ϕ2())=(0,0) and

    K1lim infξ+ϕ1(ξ)lim supξ+ϕ1(ξ)K+1,0lim infξ+ϕ2(ξ)lim supξ+ϕ2(ξ)K+2.

    Finally, we shall state the stability result of traveling waves derived in Proposition 1. Before that, let us introduce the following notations.

    Notations. C>0 denotes a generic constant, while Ci(i=1,2,) represents a specific constant. Let and denote 1-norm and -norm of the matrix (or vector), respectively. Let I be an interval, typically I=R. Denote by L1(I) the space of integrable functions defined on I, and Wk,1(I)(k0) the Sobolev space of the L1-functions f(x) defined on the interval I whose derivatives dndxnf(n=1,,k) also belong to L1(I). Let L1w(I) be the weighted L1-space with a weight function w(x)>0 and its norm is defined by

    ||f||L1w(I)=Iw(x)|f(x)|dx,

    Wk,1w(I) be the weighted Sobolev space with the norm given by

    ||f||Wk,1w(I)=ki=0Iw(x)|dif(x)dxi|dx.

    Let T>0 be a number and B be a Banach space. We denote by C([0,T];B) the space of the B-valued continuous functions on [0,T], and by L1([0,T];B) the space of the B-valued L1-functions on [0,T]. The corresponding spaces of the B-valued functions on [0,) are defined similarly. For any function f(x), its Fourier transform is defined by

    F[f](η)=ˆf(η)=Reixηf(x)dx

    and the inverse Fourier transform is given by

    F1[ˆf](x)=12πReixηˆf(η)dη,

    where i is the imaginary unit, i2=1.

    To guarantee the global stability of traveling waves of (1), we need the following additional assumptions.

    (H3) |g(u)|g(0) and |h(v)|h(0) for u,v[0,+).

    (H4) d2>d1, α>β, d2d1<αβ2 and max{h(0),g(0)}>β.

    (H5) The initial data (v10(x,s),v20(x,s))(0,0) satisfies

    limx±(v10(x,s),v20(x,s))=(v1±,v2±) uniformly in  s[τ,0].

    Consider the following function

    P2(λ,c)=d2(eλ+eλ2)cλβ+max{h(0),g(0)}eλcτ.

    Since max{h(0),g(0)}>β, it then follows from [20,Lemma 2.1] that there exists λ>0 and c>0, such that P2(λ,c)=0 and P2(λ,c)λ|(λ,c)=0. When c>c, the equation P2(λ,c)=0 has two positive real roots λ1(c) and λ2(c) with 0<λ1(c)<λ<λ2(c). When λ(λ1(c),λ2(c)), P2(λ,c)<0. Moreover, (λ1)(c)<0 and (λ2)(c)>0.

    We select the weight function w(ξ)>0 as the form

    w(ξ)=e2λξ,

    where λ>0 satisfies λ1(c)<λ<λ2(c). Now we are ready to present the main result of this paper.

    Theorem 2.2. (Global stability of traveling waves). Assume that (H1), (H3)-(H5) hold. For any given traveling wave (ϕ1(x+ct),ϕ2(x+ct)) of (1) with speed c>max{c,c} connecting (0,0) and (K1,K2), whether it is monotone or non-monotone, if the initial data satisfy

    vi0(x,s)ϕi(x+cs)Cunif[τ,0]C([τ,0];W1,1w(R)), i=1,2,s(vi0ϕi)L1([τ,0];L1w(R)), i=1,2,

    then there exists τ0>0 such that for any ττ0, the solution (v1(x,t),v2(x,t)) of (1)-(2) converges to the traveling wave (ϕ1(x+ct),ϕ2(x+ct)) as follows:

    supxR|vi(x,t)ϕi(x+ct)|Ceμt,t>0,

    where C and μ are two positive constants, and Cunif[r,T] is the uniformly continuous space, for 0<T, defined by

    Cunif[r,T]={uC([r,T]×R)such thatlimx+v(x,t)exists uniformly int[r,T]}.

    This section is devoted to proving the stability theorem, i.e., Theorem 2.2. Let (ϕ1(x+ct),ϕ2(x+ct))=(ϕ1(ξ),ϕ2(ξ)) be a given traveling wave solution with speed cc and define

    {Vi(ξ,t):=vi(x,t)ϕi(x+ct)=vi(ξct,t)ϕi(ξ), i=1,2,Vi0(ξ,s):=vi0(x,s)ϕi(x+cs)=vi0(ξcs,s)ϕ(ξ), i=1,2.

    Then it follows from (1) and (3) that Vi(ξ,t) satisfies

    {V1t+cV1ξd1D[V1]+αV1=Q1(V2(ξcτ,tτ)),V2t+cV2ξd2D[V2]+βV2=Q2(V1(ξcτ,tτ)),Vi(ξ,s)=Vi0(ξ,s), (ξ,s)R×[τ,0], i=1,2. (10)

    The nonlinear terms Q1 and Q2 are given by

    {Q1(V2):=h(ϕ2+V2)h(ϕ2)=h(˜ϕ2)V2,Q2(V1):=g(ϕ1+V1)g(ϕ1)=g(˜ϕ1)V1, (11)

    for some ˜ϕi between ϕi and ϕi+Vi, with ϕi=ϕi(ξcτi) and Vi=Vi(ξcτi,tτi).

    We first prove the existence and uniqueness of solution (V1(ξ,t),V2(ξ,t)) to the initial value problem (10) in the uniformly continuous space Cunif[τ,+)×Cunif[τ,+).

    Lemma 3.1. Assume that (H1)and(H3) hold. If the initial perturbation (V10,V20)Cunif[τ,0]×Cunif[τ,0] for cc, then the solution (V1,V2) of the perturbed equation (10) is unique and time-globally exists in Cunif[τ,+)×Cunif[τ,+).

    Proof. Let Ui(x,t)=vi(x,t)ϕi(x+ct), i=1,2. It is clear that Ui(x,t)=Vi(ξ,t), i=1,2, and satisfies

    {U1td1D[U1]+αU1=Q1(U2(x,tτ)),U2td2D[U2]+βU2=Q2(U1(x,tτ)),Ui(x,s)=vi0(x,s)ϕi(x+cs):=Ui0(x,s), (x,s)R×[τ,0], i=1,2. (12)

    Thus, the global existence and uniqueness of solution of (10) are transformed into that of (12).

    When t[0,τ], we have tτ[τ,0] and Ui(x,tτ)=Ui0(x,tτ), i=1,2, which imply that (12) is linear. Thus, the solution of (12) can be explicitly and uniquely solved by

    {U1(x,t)=e(2d1+α)tm=Im(2d1t)U10(xm,0)             +m=t0e(2d1+α)(ts)Im(2d1(ts))Q1(U20(xm,sτ))ds,U2(x,t)=e(2d2+β)tm=Im(2d2t)U20(xm,0)             +m=t0e(2d2+β)(ts)Im(2d2(ts))Q2(U10(xm,sτ))ds (13)

    for t[0,τ].

    Since Vi0(ξ,t)Cunif[τ,0], i=1,2, namely, limξ+Vi0(ξ,t) exist uniformly in t[τ,0], which implies limx+Ui0(x,t) exist uniformly in t[τ,0]. Denote Ui0(,t)=limx+Ui0(x,t), i=1,2. Taking the limit x+ to (13) yields

    limx+U1(x,t)=e(2d1+α)tm=Im(2d1t)limx+U10(xm,0)+m=t0e(2d1+α)(ts)Im(2d1(ts))limx+Q1(U20(xm,sτ))ds=eαtU10(,0)+t0eα(ts)Q1(U20(,sτ))m=e2d1(ts)Im(2d1(ts))ds=:U1(t)  uniformly in t[0,τ] (14)

    and

    limx+U2(x,t)=e(2d2+β)tm=Im(2d2t)limx+U20(xm,0)
    +m=t0e(2d2+β)(ts)Im(2d2(ts))limx+Q2(U10(xm,sτ))ds=eβtU20(,0)+t0eβ(ts)Q2(U10(,sτ))m=e2d2(ts)Im(2d2(ts))ds=:U2(t)  uniformly in t[0,τ], (15)

    where we have used (8). Thus, we obtain that (U1,U2)Cunif[τ,τ)×Cunif[τ,τ).

    When t[τ,2τ], system (12) with the initial data Ui(x,s) for s[0,τ] is still linear, because the source term Q1(U2(x,tτ)) and Q2(U1(x,tτ)) is known due to tτ[0,τ] and Ui(s,tτ) is solved in (13). Hence, the solution Ui(x,t) for t[τ,2τ] is uniquely and explicitly given by

    U1(x,t)=e(2d1+α)(tτ)m=Im(2d1(tτ))U1(xm,τ)+m=tτe(2d1+α)(ts)Im(2d1(ts))Q1(U2(xm,sτ))ds,U2(x,t)=e(2d2+β)(tτ)m=Im(2d2(tτ))U2(xm,τ)+m=tτe(2d2+β)(ts)Im(2d2(ts))Q2(U1(xm,sτ))ds.

    Similarly, by (14) and (15), we have

    limx+U1(x,t)=e(2d1+α)(tτ)m=Im(2d1(tτ))limx+U1(xm,τ)+m=tτe(2d1+α)(ts)Im(2d1(ts))limx+Q1(U2(xm,sτ))ds=eα(tτ)U1(τ)+tτeα(ts)Q1(U1(sτ))m=e2d1(ts)Im(2d1(ts))ds=:ˉU1(t)  uniformly in t[τ,2τ],

    and

    limx+U2(x,t)=e(2d2+β)(tτ)m=Im(2d2(tτ))limx+U2(xm,τ)+m=tτe(2d2+β)(ts)Im(2d2(ts))limx+Q2(U1(xm,sτ))ds=eβ(tτ)U2(τ)+tτeβ(ts)Q2(U2(sτ))m=e2d2(ts)Im(2d2(ts))ds
    =:ˉU2(t)  uniformly in t[τ,2τ].

    By repeating this procedure for t[nτ,(n+1)τ] with nZ+, we prove that there exists a unique solution (V1,V2)Cunif[τ,(n+1)τ]×Cunif[τ,(n+1)τ] for (10), and step by step, we finally prove the uniqueness and time-global existence of the solution (V1,V2)Cunif[τ,)×Cunif[τ,) for (10). The proof is complete.

    Now we state the stability result for the perturbed system (10), which automatically implies Theorem 2.2.

    Proposition 2. Assume that (H1), (H3)-(H5) hold. If

    Vi0Cunif[τ,0]C([τ,0];W1,1w(R)), i=1,2,

    and

    sVi0L1([τ,0];L1w(R)), i=1,2,

    then there exists τ0>0 such that for any ττ0, when c>max{c,c}, it holds

    supξR|Vi(ξ,t)|Ceμt,t>0, i=1,2, (16)

    for some μ>0 and C>0.

    In order to prove Proposition 2, we first investigate the decay estimate of Vi(ξ,t) at ξ=+, i=1,2.

    Lemma 3.2. Assume that Vi0Cunif[τ,0], i=1,2. Then, there exist τ0>0 and a large number x01 such that when ττ0, the solution Vi(ξ,t) of (10) satisfies

    supξ[x0,+)|Vi(ξ,t)|Ceμ1t, t>0, i=1,2,

    for some μ1>0 and C>0.

    Proof. Denote

    z+i(t):=Vi(,t), z+i0(s):=Vi0(,s), s[τ,0], i=1,2.

    Since Vi0Cunif[τ,0], i=1,2, by Lemma 3.1, we have ViCunif[τ,+), which implies

    limξ+Vi(ξ,t)=z+i(t)

    exists uniformly for t[τ,+). Taking the limit ξ+ to (10), we obtain

    {dz+1dt+αz+1h(v2+)z+2(tτ)=P1(z+2(tτ)),dz+2dt+βz+2g(v1+)z+1(tτ)=P2(z+1(tτ)),z+i(s)=z+i0(s), s[τ,0], i=1,2,

    where

    {P1(z+2)=h(v2++z+2)h(v2+)h(v2+)z+2,P2(z+1)=g(v1++z+1)g(v1+)g(v1+)z+1.

    Then by [9,Lemma 3.8], there exist positive constants τ0, μ1 and C such that when ττ0,

    |Vi(,t)|=|z+i(t)|Ceμ1t, t>0, i=1,2, (17)

    provided that |z+i0|1, i=1,2.

    By the continuity and the uniform convergence of Vi(ξ,t) as ξ+, there exists a large x01 such that (17) implies

    supξ[x0,+)|Vi(ξ,t)|Ceμ1t, t>0, i=1,2,

    provided that supξ[x0,+)|Vi0(ξ,s)|1 for s[τ,0]. Such a smallness for the initial perturbation (V10,V20) near ξ+ can be easily verified, since

    limx+(v10(x,s),v20(x,s))=(K1,K2) uniformly in s[τ,0],

    which implies

    limξ+Vi0(ξ,s)=limξ+[vi0(ξ,s)ϕi(ξ)]=KiKi=0

    uniformly for s[τ,0], i=1,2. The proof is complete.

    Next we are going to establish the a priori decay estimate of supξ(,x0]|Vi(ξ,t)| by using the anti-weighted technique [3] together with the Fourier transform. First of all, we shift Vi(ξ,t) to Vi(ξ+x0,t) by the constant x0 given in Lemma 3.2, and then introduce the following transformation

    ˜Vi(ξ,t)=w(ξ)Vi(ξ+x0,t)=eλξVi(ξ+x0,t),i=1,2.

    Substituting Vi=w1/2˜Vi to (10) yields

    {˜V1t+c˜V1ξ+c1˜V1(ξ,t)d1eλ˜V1(ξ+1,t)d1eλ˜V1(ξ1,t)=˜Q1(˜V2(ξcτ,tτ)),˜V2t+c˜V2ξ+c2˜V2(ξ,t)d2eλ˜V2(ξ+1,t)d2eλ˜V2(ξ1,t)=˜Q2(˜V1(ξcτ,tτ)),˜Vi(ξ,s)=w(ξ)Vi0(ξ+x0,s)=:˜Vi0(ξ,s), ξR,s[τ,0], i=1,2, (18)

    where

    c1=cλ+2d1+α,c2=cλ+2d2+β

    and

    ˜Q1(˜V2)=eλξQ1(V2),˜Q2(˜V1)=eλξQ2(V1).

    By (11), ˜Q1(˜V2) satisfies

    ˜Q1(˜V2(ξcτ,tτ))=eλξQ1(V2(ξcτ+x0,tτ))=eλξh(˜ϕ2)V2(ξcτ+x0,tτ)=eλcτh(˜ϕ2)˜V2(ξcτ,tτ) (19)

    and ˜Q2(˜V1) satisfies

    ˜Q2(˜V1(ξcτ,tτ))=eλcτg(˜ϕ1)˜V1(ξcτ,tτ). (20)

    By (H3), we further obtain

    |˜Q1(˜V2(ξcτ,tτ))|h(0)eλcτ|˜V2(ξcτ,tτ)|,|˜Q2(˜V1(ξcτ,tτ))|g(0)eλcτ|˜V1(ξcτ,tτ)|.

    Taking (19) and (20) into (18), one can see that the coefficients h(˜ϕ2) and g(˜ϕ1) on the right side of (18) are variable and can be negative. Thus, the classical methods, such as the monotone technique and the Fourier transform cannot be applied directly to establish the decay estimate for (˜V1,˜V2). Motivated by [15,28,17,23], we introduce a new method which can be described as follows.

    By replacing h(˜ϕ2) in the first equation of (18) with a constant h(0), and g(˜ϕ1) in the second equation of (18) with a constant g(0), we can obtain a linear delayed reaction-diffusion system

    {V+1t+cV+1ξ+c1V+1(ξ,t)d1eλV+1(ξ+1,t)d1eλV+1(ξ1,t) =h(0)eλcτV+2(ξcτ,tτ),V+2t+cV+2ξ+c2V+2(ξ,t)d2eλV+2(ξ+1,t)d2eλV+2(ξ1,t) =g(0)eλcτV+1(ξcτ,tτ), (21)

    with

    V+i(ξ,s)=w(ξ)Vi0(ξ+x0,s)=:V+i0(ξ,s), i=1,2,

    where ξR, t(0,+] and s[τ,0]. Then we investigate the decay estimate of (V+1,V+2) by applying the Fourier transform to (21);

    We prove that the solution (˜V1,˜V2) of (18) can be bounded by the solution (V+1,V+2) of (21).

    Now we are in a position to derive the decay estimate of (V+1,V+2) for the linear system (21). We first recall some properties of the solutions to the delayed ODE system.

    Lemma 3.3. ([11,Lemma 3.1]) Let z(t) be the solution to the following scalar differential equation with delay

    {ddtz(t)=Az(t)+Bz(tτ),t0,τ>0,z(s)=z0(s),s[τ,0]. (22)

    where A,BCN×N, N2, and z0(s)C1([τ,0],CN). Then

    z(t)=eA(t+τ)eB1tτz0(τ)+0τeA(ts)eB1(tτs)τ[z0(s)Az0(s)]ds,

    where B1=BeAτ and eB1tτ is the so-called delayed exponential function in the form

    eB1tτ={0,<t<τ,I,τt<0,I+B1t1!,0t<τ,I+B1t1!+B21(tτ)22!,τt<2τ,I+B1t1!+B21(tτ)22!++Bm1[t(m1)τ]mm!,(m1)τt<mτ,

    where 0,ICN×N, and 0 is zero matrix and I is unit matrix.

    Lemma 3.4. ([11,Theorem 3.1]) Suppose μ(A):=μ1(A)+μ(A)2<0, where μ1(A) and μ(A) denote the matrix measure of A induced by the matrix 1-norm 1 and -norm , respectively. If ν(B):=B+B2μ(A), then there exists a decreasing function ετ=ε(τ)(0,1) for τ>0 such that any solution of system (22) satisfies

    z(t)C0eετσt,t>0,

    where C0 is a positive constant depending on initial data z0(s),s[τ,0] and σ=|μ(A)|ν(B). In particular,

    eAteB1tτC0eετσt,t>0,

    where eB1tτ is defined in Lemma 3.3.

    From the proof of [11,Theome 3.1], one can see that

    μ1(A)=limθ0+I+θA1θ=max1jN[Re(ajj)+Nji|aij|]

    and

    μ(A)=limθ0+I+θA1θ=max1iN[Re(aii)+Nij|aij|].

    Taking the Fourier transform to (21) and denoting the Fourier transform of V+(ξ,t):=(V+1(ξ,t),V+2(ξ,t))T by ˆV+(η,t):=(ˆV+1(η,t),ˆV+2(η,t))T, we obtain

    {tˆV+1(η,t)=(c1+d1(eλ+iη+e(λ+iη))icη)ˆV+1(η,t)                 +h(0)ecτ(λ+iη)ˆV+2(η,tτ),tˆV+2(η,t)=(c2+d2(eλ+iη+e(λ+iη))icη)ˆV+2(η,t)                 +g(0)ecτ(λ+iη)ˆV+1(η,tτ),ˆV+i(η,s)=ˆV+i0(η,s), ηR, s[τ,0], i=1,2. (23)

    Let

    A(η)=(c1+d1(eλ+iη+e(λ+iη))icη00c2+d2(eλ+iη+e(λ+iη))icη)

    and

    B(η)=(0h(0)ecτ(λ+iη)g(0)ecτ(λ+iη)0).

    Then system (23) can be rewritten as

    ˆV+t(η,t)=A(η)ˆV+(η,t)+B(η)ˆV+(η,tτ). (24)

    By Lemma 3.3, the linear delayed system (24) can be solved by

    ˆV+(η,t)=eA(η)(t+τ)eB1(η)tτˆV+0(η,τ)+0τeA(η)(ts)eB1(η)(tsτ)τ[sˆV+0(η,s)A(η)ˆV+0(η,s)]ds:=I1(η,t)+0τI2(η,ts)ds, (25)

    where B1(η)=B(η)eA(η)τ. Then by taking the inverse Fourier transform to (25), one has

    V+(ξ,t) (26)
    =F1[I1](ξ,t)+0τF1[I2](ξ,ts)ds=12πeiξηeA(η)(t+τ)eB1(η)tτˆV+0(η,τ)dη   +12π0τeiξηeA(η)(ts)eB1(η)(tsτ)τ[sˆV+0(η,s)A(η)ˆV+0(η,s)]dηds. (27)

    Lemma 3.5. Let the initial data V+i0(ξ,s), i=1,2, be such that

    V+i0C([τ,0];W1,1(R)), sV+i0L1([τ,0];L1(R)), i=1,2.

    Then

    V+i(t)L(R)Ceμ2t for cmax{c,c}, i=1,2,

    where μ2>0 and C>0.

    Proof. According to (26), we shall estimate F1[I1](ξ,t) and 0τF1[I2](ξ,ts)ds, respectively. By the definition of μ() and ν(), we have

    μ(A(η))=μ1(A(η))+μ(A(η))2=max{c1+d1(eλcosη+eλcosη),c2+d2(eλcosη+eλcosη)}=c2+d2(eλcosη+eλcosη)=c2+d2(eλ+eλ)cosη=cλ+d2(eλ+eλ2)βm(η),

    where c2=cλ+2d2+β and

    m(η)=d2(1cosη)(eλ+eλ)0,

    since d2>d1, α>β and d2d1<αβ2, and

    ν(B(η))=max{h(0),g(0)}eλcτ.

    By considering λ(λ1(c),λ2(c)), we get μ(A(η))<0 and

    μ(A(η))+ν(B(η))=cλ+d2(eλ+eλ2)βm(η)+max{h(0),g(0)}eλcτ<0.

    Furthermore, we obtain

    |μ(A(η))|ν(B(η))=cλd2(eλ+eλ2)+β+m(η)max{h(0),g(0)}eλcτ=P2(λ,c)+m(η),

    where P2(λ,c)=d2(eλ+eλ2)cλβ+max{h(0),g(0)}eλcτ<0 for c>max{c,c}. It then follows from Lemma 3.4 that there exists a decreasing function ετ=ε(τ)(0,1) such that

    eA(η)(t+τ)eB1(η)tC1eετ(|μ(A(η))|ν(B(η)))tC1eετμ0teετm(η)t, (28)

    where C1 is a positive constant and μ0:=P2(λ,c)>0 with c>c. By the definition of Fourier's transform, we have

    supηRˆV+0(η,τ)RV+0(ξ,τ)dξ=2i=1V+i0(,τ)L1(R).

    Applying (28), we derive

    supξRF1[I1](ξ,t)=supξR12πeiξηeA(η)(t+τ)eB1(η)tˆV+0(η,τ)dηCeετm(η)teετμ0tˆV+0(η,τ)dηCeετμ0tsupηRˆV+0(η,τ)eετm(η)tdηCeμ2t2i=1V+i0(,τ)L1(R), (29)

    with μ2:=ετμ0.

    Note that

    supηRA(η)ˆV+0(η,s)C2i=1V+i0(,s)W1,1(R).

    Similarly, we can obtain

    supξRF1[I2](ξ,ts)=supξR12πeiξηeA(η)(ts)eB1(η)(tsτ)[sˆV+0(η,s)A(η)ˆV+0(η,s)]dηCeετm(η)(ts)eετμ0(ts)sˆV+0(η,s)A(η)ˆV+0(η,s)dηCeετμ0teετμ0ssupηRsˆV+0(η,s)A(η)ˆV+0(η,s)eετm(η)(ts)dη.

    It then follows that

    0τsupξRF1[I2](ξ,ts)dsCeετμ0t0τeετμ0ssupηRsˆV+0(η,s)A(η)ˆV+0(η,s)eετm(η)(ts)dηdsCeετμ0t0τsV+0(,s)L1(R)+V+0(,s)W1,1(R)dsCeετμ0t(sV+0(s)L1([τ,0];L1(R))+V+0(s)L1([τ,0];W1,1(R))). (30)

    Substituting (29) and (30) to (26), we obtain the following the decay rate

    2i=1V+i(t)L(R)Ceμ2t.

    This proof is complete.

    The following maximum principle is needed to obtain the crucial boundedness estimate of (˜V1,˜V2), which has been proved in [17,Lemma 3.4].

    Lemma 3.6. Let T>0. For any a1,a2R and ν>0, if the bounded function v satisfies

    {vt+a1vξ+a2vdeνv(t,ξ+1)deνv(t,ξ1)0, (t,ξ)(0,T]×R,v(0,ξ)0,ξR, (31)

    then v(t,ξ)0 for all (t,ξ)(0,T]×R.

    Lemma 3.7. When (V+10(ξ,s),V+20(ξ,s))(0,0) for (ξ,s)R×[τ,0], then (V+1(ξ,t),V+2(ξ,t))(0,0) for (ξ,t)R×[0,+).

    Proof. When t[0,τ], we have tτ[τ,0] and

    h(0)eλcτV+2(ξcτ,tτ)=h(0)eλcτV+20(ξcτ,tτ)0. (32)

    Applying (32) to the first equation of (21), we get

    {V+1t+cV+1ξ+c1V+1(ξ,t)d1eλV+1(ξ+1,t)d1eλV+1(ξ1,t)0, (ξ,t)R×[0,τ],V+10(ξ,s)0, ξR, s[τ,0].

    By Lemma 3.6, we derive

    V+1(ξ,t)0,(ξ,t)R×[0,τ]. (33)

    Similarly, we obtain

    {V+2t+cV+2ξ+c2V+2(ξ,t)d2eλV+2(ξ+1,t)d2eλV+2(ξ1,t)0, (ξ,t)R×[0,τ],V+20(ξ,s)0, ξR s[τ,0].

    Using Lemma 3.6 again, we obtain

    V+2(ξ,t)0,(ξ,t)R×[0,τ]. (34)

    When t[nτ,(n+1)τ], n=1,2,, repeating the above procedure step by step, we can similarly prove

    (V+1(ξ,t),V+2(ξ,t))(0,0),(ξ,t)R×[nτ,(n+1)τ]. (35)

    Combining (33), (34) and (31), we obtain (V+1(ξ,t),V+2(ξ,t))(0,0) for (ξ,t)R×[0,+). The proof is complete.

    Now we establish the following crucial boundedness estimate for (˜V1,˜V2).

    Lemma 3.8. Let (˜V1(ξ,t),˜V2(ξ,t)) and (V+1(ξ,t),V+2(ξ,t)) be the solutions of (18) and (21), respectively. When

    |˜Vi0(ξ,s)|V+i0(ξ,s)for(ξ,s)R×[τ,0], i=1,2, (36)

    then

    |˜Vi(ξ,t)|V+i(ξ,t)for(ξ,t)R×[0,+), i=1,2.

    Proof. First of all, we prove |˜Vi(ξ,t)|V+i(ξ,t) for t[0,τ],i=1,2. In fact, when t[0,τ], namely, tτ[τ,0], it follows from (36) that

    |˜Vi(ξcτ,tτ)|=|˜Vi0(ξcτ,tτ)|V+i0(ξcτ,tτ)=V+i(ξcτ,tτ)for (ξ,t)R×[0,τ]. (37)

    Then by |h(˜ϕ2)|<h(0) and |g(˜ϕ1)|<g(0) and (37), we get

    h(0)eλcτV+2(ξcτ,tτ)±h(˜ϕ2)eλcτ˜V2(ξcτ,tτ)h(0)eλcτV+2(ξcτ,tτ)|h(˜ϕ2)|eλcτ|˜V2(ξcτ,tτ)|0for (ξ,t)R×[0,τ] (38)

    and

    g(0)eλcτV+1(ξcτ,tτ)±g(˜ϕ1)eλcτ˜V1(ξcτ,tτ)0for (ξ,t)R×[0,τ]. (39)

    Let

    Ui(ξ,t):=V+i(ξ,t)˜Vi(ξ,t)andU+i(ξ,t):=V+i(ξ,t)+˜Vi(ξ,t),i=1,2.

    We are going to estimate U±i(ξ,t) respectively.

    From (18), (19), (21) and (38), we see that U1(ξ,t) satisfies

    {U1t+cU1ξ+c1U1(ξ,t)d1eλU1(ξ+1,t)d1eλU1(ξ1,t)0,(ξ,t)R×[0,τ],U10(ξ,s)=V+10(ξ,s)˜V10(ξ,s)0,ξR, s[τ,0].

    By Lemma 3.6, we obtain

    U1(ξ,t)0,(ξ,t)R×[0,τ],

    namely,

    ˜V1(ξ,t)V+1(ξ,t),(ξ,t)R×[0,τ]. (40)

    Similarly, one has

    {U2t+cU2ξ+c2U2(ξ,t)d2eλU2(ξ+1,t)d2eλU2(ξ1,t)0,(ξ,t)R×[0,τ],U20(ξ,s)=V+20(ξ,s)˜V20(ξ,s)0,ξR, s[τ,0].

    Applying Lemma 3.6 again, we have

    U2(ξ,t)0,(ξ,t)R×[0,τ],

    i.e.,

    ˜V2(ξ,t)V+2(ξ,t),(ξ,t)R×[0,τ]. (41)

    On the other hand, U+1(ξ,t) satisfies

    {U+1t+cU+1ξ+c1U+1(ξ,t)d1eλU+1(ξ+1,t)d1eλU+1(ξ1,t)0,(ξ,t)R×[0,τ],U10(ξ,s)=V+10(ξ,s)˜V10(ξ,s)0,ξR, s[τ,0].

    Then Lemma 3.6 implies that

    U+1(ξ,t)=V+1(ξ,t)+˜V1(ξ,t)0,(ξ,t)R×[0,τ],

    that is,

    V+1(ξ,t)˜V1(ξ,t),(ξ,t)R×[0,τ]. (42)

    Similarly, U+2(ξ,t) satisfies

    {U+2t+cU+2ξ+c2U+2(ξ,t)d2eλU+2(ξ+1,t)d2eλU+2(ξ1,t)0,(ξ,t)R×[0,τ],U20(ξ,s)=V+20(ξ,s)˜V10(ξ,s)0,ξR, s[τ,0].

    Therefore, we can prove that

    U+2(ξ,t)=V+2(ξ,t)+˜V2(ξ,t)0,(ξ,t)R×[0,τ],

    namely

    V+2(ξ,t)˜V2(ξ,t),(ξ,t)R×[0,τ]. (43)

    Combining (40) and (42), we obtain

    |˜V1(ξ,t)|V+1(ξ,t)for(ξ,t)R×[0,τ], (44)

    and combining (41) and (43), we prove

    |˜V2(ξ,t)|V+2(ξ,t)for(ξ,t)R×[0,τ]. (45)

    Next, when t[τ,2τ], namely, tτ[0,τ], based on (44) and (45), we can similarly prove

    |˜Vi(ξ,t)|V+i(ξ,t)for(ξ,t)R×[τ,2τ],i=1,2.

    Repeating this procedure, we then further prove

    |˜Vi(ξ,t)|V+i(ξ,t),(ξ,t)R×[nτ,(n+1)τ],n=1,2,,

    which implies

    |˜Vi(ξ,t)|V+i(ξ,t)for(ξ,t)R×[0,),i=1,2.

    The proof is complete.

    Let us choose V+i0(ξ,s) such that

    V+i0C([τ,0];W1,1(R)),sV+i0L1([τ,0];L1(R)),

    and

    V+i0(ξ,s)|Vi0(ξ,s)|,(ξ,s)R×[τ,0], i=1,2.

    Combining Lemmas 3.5 and 3.8, we can get the convergence rates for ˜V(ξ,t).

    Lemma 3.9. When ˜Vi0C([τ,0];W1,1(R)) and s˜Vi0L1([τ,0];L1(R)), then

    ˜Vi(t)L(R)Ceμ2t,

    for some μ2>0, i=1,2.

    Lemma 3.10. It holds that

    supξ(,x0]|Vi(ξ,t)|Ceμ2t, i=1,2,

    for some μ2>0.

    Proof. Since ˜Vi(ξ,t)=w(ξ)Vi(ξ+x0,t)=eλξVi(ξ+x0,t) and w(ξ)=eλξ1 for ξ(,0], then we obtain

    supξ(,0]|Vi(ξ+x0,t)|˜Vi(t)L(R)Ceμ2t,

    which implies

    supξ(,x0]|Vi(ξ,t)|Ceμ2t.

    Thus, the estimate for the unshifted V(ξ,t) is obtained. The proof is complete.

    Proof of Proposition 3.2. By Lemmas 3.2 and 3.10, we immediately obtain (16) for 0<μ<min{μ1,μ2}.

    We are grateful to the anonymous referee for careful reading and valuable comments which led to improvements of our original manuscript.



    [1] L. K. Johnson, Smart intelligence, Foreign Policy, (1992), 53–69.
    [2] J. Wang, C. Xu, J. Zhang, R. Zhong, Big data analytics for intelligent manufacturing systems: A review, J. Manuf Syst., (2021). https://doi.org/10.1016/j.jmsy.2021.03.005
    [3] W. H. Zijm, Towards intelligent manufacturing planning and control systems, OR-Spektrum, 22 (2000), 313–345. https://doi.org/10.1007/s002919900032 doi: 10.1007/s002919900032
    [4] W. Qi H. Su, A cybertwin based multimodal network for ecg patterns monitoring using deep learning, IEEE Trans. Industr. Inform., (2022). https://doi.org/10.1109/TII.2022.3159583
    [5] L. Monostori, J. Prohaszka, A step towards intelligent manufacturing: Modelling and monitoring of manufacturing processes through artificial neural networks, CIRP Ann., 42 (1993), 485–488. https://doi.org/10.1016/S0007-8506(07)62491-3 doi: 10.1016/S0007-8506(07)62491-3
    [6] X. Yao, J. Zhou, J. Zhang, C. R. Boër, From intelligent manufacturing to smart manufacturing for industry 4.0 driven by next generation artificial intelligence and further on, in 2017 5th international conference on enterprise systems (ES). IEEE, (2017), 311–318. https://doi.org/10.1109/ES.2017.58
    [7] J. Yi, C. Lu, G. Li, A literature review on latest developments of harmony search and its applications to intelligent manufacturing, Math. Biosci. Eng., 16 (2019), 2086–2117. https://doi.org/10.3934/mbe.2019102 doi: 10.3934/mbe.2019102
    [8] S. Shan, X. Wen, Y. Wei, Z. Wang, Y. Chen, Intelligent manufacturing in industry 4.0: A case study of sany heavy industry, Syst. Res. Behav. Sci., 37 (2020), 679–690. https://doi.org/10.1002/sres.2709 doi: 10.1002/sres.2709
    [9] H. Yoshikawa, Manufacturing and the 21st century intelligent manufacturing systems and the renaissance of the manufacturing industry, Technol. Forecast Soc. Change, 49 (1995), 195–213. https://doi.org/10.1016/0040-1625(95)00008-X doi: 10.1016/0040-1625(95)00008-X
    [10] J. Zheng, K. Chan, I. Gibson, Virtual reality, IEEE Potent., 17 (1998), 20–23.
    [11] M. J. Schuemie, P. Van Der Straaten, M. Krijn, C. A. Van Der Mast, Research on presence in virtual reality: A survey, Cyberpsychol. & Behav., 4 (2001), 183–201. https://doi.org/10.1089/109493101300117884 doi: 10.1089/109493101300117884
    [12] C. Anthes, R. J. García-Hernández, M. Wiedemann, D. Kranzlmüller, State of the art of virtual reality technology, in IEEE Aerosp. Conf.. (2016), 1–19. 10.1109/AERO.2016.7500674
    [13] F. Biocca, B. Delaney, Immersive virtual reality technology, Communication in the age of virtual reality, 15 (1995). https://doi.org/10.4324/9781410603128
    [14] T. Mazuryk, M. Gervautz, Virtual reality-history, applications, technology and future, 1996.
    [15] N.-N. Zhou, Y.-L. Deng, Virtual reality: A state-of-the-art survey, Int. J. Autom. Comput., 6 (2009), 319–325. https://doi.org/10.1007/s11633-009-0319-9 doi: 10.1007/s11633-009-0319-9
    [16] J. Egger, T. Masood, Augmented reality in support of intelligent manufacturing–a systematic literature review, Comput. Ind. Eng., 140 (2020), 106195. https://doi.org/10.1016/j.cie.2019.106195 doi: 10.1016/j.cie.2019.106195
    [17] B.-H. Li, B.-C. Hou, W.-T. Yu, X.-B. Lu, C.-W. Yang, Applications of artificial intelligence in intelligent manufacturing: a review, Front. Inform. Tech. El., 18 (2017), 86–96. https://doi.org/10.1631/FITEE.1601885 doi: 10.1631/FITEE.1601885
    [18] B. He, K.-J. Bai, Digital twin-based sustainable intelligent manufacturing: A review, Adv. Manuf., 9 (2021), 1–21. https://doi.org/10.1007/s40436-020-00302-5 doi: 10.1007/s40436-020-00302-5
    [19] G.-J. Cheng, L.-T. Liu, X.-J. Qiang, Y. Liu, Industry 4.0 development and application of intelligent manufacturing, in 2016 international conference on information system and artificial intelligence (ISAI). IEEE, (2016), 407–410. https://doi.org/10.1109/ISAI.2016.0092
    [20] G. Y. Tian, G. Yin, D. Taylor, Internet-based manufacturing: A review and a new infrastructure for distributed intelligent manufacturing, J. Intell. Manuf., 13 (2002), 323–338. https://doi.org/10.1023/A:1019907906158 doi: 10.1023/A:1019907906158
    [21] H. Su, W. Qi, J. Chen, D. Zhang, Fuzzy approximation-based task-space control of robot manipulators with remote center of motion constraint, IEEE Trans. Fuzzy Syst., 30 (2022), 1564–1573. https://doi.org/10.1109/TFUZZ.2022.3157075 doi: 10.1109/TFUZZ.2022.3157075
    [22] M.-S. Yoh, The reality of virtual reality, in Proceedings seventh international conference on virtual systems and multimedia. IEEE, (2001), 666–674. https://doi.org/10.1109/VSMM.2001.969726
    [23] V. Antoniou, F. L. Bonali, P. Nomikou, A. Tibaldi, P. Melissinos, F. P. Mariotto, et al., Integrating virtual reality and gis tools for geological mapping, data collection and analysis: An example from the metaxa mine, santorini (greece), Appl. Sci., 10 (2020), 8317. https://doi.org/10.3390/app10238317 doi: 10.3390/app10238317
    [24] A. Kunz, M. Zank, T. Nescher, K. Wegener, Virtual reality based time and motion study with support for real walking, Proced. CIRP, 57 (2016), 303–308. https://doi.org/10.1016/j.procir.2016.11.053 doi: 10.1016/j.procir.2016.11.053
    [25] M. Serras, L. G.-Sardia, B. Simes, H. lvarez, J. Arambarri, Dialogue enhanced extended reality: Interactive system for the operator 4.0, Appl. Sci., 10 (2020). https://doi.org/10.3390/app10113960
    [26] A. G. da Silva, M. V. M. Gomes, I. Winkler, Virtual reality and digital human modeling for ergonomic assessment in industrial product development: A patent and literature review, Appl. Sci., 12 (2022), 1084. https://doi.org/10.3390/app12031084 doi: 10.3390/app12031084
    [27] J. Kim, J. Jeong, Design and implementation of opc ua-based vr/ar collaboration model using cps server for vr engineering process, Appl. Sci., 12 (2022), 7534. https://doi.org/10.3390/app12157534 doi: 10.3390/app12157534
    [28] J.-d.-J. Cordero-Guridi, L. Cuautle-Gutiérrez, R.-I. Alvarez-Tamayo, S.-O. Caballero-Morales, Design and development of a i4. 0 engineering education laboratory with virtual and digital technologies based on iso/iec tr 23842-1 standard guidelines, Appl. Sci., 12 (2022), 5993. https://doi.org/10.3390/app12125993 doi: 10.3390/app12125993
    [29] H. Heinonen, A. Burova, S. Siltanen, J. Lähteenmäki, J. Hakulinen, M. Turunen, Evaluating the benefits of collaborative vr review for maintenance documentation and risk assessment, Appl. Sci., 12 (2022), 7155. https://doi.org/10.3390/app12147155 doi: 10.3390/app12147155
    [30] V. Settgast, K. Kostarakos, E. Eggeling, M. Hartbauer, T. Ullrich, Product tests in virtual reality: Lessons learned during collision avoidance development for drones, Designs, 6 (2022), 33. https://doi.org/10.3390/designs6020033 doi: 10.3390/designs6020033
    [31] D. Mourtzis, J. Angelopoulos, N. Panopoulos, Smart manufacturing and tactile internet based on 5g in industry 4.0: Challenges, applications and new trends, Electronics-Switz, 10 (2021), 3175. https://doi.org/10.3390/electronics10243175 doi: 10.3390/electronics10243175
    [32] Y. Saito, K. Kawashima, M. Hirakawa, Effectiveness of a head movement interface for steering a vehicle in a virtual reality driving simulation, Symmetry, 12 (2020), 1645. https://doi.org/10.3390/sym12101645 doi: 10.3390/sym12101645
    [33] Y.-P. Su, X.-Q. Chen, T. Zhou, C. Pretty, G. Chase, Mixed-reality-enhanced human–robot interaction with an imitation-based mapping approach for intuitive teleoperation of a robotic arm-hand system, Appl. Sci., 12 (2022), 4740. https://doi.org/10.3390/app12094740 doi: 10.3390/app12094740
    [34] F. Arena, M. Collotta, G. Pau, F. Termine, An overview of augmented reality, Computers, 11 (2022), 28. https://doi.org/10.3390/computers11020028 doi: 10.3390/computers11020028
    [35] P. C. Thomas, W. David, Augmented reality: An application of heads-up display technology to manual manufacturing processes, in Hawaii international conference on system sciences, 2. ACM SIGCHI Bulletin New York, NY, USA, 1992.
    [36] J. Safari Bazargani, A. Sadeghi-Niaraki, S.-M. Choi, Design, implementation, and evaluation of an immersive virtual reality-based educational game for learning topology relations at schools: A case study, Sustainability-Basel, 13 (2021), 13066. https://doi.org/10.3390/su132313066 doi: 10.3390/su132313066
    [37] K. Židek, J. Pitel', M. Balog, A. Hošovskỳ, V. Hladkỳ, P. Lazorík, et al., CNN training using 3d virtual models for assisted assembly with mixed reality and collaborative robots, Appl. Sci., 11 (2021), 4269. https://doi.org/10.3390/app11094269 doi: 10.3390/app11094269
    [38] S. Mandal, Brief introduction of virtual reality & its challenges, Int. J. Sci. Eng. Res., 4 (2013), 304–309.
    [39] D. Rose, N. Foreman, Virtual reality. The Psycho., (1999).
    [40] G. Riva, C. Malighetti, A. Chirico, D. Di Lernia, F. Mantovani, A. Dakanalis, Virtual reality, in Rehabilitation interventions in the patient with obesity. Springer, (2020), 189–204.
    [41] J. N. Latta, D. J. Oberg, A conceptual virtual reality model, IEEE Comput. Graph. Appl., 14 (1994), 23–29. https://doi.org/10.1109/38.250915 doi: 10.1109/38.250915
    [42] J. Lanier, Virtual reality: The promise of the future. Interactive Learning International, 8 (1992), 275–79.
    [43] S. Serafin, C. Erkut, J. Kojs, N. C. Nilsson, R. Nordahl, Virtual reality musical instruments: State of the art, design principles, and future directions, Comput. Music. J., 40 (2016). https://doi.org/10.1162/COMJ_a_00372
    [44] W. Qi, H. Su, A. Aliverti, A smartphone-based adaptive recognition and real-time monitoring system for human activities, IEEE Trans. Hum. Mach. Syst., 50 (2020), 414 - 423. https://doi.org/10.1109/THMS.2020.2984181 doi: 10.1109/THMS.2020.2984181
    [45] P. Kopacek, Intelligent manufacturing: present state and future trends, J. Intell. Robot. Syst., 26 (1999), 217–229. https://doi.org/10.1023/A:1008168605803 doi: 10.1023/A:1008168605803
    [46] Y. Feng, Y. Zhao, H. Zheng, Z. Li, J. Tan, Data-driven product design toward intelligent manufacturing: A review, Int. J. Adv. Robot. Syst., 17 (2020), 1729881420911257. https://doi.org/10.1177/1729881420911257 doi: 10.1177/1729881420911257
    [47] H. Su, W. Qi, Y. Hu, H. R. Karimi, G. Ferrigno, E. De Momi, An incremental learning framework for human-like redundancy optimization of anthropomorphic manipulators, IEEE Trans. Industr. Inform., 18 (2020), 1864–1872. https://doi.org/10.1109/TII.2020.3036693 doi: 10.1109/TII.2020.3036693
    [48] E. Hozdić, Smart factory for industry 4.0: A review, Int. J. Adv. Manuf. Technol., 7 (2015), 28–35.
    [49] R. Burke, A. Mussomeli, S. Laaper, M. Hartigan, B. Sniderman, The smart factory: Responsive, adaptive, connected manufacturing, Deloitte Insights, 31 (2017), 1–10.
    [50] R. Y. Zhong, X. Xu, E. Klotz, S. T. Newman, Intelligent manufacturing in the context of industry 4.0: a review, Engineering-Prc, 3 (2017), 616–630. https://doi.org/10.1016/J.ENG.2017.05.015 doi: 10.1016/J.ENG.2017.05.015
    [51] A. Kusiak, Intelligent manufacturing, System, Prentice-Hall, Englewood Cliffs, NJ, (1990).
    [52] G. Rzevski, A framework for designing intelligent manufacturing systems, Comput. Ind., 34 (1997), 211–219. https://doi.org/10.1016/S0166-3615(97)00056-0 doi: 10.1016/S0166-3615(97)00056-0
    [53] E. Oztemel, Intelligent manufacturing systems, in Artificial intelligence techniques for networked manufacturing enterprises management. Springer, (2010), pp. 1–41. https://doi.org/10.1007/978-1-84996-119-6_1
    [54] J. Zhou, P. Li, Y. Zhou, B. Wang, J. Zang, L. Meng, Toward new-generation intelligent manufacturing, Engineering-Prc, 4 (2018), 11–20. https://doi.org/10.1016/j.eng.2018.01.002 doi: 10.1016/j.eng.2018.01.002
    [55] R. Y. Zhong, X. Xu, E. Klotz, S. T. Newman, Intelligent manufacturing in the context of industry 4.0: a review, Engineering-Prc, 3 (2017), 616–630. https://doi.org/10.1016/J.ENG.2017.05.015 doi: 10.1016/J.ENG.2017.05.015
    [56] H. S. Kang, J. Y. Lee, S. Choi, H. Kim, J. H. Park, J. Y. Son, B. H. Kim, S. D. Noh, Smart manufacturing: Past research, present findings, and future directions, Int. J. Pr. Eng. Man-Gt., 3 (2016), 111–128. https://doi.org/10.1007/s40684-016-0015-5 doi: 10.1007/s40684-016-0015-5
    [57] R. Jardim-Goncalves, D. Romero, A. Grilo, Factories of the future: challenges and leading innovations in intelligent manufacturing, Int. J. Comput. Integr. Manuf., 30 (2017), 4–14.
    [58] A. Kusiak, Smart manufacturing, Int. J. Prod. Res., 56 (2018), 508–517. https://doi.org/10.1080/00207543.2017.1351644
    [59] B. Wang, F. Tao, X. Fang, C. Liu, Y. Liu, T. Freiheit, Smart manufacturing and intelligent manufacturing: A comparative review, Engineering-Prc, 7 (2021), 738–757. https://doi.org/10.1016/j.eng.2020.07.017 doi: 10.1016/j.eng.2020.07.017
    [60] P. Zheng, Z. Sang, R. Y. Zhong, Y. Liu, C. Liu, K. Mubarok, et al., Smart manufacturing systems for industry 4.0: Conceptual framework, scenarios, and future perspectives, Front. Mech. Eng., 13 (2018), 137–150. https://doi.org/10.1007/s11465-018-0499-5 doi: 10.1007/s11465-018-0499-5
    [61] P. Osterrieder, L. Budde, T. Friedli, The smart factory as a key construct of industry 4.0: A systematic literature review, Int. J. Prod. Econ., 221 107476. https://doi.org/10.1016/j.ijpe.2019.08.011
    [62] D. Guo, M. Li, R. Zhong, G. Q. Huang, Graduation intelligent manufacturing system (gims): an industry 4.0 paradigm for production and operations management, Ind. Manage. Data Syst., (2020). https://doi.org/10.1108/IMDS-08-2020-0489
    [63] A. Barari, M. de Sales Guerra Tsuzuki, Y. Cohen, M. Macchi, Intelligent manufacturing systems towards industry 4.0 era, J. Intell. Manuf., 32 (2021), 1793–1796. https://doi.org/10.1007/s10845-021-01769-0 doi: 10.1007/s10845-021-01769-0
    [64] C. Christo, C. Cardeira, Trends in intelligent manufacturing systems, in 2007 IEEE International Symposium on Industrial Electronics-Switz.. IEEE, (2007), 3209–3214. https://doi.org/10.1109/ISIE.2007.4375129
    [65] M.-P. Pacaux-Lemoine, D. Trentesaux, G. Z. Rey, P. Millot, Designing intelligent manufacturing systems through human-machine cooperation principles: A human-centered approach, Comput. Ind. Eng., 111 (2017), 581–595. https://doi.org/10.1016/j.cie.2017.05.014 doi: 10.1016/j.cie.2017.05.014
    [66] W. F. Gaughran, S. Burke, P. Phelan, Intelligent manufacturing and environmental sustainability, Robot. Comput. Integr. Manuf., 23 (2007), 704–711. https://doi.org/10.1016/j.rcim.2007.02.016 doi: 10.1016/j.rcim.2007.02.016
    [67] Y. Boas, Overview of virtual reality technologies, in Inter. Mult. Confer., 2013 (2013).
    [68] lvaro Segura, H. V. Diez, I. Barandiaran, A. Arbelaiz, H. lvarez, B. Simes, J. Posada, A. Garca-Alonso, R. Ugarte, Visual computing technologies to support the operator 4.0, Comput. Ind. Eng., 139 (2020), 105550. https://doi.org/10.1016/j.cie.2018.11.060 doi: 10.1016/j.cie.2018.11.060
    [69] D. Romero, J. Stahre, T. Wuest, O. Noran, P. Bernus, Fasth, Fast-Berglund, D. Gorecky, Towards an operator 4.0 typology: A human-centric perspective on the fourth industrial revolution technologies, 10 (2016).
    [70] H. Qiao, J. Chen, X. Huang, A survey of brain-inspired intelligent robots: Integration of vision, decision, motion control, and musculoskeletal systems, " IEEE T. Cybernetics, 52 (2022), 11267 - 11280. https://doi.org/10.1109/TCYB.2021.3071312
    [71] F. Firyaguna, J. John, M. O. Khyam, D. Pesch, E. Armstrong, H. Claussen, H. V. Poor et al., Towards industry 5.0: Intelligent reflecting surface (irs) in smart manufacturing, arXiv preprint arXiv: 2201.02214, (2022). https://doi.org/10.1109/MCOM.001.2200016
    [72] A. M. Almassri, W. Wan Hasan, S. A. Ahmad, A. J. Ishak, A. Ghazali, D. Talib, C. Wada, Pressure sensor: state of the art, design, and application for robotic hand, J. Sensors, 2015 (2015). https://doi.org/10.1155/2015/846487
    [73] B. Munari, Design as art. Penguin UK, (2008).
    [74] B. De La Harpe, J. F. Peterson, N. Frankham, R. Zehner, D. Neale, E. Musgrave, R. McDermott, Assessment focus in studio: What is most prominent in architecture, art and design? IJADE., 28 (2009), 37–51. https://doi.org/10.1111/j.1476-8070.2009.01591.x
    [75] C. Gray, J. Malins, Visualizing research: A guide to the research process in art and design. Routledge, (2016).
    [76] M. Barnard, Art, design and visual culture: An introduction. Bloomsbury Publishing, (1998).
    [77] C. Crouch, Modernism in art, design and architecture. Bloomsbury Publishing, (1998).
    [78] M. Biggs, The role of the artefact in art and design research, Int. J. Des. Sci. Technol., 2002.
    [79] H. Su, W. Qi, Y. Schmirander, S. E. Ovur, S. Cai, X. Xiong, A human activity-aware shared control solution for medical human–robot interaction, Assembly Autom., (2022) ahead-of-print. https://doi.org/10.1108/AA-12-2021-0174
    [80] R. D. Gandhi, D. S. Patel, Virtual reality–opportunities and challenges, Virtual Real., 5 (2018).
    [81] A. J. Trappey, C. V. Trappey, M.-H. Chao, C.-T. Wu, Vr-enabled engineering consultation chatbot for integrated and intelligent manufacturing services, J. Ind. Inf. Integrat., 26 (2022), 100331. https://doi.org/10.1016/j.jii.2022.100331 doi: 10.1016/j.jii.2022.100331
    [82] K. Valaskova, M. Nagy, S. Zabojnik, G. Lăzăroiu, Industry 4.0 wireless networks and cyber-physical smart manufacturing systems as accelerators of value-added growth in slovak exports, Mathematics-Basel, 10 (2022), 2452. https://doi.org/10.3390/math10142452 doi: 10.3390/math10142452
    [83] J. de Assis Dornelles, N. F. Ayala, A. G. Frank, Smart working in industry 4.0: How digital technologies enhance manufacturing workers' activities, Comput. Ind. Eng., 163 (2022), 107804. https://doi.org/10.1016/j.cie.2021.107804 doi: 10.1016/j.cie.2021.107804
    [84] V. Tripathi, S. Chattopadhyaya, A. K. Mukhopadhyay, S. Sharma, C. Li, S. Singh, W. U. Hussan, B. Salah, W. Saleem, A. Mohamed, A sustainable productive method for enhancing operational excellence in shop floor management for industry 4.0 using hybrid integration of lean and smart manufacturing: An ingenious case study, Sustainability-Basel, 14 (2022), 7452. https://doi.org/10.3390/su14127452 doi: 10.3390/su14127452
    [85] S. M. M. Sajadieh, Y. H. Son, S. D. Noh, A conceptual definition and future directions of urban smart factory for sustainable manufacturing, Sustainability-Basel, 14 (2022), 1221. https://doi.org/10.3390/su14031221 doi: 10.3390/su14031221
    [86] Y. H. Son, G.-Y. Kim, H. C. Kim, C. Jun, S. D. Noh, Past, present, and future research of digital twin for smart manufacturing, J. Comput. Des. Eng., 9 (2022), 1–23. https://doi.org/10.1093/jcde/qwab067 doi: 10.1093/jcde/qwab067
    [87] G. Moiceanu, G. Paraschiv, Digital twin and smart manufacturing in industries: A bibliometric analysis with a focus on industry 4.0, Sensors-Basel, 22 (2022), 1388. https://doi.org/10.3390/s22041388 doi: 10.3390/s22041388
    [88] K. Cheng, Q. Wang, D. Yang, Q. Dai, M. Wang, Digital-twins-driven semi-physical simulation for testing and evaluation of industrial software in a smart manufacturing system, Machines, 10 (2022), 388. https://doi.org/10.3390/machines10050388 doi: 10.3390/machines10050388
    [89] S. Arjun, L. Murthy, P. Biswas, Interactive sensor dashboard for smart manufacturing, Procedia Comput. Sci., 200 (2022), 49–61. https://doi.org/10.1016/j.procs.2022.01.204 doi: 10.1016/j.procs.2022.01.204
    [90] J. Yang, Y. H. Son, D. Lee, S. D. Noh, Digital twin-based integrated assessment of flexible and reconfigurable automotive part production lines, Machines, 10 (2022), 75. https://doi.org/10.3390/machines10020075 doi: 10.3390/machines10020075
    [91] J. Friederich, D. P. Francis, S. Lazarova-Molnar, N. Mohamed, A framework for data-driven digital twins for smart manufacturing, Comput. Ind., 136 (2022), 103586. https://doi.org/10.1016/j.compind.2021.103586 doi: 10.1016/j.compind.2021.103586
    [92] L. Li, B. Lei, C. Mao, Digital twin in smart manufacturing, J. Ind. Inf. Integr., 26 (2022), 100289. https://doi.org/10.1016/j.jii.2021.100289 doi: 10.1016/j.jii.2021.100289
    [93] D. Nåfors, B. Johansson, Virtual engineering using realistic virtual models in brownfield factory layout planning, Sustainability-Basel, 13 (2021), 11102. https://doi.org/10.3390/su131911102 doi: 10.3390/su131911102
    [94] A. Geiger, E. Brandenburg, R. Stark, Natural virtual reality user interface to define assembly sequences for digital human models, Appl. System Innov., 3 (2020), 15. https://doi.org/10.3390/asi3010015 doi: 10.3390/asi3010015
    [95] G. Gabajova, B. Furmannova, I. Medvecka, P. Grznar, M. Krajčovič, R. Furmann, Virtual training application by use of augmented and virtual reality under university technology enhanced learning in slovakia, Sustainability-Basel, 11 (2019), 6677. https://doi.org/10.3390/su11236677 doi: 10.3390/su11236677
    [96] W. Qi, S. E. Ovur, Z. Li, A. Marzullo, R. Song, Multi-sensor guided hand gesture recognition for a teleoperated robot using a recurrent neural network, IEEE Robot Autom Lett., 6 (2021), 6039–6045. https://doi.org/10.1109/LRA.2021.3089999 doi: 10.1109/LRA.2021.3089999
    [97] L. Pérez, S. Rodríguez-Jiménez, N. Rodríguez, R. Usamentiaga, D. F. García, Digital twin and virtual reality based methodology for multi-robot manufacturing cell commissioning, Appl. Sci., 10 (2020), 3633. https://doi.org/10.3390/app10103633 doi: 10.3390/app10103633
    [98] J. Mora-Serrano, F. Muñoz-La Rivera, I. Valero, Factors for the automation of the creation of virtual reality experiences to raise awareness of occupational hazards on construction sites, Electronics-Switz., 10 (2021), 1355. https://doi.org/10.3390/electronics10111355 doi: 10.3390/electronics10111355
    [99] C. McDonald, K. A. Campbell, C. Benson, M. J. Davis, C. J. Frost, Workforce development and multiagency collaborations: A presentation of two case studies in child welfare, Sustainability-Basel, 13 (2021), 10190. https://doi.org/10.3390/su131810190 doi: 10.3390/su131810190
    [100] Z. Xu, N. Zheng, Incorporating virtual reality technology in safety training solution for construction site of urban cities, Sustainability-Basel, 13 (2020), 243. https://doi.org/10.3390/su13010243 doi: 10.3390/su13010243
    [101] L. Frizziero, L. Galletti, L. Magnani, E. G. Meazza, M. Freddi, Blitz vision: Development of a new full-electric sports sedan using qfd, sde and virtual prototyping, Inventions, 7 (2022), 41. https://doi.org/10.3390/inventions7020041 doi: 10.3390/inventions7020041
    [102] N. Lyons, Deep learning-based computer vision algorithms, immersive analytics and simulation software, and virtual reality modeling tools in digital twin-driven smart manufacturing, Econom. Manag. Financ. Markets, 17 (2022).
    [103] H. Qiao, S. Zhong, Z. Chen, H. Wang, Improving performance of robots using human-inspired approaches: A survey, Sci. China Inf. Sci., 65 (2022), 221201. https://doi.org/10.1007/s11432-022-3606-1 doi: 10.1007/s11432-022-3606-1
    [104] H. Su, A. Mariani, S. E. Ovur, A. Menciassi, G. Ferrigno, E. De Momi, Toward teaching by demonstration for robot-assisted minimally invasive surgery, IEEE Trans. Autom, 18 (2021), 484 - 494. https://doi.org/10.1109/TASE.2020.3045655 doi: 10.1109/TASE.2020.3045655
    [105] H. Su, W. Qi, Z. Li, Z. Chen, G. Ferrigno, E. De Momi, Deep neural network approach in EMG-based force estimation for human–robot interaction, IEEE Trans. Artif. Intell., 2 (2021), 404 - 412. https://doi.org/10.1109/TAI.2021.3066565 doi: 10.1109/TAI.2021.3066565
    [106] A. A. Malik, T. Masood, A. Bilberg, Virtual reality in manufacturing: immersive and collaborative artificial-reality in design of human-robot workspace, Int. J. Comput. Integr. Manuf., 33 (2020), 22–37. https://doi.org/10.1080/0951192X.2019.1690685 doi: 10.1080/0951192X.2019.1690685
    [107] A. Corallo, A. M. Crespino, M. Lazoi, M. Lezzi, Model-based big data analytics-as-a-service framework in smart manufacturing: A case study, Robot. Comput. Integr. Manuf., 76 (2022), 102331. https://doi.org/10.1016/j.rcim.2022.102331 doi: 10.1016/j.rcim.2022.102331
    [108] Y.-M. Tang, G. T. S. Ho, Y.-Y. Lau, S.-Y. Tsui, Integrated smart warehouse and manufacturing management with demand forecasting in small-scale cyclical industries, Machines, 10 (2022), 472. https://doi.org/10.3390/machines10060472 doi: 10.3390/machines10060472
    [109] M. Samardžić, D. Stefanović, U. Marjanović, Transformation towards smart working: Research proposal, in 2022 21st International Symposium INFOTEH-JAHORINA (INFOTEH). IEEE, (2022), 1–6. https://doi.org/10.1109/INFOTEH53737.2022.9751256
    [110] T. Caporaso, S. Grazioso, G. Di Gironimo, Development of an integrated virtual reality system with wearable sensors for ergonomic evaluation of human–robot cooperative workplaces, Sensors-Basel, 22 (2022), 2413. https://doi.org/10.3390/s22062413 doi: 10.3390/s22062413
    [111] W. Qi, N. Wang, H. Su, A. Aliverti DCNN based human activity recognition framework with depth vision guiding, Neurocomputing, 486 (2022), 261–271. https://doi.org/10.1016/j.neucom.2021.11.044 doi: 10.1016/j.neucom.2021.11.044
    [112] W. Zhu, X. Fan, Y. Zhang, Applications and research trends of digital human models in the manufacturing industry, VRIH, 1 (2019), 558–579. https://doi.org/10.1016/j.vrih.2019.09.005 doi: 10.1016/j.vrih.2019.09.005
    [113] O. Robert, P. Iztok, B. Borut, Real-time manufacturing optimization with a simulation model and virtual reality, Procedia Manuf., 38 (2019), 1103–1110. https://doi.org/10.1016/j.promfg.2020.01.198 doi: 10.1016/j.promfg.2020.01.198
    [114] I. Kačerová, J. Kubr, P. Hořejší, J. Kleinová, Ergonomic design of a workplace using virtual reality and a motion capture suit, Appl. Sci., 12 (2022), 2150. https://doi.org/10.3390/app12042150 doi: 10.3390/app12042150
    [115] M. Woschank, D. Steinwiedder, A. Kaiblinger, P. Miklautsch, C. Pacher, H. Zsifkovits, The integration of smart systems in the context of industrial logistics in manufacturing enterprises, Procedia Comput. Sci., 200 (2022), 727–737. https://doi.org/10.1016/j.procs.2022.01.271 doi: 10.1016/j.procs.2022.01.271
    [116] A. Umbrico, A. Orlandini, A. Cesta, M. Faroni, M. Beschi, N. Pedrocchi, A. Scala, P. Tavormina, S. Koukas, A. Zalonis et al., Design of advanced human–robot collaborative cells for personalized human–robot collaborations, Appl. Sci., 12 (2022), 6839. https://doi.org/10.3390/app12146839 doi: 10.3390/app12146839
    [117] W. Qi, A. Aliverti, A multimodal wearable system for continuous and real-time breathing pattern monitoring during daily activity, IEEE JBHI., 24 (2019), 2199–2207. https://doi.org/10.1109/JBHI.2019.2963048 doi: 10.1109/JBHI.2019.2963048
    [118] J. M. Runji, Y.-J. Lee, C.-H. Chu, User requirements analysis on augmented reality-based maintenance in manufacturing, J. Comput. Inf. Sci. Eng., 22 (2022), 050901. https://doi.org/10.1115/1.4053410 doi: 10.1115/1.4053410
    [119] D. Wuttke, A. Upadhyay, E. Siemsen, A. Wuttke-Linnemann, Seeing the bigger picture? ramping up production with the use of augmented reality, Manuf. Serv. Oper. Manag., (2022). https://doi.org/10.1287/msom.2021.1070
    [120] M. Catalano, A. Chiurco, C. Fusto, L. Gazzaneo, F. Longo, G. Mirabelli, L. Nicoletti, V. Solina, S. Talarico, A digital twin-driven and conceptual framework for enabling extended reality applications: A case study of a brake discs manufacturer, Procedia Comput. Sci., 200 (2022), 1885–1893. https://doi.org/10.1016/j.procs.2022.01.389 doi: 10.1016/j.procs.2022.01.389
    [121] J. S. Devagiri, S. Paheding, Q. Niyaz, X. Yang, S. Smith, Augmented reality and artificial intelligence in industry: Trends, tools, and future challenges, Expert Syst. Appl., (2022), 118002. https://doi.org/10.1016/j.eswa.2022.118002
    [122] P. T. Ho, J. A. Albajez, J. Santolaria, J. A. Yagüe-Fabra, Study of augmented reality based manufacturing for further integration of quality control 4.0: A systematic literature review, Appl. Sci., 12 (2022), 1961. https://doi.org/10.3390/app12041961 doi: 10.3390/app12041961
    [123] Z.-H. Lai, W. Tao, M. C. Leu, Z. Yin, Smart augmented reality instructional system for mechanical assembly towards worker-centered intelligent manufacturing, J. Manuf. Syst., 55 (2020), 69–81. https://doi.org/10.1016/j.jmsy.2020.02.010 doi: 10.1016/j.jmsy.2020.02.010
    [124] J. Xiong, E.-L. Hsiang, Z. He, T. Zhan, S.-T. Wu, Augmented reality and virtual reality displays: emerging technologies and future perspectives, Light Sci. Appl., 10 (2021), 1–30. https://doi.org/10.1038/s41377-021-00658-8 doi: 10.1038/s41377-021-00658-8
    [125] M.-G. Kim, J. Kim, S. Y. Chung, M. Jin, M. J. Hwang, Robot-based automation for upper and sole manufacturing in shoe production, Machines, 10 (2022), 255. https://doi.org/10.3390/machines10040255 doi: 10.3390/machines10040255
    [126] P. Grefen, I. Vanderfeesten, K. Traganos, Z. Domagala-Schmidt, J. van der Vleuten, Advancing smart manufacturing in europe: Experiences from two decades of research and innovation projects, Machines, 10 (2022), 45. https://doi.org/10.3390/machines10010045 doi: 10.3390/machines10010045
    [127] Y. Zhou, J. Zang, Z. Miao, T. Minshall, Upgrading pathways of intelligent manufacturing in china: Transitioning across technological paradigms, Engineering-Prc, 5 (2019), 691–701. https://doi.org/10.1016/j.eng.2019.07.016 doi: 10.1016/j.eng.2019.07.016
    [128] K. S. Kiangala, Z. Wang, An experimental safety response mechanism for an autonomous moving robot in a smart manufacturing environment using q-learning algorithm and speech recognition, Sensors-Basel, 22 (2022), 941. https://doi.org/10.3390/s22030941 doi: 10.3390/s22030941
    [129] S. Fernandes, Which way to cope with covid-19 challenges? contributions of the iot for smart city projects, Big Data Cogn. Comput., 5 (2021), 26. https://doi.org/10.3390/bdcc5020026 doi: 10.3390/bdcc5020026
    [130] C. Thomay, U. Bodin, H. Isakovic, R. Lasch, N. Race, C. Schmittner, G. Schneider, Z. Szepessy, M. Tauber, Z. Wang, Towards adaptive quality assurance in industrial applications, in 2022 IEEE/IFIP NOMS.. IEEE, (2022), 1–6. https://doi.org/10.1109/NOMS54207.2022.9789928
    [131] D. Stadnicka, P. Litwin, D. Antonelli, Human factor in intelligent manufacturing systems-knowledge acquisition and motivation, Proced. CIRP, 79 (2019), 718–723. https://doi.org/10.1016/j.procir.2019.02.023 doi: 10.1016/j.procir.2019.02.023
    [132] H.-X. Li, H. Si, Control for intelligent manufacturing: A multiscale challenge, Engineering-Prc, 3 (2017), 608–615. https://doi.org/10.1016/J.ENG.2017.05.016 doi: 10.1016/J.ENG.2017.05.016
    [133] T. Kalsoom, N. Ramzan, S. Ahmed, M. Ur-Rehman, Advances in sensor technologies in the era of smart factory and industry 4.0, Sensors-Basel, 20 (2020), 6783. https://doi.org/10.3390/s20236783 doi: 10.3390/s20236783
    [134] J. Radianti, T. A. Majchrzak, J. Fromm, I. Wohlgenannt, A systematic review of immersive virtual reality applications for higher education: Design elements, lessons learned, and research agenda, Comput. Educ., 147 (2020), COMPUT EDUC103778. https://doi.org/10.1016/j.compedu.2019.103778 doi: 10.1016/j.compedu.2019.103778
    [135] D. Kamińska, T. Sapiński, S. Wiak, T. Tikk, R. E. Haamer, E. Avots, A. Helmi, C. Ozcinar, G. Anbarjafari, Virtual reality and its applications in education: Survey, Information, 10 (2019), 318. https://doi.org/10.3390/info10100318 doi: 10.3390/info10100318
    [136] T. Joda, G. Gallucci, D. Wismeijer, N. U. Zitzmann, Augmented and virtual reality in dental medicine: A systematic review, Comput. Biol. Med., 108 (2019), 93–100. https://doi.org/10.1016/j.compbiomed.2019.03.012 doi: 10.1016/j.compbiomed.2019.03.012
    [137] C. Li, Y. Chen, Y. Shang, A review of industrial big data for decision making in intelligent manufacturing, J. Eng. Sci. Technol., (2021). https://doi.org/10.1016/j.jestch.2021.06.001
    [138] L. Zhou, Z. Jiang, N. Geng, Y. Niu, F. Cui, K. Liu, N. Qi, Production and operations management for intelligent manufacturing: a systematic literature review, Int. J. Prod. Res., 60 (2022), 808–846. https://doi.org/10.1080/00207543.2021.2017055 doi: 10.1080/00207543.2021.2017055
    [139] L. Adriana Crdenas-Robledo, scar Hernndez-Uribe, C. Reta, J. Antonio Cantoral-Ceballos, Extended reality applications in industry 4.0. a systematic literature review, Telemat. Inform., 73 (2022), 101863. https://doi.org/10.1016/j.tele.2022.101863 doi: 10.1016/j.tele.2022.101863
    [140] Z. Wang, X. Bai, S. Zhang, M. Billinghurst, W. He, P. Wang, W. Lan, H. Min, Y. Chen, A comprehensive review of augmented reality-based instruction in manual assembly, training and repair, Robot. Comput. Integr. Manuf., 78 (2022), 102407. https://doi.org/10.1016/j.rcim.2022.102407 doi: 10.1016/j.rcim.2022.102407
    [141] N. Kumar, S. C. Lee, Human-machine interface in smart factory: A systematic literature review, Technol. Forecast. Soc. Change, 174 (2022), 121284. https://doi.org/10.1016/j.techfore.2021.121284 doi: 10.1016/j.techfore.2021.121284
    [142] M. Javaid, A. Haleem, R. P. Singh, R. Suman, Enabling flexible manufacturing system (fms) through the applications of industry 4.0 technologies, Int. Things Cyber-Phys. Syst., (2022). https://doi.org/10.1016/j.iotcps.2022.05.005
    [143] A. Künz, S. Rosmann, E. Loria, J. Pirker, The potential of augmented reality for digital twins: A literature review, in 2022 IEEE Conference on Virtual Reality and 3D User Interfaces (VR). IEEE, (2022), 389–398. https://doi.org/10.1109/VR51125.2022.00058
    [144] I. Shah, C. Doshi, M. Patel, S. Tanwar, W.-C. Hong, R. Sharma, A comprehensive review of the technological solutions to analyse the effects of pandemic outbreak on human lives, Medicina (Kaunas), 58 (2022), 311. https://doi.org/10.3390/medicina58020311 doi: 10.3390/medicina58020311
    [145] R. P. Singh, M. Javaid, R. Kataria, M. Tyagi, A. Haleem, R. Suman, Significant applications of virtual reality for covid-19 pandemic, Diabetes Metab. Syndr., 14 (2020), 661–664. https://doi.org/10.1016/j.dsx.2020.05.011 doi: 10.1016/j.dsx.2020.05.011
    [146] A. O. Kwok, S. G. Koh, Covid-19 and extended reality (xr), Curr. Issues Tour., 24 (2021), 1935–1940. https://doi.org/10.1080/13683500.2020.1798896 doi: 10.1080/13683500.2020.1798896
    [147] G. Czifra, Z. Molnár et al., Covid-19, industry 4.0, Research papers faculty of materials science and technology slovak university of technology, 28 (2020), 36–45. https://doi.org/10.2478/rput-2020-0005
    [148] Q. Yu-ming, D. San-peng et al., Research on intelligent manufacturing flexible production line system based on digital twin, in 2020 35th Youth Academic Annual Conference of Chinese Association of Automation (YAC), IEEE, (2020), 854–862. https://doi.org/10.1109/YAC51587.2020.9337500
    [149] L. O. Alpala, D. J. Quiroga-Parra, J. C. Torres, D. H. Peluffo-Ordóñez, Smart factory using virtual reality and online multi-user: Towards a metaverse for experimental frameworks, Appl. Sci., 12 (2022), 6258. https://doi.org/10.3390/app12126258 doi: 10.3390/app12126258
    [150] E. Chang, H. T. Kim, B. Yoo, Virtual reality sickness: A review of causes and measurements, Int. J. Hum-Comput. Int., 36 (2020), 1658–1682. https://doi.org/10.1080/10447318.2020.1778351 doi: 10.1080/10447318.2020.1778351
    [151] H. Su, W. Qi, C. Yang, J. Sandoval, G. Ferrigno, E. De Momi, Deep neural network approach in robot tool dynamics identification for bilateral teleoperation, IEEE Robot. Autom. Lett., 5 (2020), 2943–2949. https://doi.org/10.1109/LRA.2020.2974445 doi: 10.1109/LRA.2020.2974445
    [152] H. Su, Y. Hu, H. R. Karimi, A. Knoll, G. Ferrigno, E. De Momi, Improved recurrent neural network-based manipulator control with remote center of motion constraints: Experimental results, Neural Netw., 131 (2020), 291–299. https://doi.org/10.1016/j.neunet.2020.07.033 doi: 10.1016/j.neunet.2020.07.033
    [153] S. Phuyal, D. Bista, R. Bista, Challenges, opportunities and future directions of smart manufacturing: A state of art review, Sustain. Fut., 2 (2020), 100023. https://doi.org/10.1016/j.sftr.2020.100023 doi: 10.1016/j.sftr.2020.100023
  • This article has been cited by:

    1. Xing-Xing Yang, Guo-Bao Zhang, Yu-Cai Hao, Existence and stability of traveling wavefronts for a discrete diffusion system with nonlocal delay effects, 2023, 0, 1531-3492, 0, 10.3934/dcdsb.2023160
    2. Zhi-Jiao Yang, Guo-Bao Zhang, Ge Tian, Global stability of traveling wave solutions for a discrete diffusion epidemic model with nonlocal delay effects, 2025, 66, 0022-2488, 10.1063/5.0202813
    3. Jiao Dang, Guo-Bao Zhang, Ge Tian, Wave Propagation for a Discrete Diffusive Mosquito-Borne Epidemic Model, 2024, 23, 1575-5460, 10.1007/s12346-024-00964-7
  • Reader Comments
  • © 2023 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(5427) PDF downloads(416) Cited by(10)

Figures and Tables

Figures(9)  /  Tables(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog