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Research article

Nonlinear extended state observer based control for the teleoperation of robotic systems with flexible joints

  • Received: 07 September 2023 Revised: 15 December 2023 Accepted: 20 December 2023 Published: 25 December 2023
  • The control of robot manipulator pose is significantly complicated by the uncertainties arising from flexible joints, presenting substantial challenges in incorporating practical operational constraints. These challenges are further exacerbated in teleoperation scenarios, where factors such as synchronization and external disturbances further amplify the difficulties. At the core of this research is the introduction of a pioneering teleoperation controller, ingeniously integrating a nonlinear extended state observer (ESO) with the barrier Lyapunov function (BLF) while effectively accommodating a steady time delay. The controller in our study demonstrates exceptional proficiency in accurately estimating uncertainties arising from both flexible joints and external disturbances using the nonlinear ESO. Refined estimates, in conjunction with operational constraints of the system, are integrated into our BLF-based controller. Consequently, a synchronized control mechanism for teleoperation is achieved, exhibiting promising performance. Importantly, our experimental findings provide substantial evidence that our proposed approach effectively reduces the tracking error of the teleoperation system to within 0.02 rad. This advancement highlights the potential of our controller in significantly enhancing the precision and reliability of teleoperated robot manipulators.

    Citation: Yongli Yan, Fucai Liu, Teng Ren, Li Ding. Nonlinear extended state observer based control for the teleoperation of robotic systems with flexible joints[J]. Mathematical Biosciences and Engineering, 2024, 21(1): 1203-1227. doi: 10.3934/mbe.2024051

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  • The control of robot manipulator pose is significantly complicated by the uncertainties arising from flexible joints, presenting substantial challenges in incorporating practical operational constraints. These challenges are further exacerbated in teleoperation scenarios, where factors such as synchronization and external disturbances further amplify the difficulties. At the core of this research is the introduction of a pioneering teleoperation controller, ingeniously integrating a nonlinear extended state observer (ESO) with the barrier Lyapunov function (BLF) while effectively accommodating a steady time delay. The controller in our study demonstrates exceptional proficiency in accurately estimating uncertainties arising from both flexible joints and external disturbances using the nonlinear ESO. Refined estimates, in conjunction with operational constraints of the system, are integrated into our BLF-based controller. Consequently, a synchronized control mechanism for teleoperation is achieved, exhibiting promising performance. Importantly, our experimental findings provide substantial evidence that our proposed approach effectively reduces the tracking error of the teleoperation system to within 0.02 rad. This advancement highlights the potential of our controller in significantly enhancing the precision and reliability of teleoperated robot manipulators.



    ()m,()s Subscripts {m, s} denote the master robot and the slave robot, respectively.
    (ˆ) Diacritical mark wedge denotes the estimation, for example, ˆΘ is the estimation of Θ.
    (˜) Diacritical mark tilde denotes the estimation error, for example ˜Θ=ΘˆΘ.
    qi,˙qi,¨qi n-dimensional joint position, velocity, acceleration, i = m, s.
    Mi(qi) Robotic inertia matrix, i = m, s.
    Ci(qi,˙qi) Coriolis/centrifugal matrix, i = m, s.
    fi(qi) Viscous friction force vector, i = m, s.
    Bi(qi) Bounded external disturbance, i = m, s.
    Gi(qi) Gravity torque, i = m, s.
    τi Control torque, i = m, s.
    τh,τe Forces exerted on the end-effectors of the master and slave robots by the human operator and environment, respectively.
    θi,˙θi,¨θi n-dimensional angular position, velocity, acceleration, i = m, s.
    Js Jacobian matrix of the slave robot.
    τf Friction caused by the motor shaft, where the subscript {f} represents variables associated with friction caused by the motor shaft.
    τc Torque generated by the flexibility of the joint, where the subscript {c} represents variables associated with the joint.
    Ks Equivalent stiffness coefficient of the joint in the slave robot.
    Rn,Rn×m n-dimensional real space and n × m dimensional real matrix space, respectively.
    Ti Communication delays, i = m, s.
    Zik State variable of ESO, i = m, s, k = 1,2,3.
    eik State estimation error of ESO, i = m, s, k = 1,2,3.

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    The implementation of teleoperation within robotic systems extends the ability of humans to perform specific operations personally in remote environments. Teleoperation is conducted by remote operators transmitting control signals to robotic systems through communication channels, and receiving the force interaction data from the robotic system via the same communication channels. Currently, teleoperation systems find extensive applications in diverse fields including remote surgery and deep-sea exploration [1,2,3].

    Increasing interest has been generated in recent years for developing robotic systems with flexible manipulator joints. This is mainly due to two reasons [4,5]. First, the increasing complexity and refinement of industrial tasks has necessitated the implementation of elastic drive transmission systems in industrial robots, such as harmonic gear transmission, joint torque sensors, and belt pulleys [6]. These must be treated as flexible-joint systems to mitigate the negative impact of end effector vibrations on system stability [7]. Second, flexible components have been introduced in new robotic manipulator designs for obtaining more human-compatible mobility characteristics that facilitate human-robot interactions, such as high flexibility, strong adaptability, and strong robustness [8]. However, the flexibility of these joints renders the dynamic behavior of robotic systems increasingly intricate and challenging to model. This complexity primarily stems from nonlinearity, uncertainty, damping effects, and vibration modes of flexible joints, presenting a significant challenge in achieving satisfactory control performance in remote operation systems.

    Since Spong proposed the flexible joint model in 1987, more and more control strategies have been proposed for manipulator systems with flexible joints [9]. Spong [10], Ghorbel et al. [11], and Chang and Daniel [12] designed an adaptive controller based on singular perturbation control. However, although this method can simplify the analysis, it may also oversimplify the system dynamics, potentially ignoring the key nonlinearity and complexity existing in the real scene [13]. Nam et al. [14] introduced finite time control based on sliding mode control and designed a control strategy that enables a flexible joint manipulator system to achieve effective tracking. However, the presence of buffeting reduces the tracking effect of the system. In addition, backstepping control is also a highly effective control strategy for addressing the tracking control problem in flexible joint manipulator systems. There have been many research achievements based on backstepping to solve control issues in flexible manipulator systems. Huang et al. [15] designed the controller of the single-link flexible manipulator system based on the reverse step method, but this control strategy is only applicable to a system with known state parameters. Cheng et al. [16] combined the singular perturbation control method and adaptive control method on the basis of backstep method to design a control strategy suitable for a flexible manipulator system. However, this control strategy has the disadvantage of too having complicated of a calculation. To tackle this issue, Sahu et al. [17] proposed a backstepping control strategy that combines extended state observer (ESO) with backstepping control. This approach not only effectively addresses the computational complexity in traditional backstepping but also efficiently handles the uncertainty and interference of the internal model, significantly improving the tracking performance of the system.

    ESO is a part of active disturbance rejection control (ADRC) theory that enables real-time estimation of internal state and unmodeled dynamics of a system [18,19]. Recently, ESO has been widely applied, such as in tracked vehicle control [20], quadrotor control [21], DC-link voltage control [22], and permanent magnet synchronous linear motor (PMSLM) servo system control [23]. ESO can be categorized into linear and nonlinear types. Linear ESO can exhibit peaking phenomena when dealing with nonlinearities and strong couplings, thereby potentially compromising the observer's accuracy to some extent [24]. Consequently, in this study, nonlinear ESO was chosen to address the uncertainty in the internal model and external disturbances of the flexible-joint teleoperation system, enhancing its robustness and accuracy.

    Furthermore, in real-world engineering applications, the system may face limitations due to factors such as temperature and space. If the system's operation exceeds predefined boundaries, it can adversely affect the system's performance and, in extreme cases, potentially lead to system failures, endangering the personal safety of the operators. The barrier Lyapunov function (BLF) is a standard method developed for implementing constrained control problems of this nature [25]. Wang [26] and Li et al. [27] used the barrier Lyapunov function to ensure that the input and output of a nonlinear system are controlled within a preset interval. Using the above research results, Zhang [28] and Yu et al. [29] applied the barrier Lyapunov function to a robot arm system with limited output, and ensured that the output of the system could be controlled within a preset interval.

    While considerable progress has been made in the control of robotic systems with flexible joints, these developments have been rarely applied in the design of teleoperation systems for remote robotic control. Moreover, teleoperation systems not only must address the many uncertainties in the pose control of robotic manipulators, but also account for complexities in the actual working environments, and the inevitability of external interference. Accordingly, these issues remain poorly addressed by the current state of teleoperation system development.

    In summary, to achieve effective tracking of joint position trajectories for a state-constrained flexible-joint teleoperation system, this paper proposes a control strategy based on the combination of nonlinear ESO and the Barrier Lyapunov Function. The main contributions in this work are summarized as follows: 1) The paper introduces a novel teleoperation controller that employs a nonlinear Extended State Observer (ESO) to estimate internal uncertainties related to flexible joints and external disturbances. This inclusion enhances the accuracy of the control system by actively predicting and compensating for disturbances in the input channel. 2) The research effectively addresses practical operational constraints on the system output states by leveraging the Barrier Lyapunov Function (BLF). This application ensures that the teleoperation controller operates within defined limitations, leading to more reliable and controlled system behavior. 3) The study focuses on a robotic system that combines a rigid master manipulator and a flexible slave manipulator. By enforcing constraints on the system output states using the BLF, synchronized control is achieved between these two distinct components. This synchronization is critical for precise and coordinated control in teleoperation scenarios.

    The remainder of this paper is organized as follows: Section 2 presents the pertinent preliminary discussion. Section 3 elucidates the nonlinear ESO and stability analyses of the proposed teleoperation control system. Following this, Section 4 provides a detailed account of the experimental results. Finally, concluding remarks and future directions are outlined in Section 5.

    The rigid master manipulator is assumed to have n degrees of freedom, and its nonlinear dynamic behavior is defined as follows.

    Mm(qm)¨qm+Cm(qm,˙qm)˙qm+fm(qm)+Bm(qm)+Gm(qm)=τm+τh (1)

    Similarly, the flexible slave manipulator is assumed to have n degrees of freedom, and its nonlinear dynamic behavior is defined as follows [9].

    {Ms(qs)¨qs+Cs(qs,˙qs)˙qs+fs(qs)+Bs(qs)+Gs(qs)=τcτeJs¨θs+τc=τs+τf (2)

    Here, for all i=m,s, qiRn, Mi(qi)Rn×n, Ci(˙qi,qi)Rn×n, fi(qi)Rn, Bi(qi)Rn, Gi(qi)Rn, JsRn×n, θsRn, τiRn, τcRn, τhRn, τeRn, τfRn, τc=Ks(θsqs), KsRn×n.

    For convenience, we make the following substitutions in the system variables: Xm1=qm, Xm2=˙qm, Xs1=qs, Xs2=˙qs, Xs3=θs, and Xs4=˙θs. Then, the following correspondences are applied.

    {˙Xm1=Xm2˙Xm2=Fm+M1mτm (3)
    {˙Xs1=Xs2˙Xs2=Fs+M1sτc˙Xs3=Xs4˙Xs4=Ff+J1sτs (4)

    The variables Fm, Fs, and Ff in (3) and (4) represent lumped uncertainties in the system, which contain internal uncertain parts and external perturbations, which are defined as follows.

    Fm=M1m(CmXm2+fm(Xm1)+Bm(Xm1)+Gm(Xm1)τh){Fs=M1s(CsXs2+fs(Xs1)+Bs(Xs1)+Gs(Xs1)+τe)Ff = J1s(τfτc) (5)

    By treating uncertainties such as τc as total perturbations, the design of the control system can be simplified, resulting in reduced computational load. This approach enhances the system's robustness to various disturbances while maintaining the effectiveness and flexibility of the control strategy.

    The nonlinear ESO is applied for estimating the interference terms Fm and Fs. In this subsection, we design the observer for Fm only, and subsequently analyze its error estimation stability. The Fs and Ff estimator design and stability are entirely equivalent and are therefore not presented herein.

    First, the nonlinear ESO is designed by introducing the following expansion terms:

    {Xm3=Fm˙Xm3=hm (6)

    where hm denotes the derivative of external disturbances. Before designing the ESO, the following assumption about external disturbances is given in this paper.

    Assumption 1: For the external disturbance in the nonlinear bilateral teleoperation system, there exists a positive unknown constant L such that |hm|L holds.

    We define the estimation error as

    emk=ZmkXmk,k=1,2,3, (7)

    where Zm1, Zm2, and Zm3 are the estimated values of Xm1, Xm2, and Xm3, respectively, and introduce new parametric variables

    ηmk=emkε3k,k=1,2,3, (8)

    where ε is an appropriate positive definite parameter. Then, the nonlinear ESO is designed in the following form:

    {˙Zm1=Zm2βm1ε|ηm1|32sgn(ηm1)˙Zm2=Zm3βm2|ηm1|12sgn(ηm1)+M1mτm˙Zm3=βm3ε1sgn(ηm1) (9)

    Here, sgn(·) is the signum function that returns the sign of its argument, and βm1, βm2, and βm3 are appropriate positive definite parameters. Combining Eqs (6)–(9), the estimation error of the system can be obtained as follows.

    {˙em1=em2βm1ε|ηm1|32sgn(ηm1)˙em2=em3βm2|ηm1|12sgn(ηm1)˙em3=hmβm3ε1sgn(ηm1) (10)

    Finally, both sides of the three expressions in error system (10) are multiplied from top to bottom by 1ε, 1, and ε, respectively, which yields the substitution of error system (10) as follows:

    {ε˙ηm1=ηm2βm1|ηm1|32sgn(ηm1)ε˙ηm2=ηm3βm2|ηm1|12sgn(ηm1)ε˙ηm3=εhmβm3sgn(ηm1) (11)

    Theorem 1: For master manipulator system (1) and nonlinear error system (11), the estimation erroremk(k=1,2,3) will converge to the neighborhood of the origin if there exist positive definite parametersβm1, βm2, and βm3 such that the following inequality is guaranteed.

    Q=[1εk8βm11εk11εk40(1εk4k29k26)1εk5+1εk21ε(k7+k4βm1)(1εk6k23)1εk5βm11εk7βm1000001ε(k8k2βm2)001εk6βm20001εk1βm1001ε(k9βm1k4βm2)0k91εβm3]0 (12)

    where k1 – k9 are appropriate positive definite constants and the following inequality group holds:

    {k1k4k50k2k4k60k3k5k60 (13)

    Proof: The proof is based on the following Lyapunov function:

    V=12k1η2m1+12k2η2m2+12k3η2m3k4ηm1ηm2+k5ηm1ηm3k6ηm2ηm3+25k7|ηm1|52+23k8|ηm1|32+k9|ηm1| (14)

    From Young's inequality, the following inequality group can be obtained:

    {k4ηm1ηm2k42(η2m1+η2m2)k5ηm1ηm3k52(η2m1+η2m3)k6ηm2ηm3k62(η2m2+η2m3) (15)

    Substituting Eq (15) into Eq (14), the following inequality can be obtained:

    V12(k1k4k5)η2m1+12(k2k4k6)η2m2+12(k3k5k6)η2m3+25k7|ηm1|52+23k8|ηm1|32+k9|ηm1| (16)

    When positive definite constants k1k9 satisfy inequality group (13), the Lyapunov function (14) is positive semidefinite.

    From the converted error system (11), taking the derivative of (14) with respect to time yields the following:

    ˙V=k1ηm1˙ηm1+k2ηm2˙ηm2+k3ηm3˙ηm3k4˙ηm1ηm2k4ηm1˙ηm2+k5˙ηm1ηm3+k5ηm1˙ηm3k6˙ηm2ηm3k6ηm2˙ηm3+k7|ηm1|32sgn(ηm1)˙ηm1+k8|ηm1|12sgn(ηm1)˙ηm1+k9sgn(ηm1)˙ηm1 (17)

    According to error system (11), we can reconfigure (17) as follows:

    ˙V=[k1ηm1k4ηm2+k5ηm3+k7|ηm1|32sgn(ηm1)+k8|ηm1|12sgn(ηm1)+k9sgn(ηm1)]1ε[ηm2βm1|ηm1|32sgn(ηm1)]+[k2ηm2k4ηm1k6ηm3]1ε[ηm3βm2|ηm1|12sgn(ηm1)]+[k3ηm3+k5ηm1k6ηm2]1ε[εhmβm3sgn(ηm1)] (18)

    This is further expanded as follows:

    ˙V=1εk1ηm1ηm21εk4η2m2+1εk5ηm2ηm3+k7|ηm1|32sgn(ηm1)ηm2+1εk8|ηm1|12sgn(ηm1)ηm2+1εk9sgn(ηm1)ηm21εk1βm1|ηm1|52+1εk4βm1|ηm1|32sgn(ηm1)ηm21εk5βm1|ηm1|32sgn(ηm1)ηm31εk8βm1η2m11εk9βm1|ηm1|32+1εk2ηm2ηm31εk4ηm1ηm31εk6η2m31εk2βm2|ηm1|12sgn(ηm1)ηm2+1εk4βm2|ηm1|321εk7βm1|ηm1|3+1εk6βm2|ηm1|12sgn(ηm1)ηm3(1εk6ηm21εk3ηm3)[εhmβm3sgn(ηm1)]+1εk5ηm1[εhmβm3sgn(ηm1)] (19)

    This can be transformed into the following inequality:

    ˙V1εk8βm1η2m1(1εk4k29k26)η2m2(1εk6k23)η2m3+1εk1ηm1ηm2+(1εk2+1εk5)ηm2ηm31εk7βm1|ηm1|31εk4ηm1ηm31εk1βm1|ηm1|52(1εk9βm11εk4βm2)|ηm1|32(1εk5βm3k5)|ηm1|+(1εk7+1εk4βm1)|ηm1|32sgn(ηm1)ηm21εk5βm1|ηm1|32sgn(ηm1)ηm3+(1εk81εk2βm2)|ηm1|12sgn(ηm1)ηm2+1εk6βm2|ηm1|12sgn(ηm1)ηm3+1ε2[εhmβm3sgn(ηm1)]2+14ε2 (20)

    Therefore,

    ˙Vξ(t)QξT(t)+1ε2[εhmβm3sgn(ηm1)]2+14ε2 (21)

    where Q is shown in Eq (12), and ξ(t) is applied as the following definitions and conditions:

    ξ(t)=[ηm1ηm2ηm3|ηm1|32sgn(ηm1)|ηm1|54sgn(ηm1)|ηm1|34sgn(ηm1)|ηm1|12sgn(ηm1)]

    Based on the Lyapunov stability principle, the solutions of the inequality Q0 represent the appropriate positive definite constants k1k9 and parameters βm1, βm2, and βm3. Therefore, the estimation stability of error system (10) is assured because the estimation error converges to the neighborhood of the origin.

    In this section, the estimations of interference factors obtained by the nonlinear ESO are used for compensation in the BLF-based controller in conjunction with the constraints on the state variables. Here, the controllers are designed separately for the master and slave manipulators because these are heterogeneous systems.

    For the master manipulator system (1) with the applied correspondences (3), the position and speed Xmj, j = 1, 2 of the manipulator system are assumed to be bounded in actual applications, and satisfy |Xmj|<kmj, where kmj is a positive definite constant. The errors associated with Xm1 and Xm2 are assumed to be respectively definable as χm1=Xm1Xmd and χm2=Xm2αm1, where Xmd=qs(tTs) is the tracking trajectory of the master manipulator, that is, the reference signal, Ts is the transmission delay of the signal from the end to the master, and αm1 is the virtual controller. In this study, the virtual controller is an intermediate design variable used to simplify the control problem of complex systems by decomposing high-order systems into lower-order subsystems for step-by-step design and analysis. These assumptions are specified and validated as follows.

    Assumption 2: Assume that appropriate positive definite constants kmj exist that ensure that |χmj|<kmj is satisfied.

    Assumption 3: Assume that appropriate positive definite constants Am0 and Am1 exist that can guarantee that the conditions |Xmd|Am0<km1 and |˙Xmd|Am1 are satisfied.

    Step 1: Select the following positive definite candidate BLF:

    Vm1=12log(k2m1k2m1χ2m1) (22)

    Taking the derivative of Vm1 with respect to time yields the following:

    ˙Vm1=χm1˙χm1k2m1χ2m1=kχm1(χm2+αm1˙Xmd) (23)

    Here, kχm1=χm1k2m1χ2m1. The virtual controller αm1 is designed as follows:

    αm1=Km1χm1+˙Xmd (24)

    Substituting (24) in (23) yields the following:

    ˙Vm1=kχm1Km1χm1+kχm1χm2 (25)

    Step 2: Select the following positive definite candidate BLF:

    Vm=Vm1+12log(k2m2k2m2χ2m2) (26)

    According to (3) and (25), taking the derivative of Vm with respect to time yields the following.

    ˙Vm=˙Vm1+χm2˙χm2k2m2χ2m2=kχm1Km1χm1+kχm1χm2+kχm2(˙Xm2˙αm1)=kχm1Km1χm1+kχm1kχm2(k2m2χ2m2)+kχm2(M1mτm+Fm˙αm1) (27)

    Here, kχm2=χm2k2m2χ2m2. The derivative of αm1 with respect to time is approximated by the tracking differentiator. The tracking differentiator is an algorithm for estimating signal derivatives based on PID control and improved by Han [18], which can accurately track and output the differential value of signals in the presence of noise. The details are as follows:

    ˙vm1=vm2˙vm2=r|vm1vm0|12sign(vm1vm0)vm2 (28)

    Setting vm0=αm1 indicates that vm1 and vm2 are the estimated values of αm1 and its derivative ˙αm1, respectively (i.e., ˆαm1=vm1 and ˆ˙αm1=vm2). Therefore, the estimation error ˜˙αm1=˙αm1ˆ˙αm1 is assumed to be bounded, and satisfies the condition |˜˙αm1|μm1, where μm1 is an appropriate positive definite constant. Finally, the input controller τm of the master manipulator is designed as:

    τm=Mm[Km2χm2kχm2(k2m2χ2m2)ˆFm+ˆ˙αm1] (29)

    where ˆFm is the value of Fm estimated by the nonlinear ESO presented in the previous section. Therefore, (27) can be expressed as follows.

    ˙Vm=kχm1Km1χm1kχm2Km2χm2+kχm2˜Fm+kχm2˜˙αm1 (30)

    where ˜Fm is the estimation error of Fm estimated by the nonlinear ESO presented in the previous section. For the slave manipulator system (2) with the applied correspondences (4), the position, angular position, speed, and angular speed Xsj, j = 1, 2, 3, 4 of the system are assumed to be bounded in practical applications, and satisfy the condition |Xsj|<ksj, where ksj is a positive definite constant. The errors in Xs1, Xs2, Xs3, and Xs4 are assumed to be respectively definable as χs1=Xs1Xsd, χs2=Xs2αs1, χs3=Xs3αs2, and χs4=Xs4αs3, where the reference signal Xsd=qm(tTm) is the tracking trajectory of the slave manipulator that accounts for the transmission delay Tm of the signal from the master manipulator to the slave manipulator, and the control signal αsj,j=1,2,3 of the virtual controller. These assumptions are specified and validated as follows.

    Assumption 4: Assume that appropriate positive definite constants ksj exist that ensure that |Xsj|<ksj is satisfied.

    Assumption 5: Assume that appropriate positive definite constants As0 and As1 exist that ensure that the conditions |Xsd|As0<ˉks1 and |˙Xsd|As1 are satisfied.

    Step 1: Select the following positive definite candidate BLF:

    Vs1=12log(k2s1k2s1χ2s1) (31)

    The derivative of Vs1 with respect to time can be obtained as follows:

    ˙Vs1=χs1˙χs1k2s1χ2s1=kχs1(χs2+αs1˙Xsd) (32)

    Here, kχs1=χs1k2s1χ2s1, while the other terms kχsj=χsjk2sjχ2sj,j=2,3,4, will be used below. The virtual controller αs1 is designed as follows.

    αs1=Ks1χs1+˙Xsd (33)

    Substituting (33) in (32) yields the following:

    ˙Vs1=kχs1Ks1χs1+kχs1χs2 (34)

    Step 2: Select the following positive definite candidate BLF:

    Vs2=Vs1+12log(k2s2k2s2χ2s2) (35)

    According to Eqs (4) and (34), taking the derivative of Vs2 with respect to time yields the following:

    ˙Vs2=˙Vs1+χs2˙χs2k2s2χ2s2=kχs1Ks1χs1+kχs1χs2+kχs2(˙Xs2˙αs1)=kχs1Ks1χs1+kχs1kχs2(k2s2χ2s2)+kχs2(M1sKs(χs3+αs2Xs1)+Fs˙αs1) (36)

    Again, the estimated values of αs1 and ˙αs1 are respectively obtained from vs1 and vs2 (i.e., ˆαs1=vs1 and ˆ˙αs1=vs2) using the tracking differentiator (28) with vs0=αs1. Therefore, the estimation error ˜˙αs1=˙αs1ˆ˙αs1 is assumed to be bounded and satisfies |˜˙αs1|μs1, where μs1 is an appropriate positive definite constant. Then, the virtual controller αs2 of the slave manipulator is designed as:

    αs2=1KsMs[Ks2χs2+ˆ˙αs1kχs2(k2s2χ2s2)ˆFs]+Xs1 (37)

    where ˆFs is the value of Fs estimated by the nonlinear ESO. Therefore, substituting (37) in (36) yields.

    ˙Vs2=kχs1Ks1χs1kχs2Ks2χs2+M1sKskχs2χs3+kχs2˜Fs+kχs2˜˙αs1 (38)

    where ˜Fs is the estimation error of Fs estimated by the nonlinear ESO presented in the previous section.

    Step 3: Select the following positive definite candidate BLF:

    Vs3=Vs2+12log(k2s3k2s3χ2s3) (39)

    According to Eqs (4) and (38), the derivative of Vs3 with respect to time can be obtained as follows:

    ˙Vs3=˙Vs2+χs3˙χs3k2s3χ2s3=kχs1Ks1χs1kχs2Ks2χs2+M1sKskχs2χs3+kχs2˜Fs+kχs2˜˙αs1+kχs3(χs4+αs3˙αs2) (40)

    The values of ˆαs2=vs1 and ˆ˙αs2=vs2 are obtained using the tracking differentiator (28) with vs0=αs2. Therefore, the estimation error ˜˙αs2=˙αs2ˆ˙αs2 is assumed to be bounded and satisfies |˜˙αs2|μs2, where μs2 is an appropriate positive definite constant. Then, the virtual controller αs3 of the slave manipulator is designed as follows:

    αs3=Ks3χs3+ˆ˙αs2M1sKskχs2(k2s3χ2s3) (41)

    Therefore, (40) can be expressed as follows:

    ˙Vs3=kχs1Ks1χs1kχs2Ks2χs2kχs3Ks3χs3+kχs3χs4+kχs2˜Fs+kχs2˜˙αs1+kχs3˜˙αs2 (42)

    Step 4: Select the following positive definite candidate BLF:

    Vs=Vs3+12log(k2s4k2s4χ2s4) (43)

    According to Eqs (4) and (42), the derivative of Vs with respect to time can be obtained as follows:

    ˙Vs=˙Vs3+χs4˙χs4k2s4χ2s4=kχs1Ks1χs1kχs2Ks2χs2kχs3Ks3χs3+kχs3kχs4(k2s4χ2s4)+kχs2˜Fs+kχs2˜˙αs1+kχs3˜˙αs2+kχs4(J1sτs+Ff˙αs3) (44)

    The values of ˆαs3=vs1 and ˆ˙αs3=vs2 are obtained using the tracking differentiator (28) with vs0=αs3. Therefore, the estimation error ˜˙αs3=˙αs3ˆ˙αs3 is assumed to be bounded and satisfies |˜˙αs3|μs3, where μs3 is an appropriate positive definite constant. Then, the input controller τs of the slave manipulator is designed as:

    τs=Js(Ks4χs4+ˆ˙αs3kχs3(k2s4χ2s4)ˆFf) (45)

    where ˆFf is the value of Ff estimated by the nonlinear ESO. Therefore, (44) can be expressed as:

    ˙Vs=kχs1Ks1χs1kχs2Ks2χs2kχv3Ks3χs3kχs4Ks4χs4+kχs2˜Fs+kχs2˜˙αs1+kχs3˜˙αs2+kχs4˜˙αs3+kχs4˜Ff (46)

    where ˜Ff is the estimation error of Ff estimated by the nonlinear ESO presented in the previous section. The control block diagram of the teleoperation system with controllers (29) and (45) is presented in Figure 1. Figure 1 delineates the advanced teleoperation control framework between a Master and Slave robot. Here, both robots employ ESOs to estimate uncertainties and a BLF-based controller for precision. The network delay in command transmission is explicitly represented, showcasing the integrated approach in handling flexible joint uncertainties. Both robots operate within full state constraints, highlighting the strategy's emphasis on synchronized and reliable robotic interactions.

    Figure 1.  Control block diagram of the teleoperation system with input controllers τm and τs.

    Theorem 2: Assume that the teleoperation system satisfies Assumptions 2–5. Then, under the virtual controller αij, where j = 1, 2 when i = m, and j = 1, 2, 3, 4 when i = s, and the input controllers τm and τs, all variables of the closed-loop system are bounded and do not exceed their constraints if two positive definite constants a and b exist that satisfy the following conditions.

    a=min (47)

    Proof: Choose the following Lyapunov function:

    V = {V_{\text{m}}} + {V_{\text{s}}} (48)

    When \left| {{\chi _{ij}}} \right| < {k_{ij}} , positive definite constants must exist that satisfy the condition \frac{{k_{ij}^2}}{{k_{ij}^2 - \chi _{ij}^2}} > \frac{{\chi _{ij}^2}}{{k_{ij}^2 - \chi _{ij}^2}} . Therefore, according to Eqs (30) and (46), the derivative of V with respect to time can be obtained as follows:

    \dot V \leqslant - aV + b (49)

    Multiplying both sides of inequality (49) by {e^{at}} yields the following form:

    {{d\left( V \right){e^{at}}} \mathord{\left/ {\vphantom {{d\left( V \right){e^{at}}} {dt}}} \right. } {dt}} \leqslant b{e^{at}} (50)

    Furthermore, integrating (50) over the interval \left[{0, t} \right] yields the following:

    V\left( t \right) \leqslant \left( {V\left( 0 \right) - \frac{b}{a}} \right){e^{ - at}} + \frac{b}{a} \leqslant V\left( 0 \right) + \frac{b}{a} (51)

    Given that {\hat F_{\text{m}}} , {\hat F_{\text{s}}} , {\hat F_{\text{f}}} and {\hat \alpha _{ij}} are bounded, \left| {{X_{i1}}} \right| < {k_{i1}} + {A_{i0}} < {\bar k_{i1}} can be obtained from {\chi _{i1}} = {X_{i1}} - {X_{i{\text{d}}}} and {X_{i{\text{d}}}} \leqslant {A_{i0}} . The boundedness of {\chi _{i1}} and {\dot X_{i{\text{d}}}} ensures that {\alpha _{i1}} is bounded and {\alpha _{i1}} \leqslant {\bar \alpha _{i1}} . Similarly, \left| {{X_{i2}}} \right| < {k_{i2}} + {\bar \alpha _{i1}} < {k_{i2}} can be obtained from {\chi _{i2}} = {X_{i2}} - {\alpha _{i1}} , and \left| {{X_{{\text{s}}3}}} \right| < {k_{{\text{s}}3}} , and \left| {{X_{{\text{s}}4}}} \right| < {k_{{\text{s}}4}} can be obtained in the same way. Furthermore, the input controller signals {\tau _{\text{m}}} and {\tau _{\text{s}}} are bounded according to their definitions in Eqs (29) and (45), respectively, where {\tau _{\text{m}}} is a function of {\chi _{{\text{m}}2}} , {\hat F_{\text{m}}} , and {\widehat {\dot \alpha }_{{\text{m}}1}} , and {\tau _{\text{s}}} is a function of {\chi _{{\text{s}}4}} , {\hat F_{\text{f}}} , and {\widehat {\dot \alpha }_{{\text{s}}3}} . Therefore, all signals of the system, namely {\tau _{\text{m}}} , {\tau _{\text{s}}} , and {X_{ij}} , are bounded and meet their constraints.

    The experimental teleoperation platform employed for testing the effectiveness of the proposed control strategy is presented in Figure 2, and consisted of two Phantom Premium 1.5HF (SensAble Technologies, Inc.) three-degrees-of-freedom robotic arms. The Phantom Premium 1.5HF arm has three rotating joints, each consisting of three parts of a gear motor encoder, which can feel or follow the movement of the controlled object on three perpendicular axes in space. The controllers and all software were implemented in MATLAB, and the flexible joint from the slave manipulator was realized by a virtual module in MATLAB.

    Figure 2.  Experimental platform employed for testing the teleoperation system.

    Angle sensors were installed at each joint on the master and slave end of the three-degree-of-freedom manipulator to measure the required angle. For the process of experimental verification, the delays were set to Tm = Ts = 200 ms. The appropriate parameters were obtained through experiments and set as: \varepsilon = 1.05, {\beta _{{\text{m}}1}} = {\beta _{{\text{s}}1}} = \left[{\begin{array}{*{20}{c}} {210} & 0 & 0 \\ 0 & {250} & 0 \\ 0 & 0 & {200} \end{array}} \right], {\beta _{{\text{m}}2}} = {\beta _{{\text{s}}2}} = \left[{\begin{array}{*{20}{c}} {2.5} & 0 & 0 \\ 0 & {1.5} & 0 \\ 0 & 0 & {2.5} \end{array}} \right], {\beta _{{\text{m}}3}} = {\beta _{{\text{s}}3}} = \left[{\begin{array}{*{20}{c}} {0.0015} & 0 & 0 \\ 0 & {0.0025} & 0 \\ 0 & 0 & {0.0025} \end{array}} \right]. The parameters of the master controller are selected as: {K_{{\text{m}}1}} = \left[{\begin{array}{*{20}{c}} 5 & 0 & 0 \\ 0 & {7.8} & 0 \\ 0 & 0 & {6.8} \end{array}} \right], {K_{{\text{m}}2}} = \left[{\begin{array}{*{20}{c}} {0.71} & 0 & 0 \\ 0 & {0.3} & 0 \\ 0 & 0 & {0.3} \end{array}} \right], {k_{{\text{m}}1}} = \left[{\begin{array}{*{20}{c}} {1.4} \\ {1.4} \\ {1.3} \end{array}} \right], {k_{{\text{m}}2}} = \left[{\begin{array}{*{20}{c}} {1.5} \\ {2.8} \\ {1.5} \end{array}} \right]. The stiffness coefficient for the flexible joint of the slave manipulator was set as {K_{\text{s}}} = \left[{\begin{array}{*{20}{c}} {12} & 0 & 0 \\ 0 & {12} & 0 \\ 0 & 0 & {12} \end{array}} \right], while the slave controller parameters were set as {K_{{\text{s}}1}} = \left[{\begin{array}{*{20}{c}} {1.1} & 0 & 0 \\ 0 & {0.9} & 0 \\ 0 & 0 & {0.9} \end{array}} \right], {K_{{\text{s}}2}} = \left[{\begin{array}{*{20}{c}} {0.7} & 0 & 0 \\ 0 & {0.5} & 0 \\ 0 & 0 & {0.7} \end{array}} \right], {K_{{\text{s}}3}} = \left[{\begin{array}{*{20}{c}} {3.1} & 0 & 0 \\ 0 & {0.9} & 0 \\ 0 & 0 & {0.9} \end{array}} \right], {K_{{\text{s}}4}} = \left[{\begin{array}{*{20}{c}} {14.5} & 0 & 0 \\ 0 & {19.5} & 0 \\ 0 & 0 & {19.5} \end{array}} \right], {k_{{\text{s}}1}} = \left[{\begin{array}{*{20}{c}} {0.4} \\ {0.5} \\ {0.5} \end{array}} \right], {k_{{\text{s}}2}} = \left[{\begin{array}{*{20}{c}} {0.5} \\ {0.5} \\ {0.5} \end{array}} \right], {k_{{\text{s}}3}} = \left[{\begin{array}{*{20}{c}} {0.4} \\ {0.4} \\ {0.4} \end{array}} \right], {k_{{\text{s}}4}} = \left[{\begin{array}{*{20}{c}} {0.5} \\ {0.3} \\ {0.3} \end{array}} \right].

    In the experimental evaluation, Figures 35 delve into the position estimation errors of both the master and slave manipulators over three degrees of freedom. Specifically, Figure 3 showcases the master manipulator's position errors, which consistently remain below ± 0.02 rad. Figures 4 and 5, representing the slave manipulator's errors at its motors and chain rod respectively, confirm errors below ± 0.05 rad. These figures underline the nonlinear ESO's superior capability in accurately estimating the positions of both manipulators.

    Figure 3.  Position estimation error of the master manipulator.
    Figure 4.  Position estimation error of the slave manipulator at the motor.
    Figure 5.  Position estimation error of the slave manipulator at the chain rod.

    Figure 6, illustrating the control inputs, highlights a synchronized control approach between the master and slave manipulators. Their control signals exhibit sine wave-like consistencies, reinforcing the effectiveness of the implemented synchronization.

    Figure 6.  Control input from the master and the slave manipulator. (In the figure, the solid line represents the master manipulator and the dotted line represents the slave manipulator.

    The master and slave manipulator positions over time are depicted in Figure 7. Notably, their positions mirror each other closely, with a mere average deviation of ± 0.02 rad as substantiated in Figure 8. Upon halting the manipulator motion after 50 seconds, the tracking error rapidly zeroes out, underscoring the controller's adeptness in position synchronization.

    Figure 7.  Positions of the master and slave manipulators over time. (In the figure, the solid line represents the master manipulator and the dotted line represents the slave manipulator.
    Figure 8.  Tracking errors between the positions of the master and slave manipulators.

    Figures 9 and 10 further emphasize the system's stability. Despite initially larger tracking errors, they swiftly reduce to a maximum of ± 0.05 rad and approach zero when the motion is halted after 50 seconds, indicating minimal influence from external forces.

    Figure 9.  Motor and chain rod positions of the slave manipulator over time. (The solid and dotted lines in the figure represent motor and chain rod positions of the slave manipulator, respectively.
    Figure 10.  Tracking errors between the positions of the motor and chain rod of the slave manipulator.

    To summarize, these experimental findings bolster the merits of the proposed control strategy. It not only achieves precise position estimations and effective synchronization, but also remains resilient against potential disturbances. This culmination holds significant promise for enhancing teleoperation systems in intricate operational landscapes.

    This research marks a significant stride in the realm of teleoperation systems specifically designed for robotic manipulators with flexible joints. Our primary focus revolves around investigating the feasibility of a pose control strategy for teleoperated systems with flexible joints. This strategy is anchored on the integration of a nonlinear Extended State Observer (ESO) with a Barrier Lyapunov Function (BLF).

    The efficacy of this novel approach is convincingly demonstrated by our experimental findings. Specifically, in a master-slave operating system, the slave exhibits exceptional synchronization capabilities with high precision. The position estimation errors in the master-slave operation consistently remain within ± 0.05 radians, particularly at the chain rod where it is even less than ± 0.01 radians. Even in complex scenarios involving flexible joints, this level of precision showcases the robustness and potential of our integration strategy.

    It is essential to highlight that the crux of this study was to explore the feasibility of the aforementioned control strategy. Therefore, instead of comparing it with existing methodologies, we have dedicated our efforts to comprehensively understand and elucidate its intrinsic merits.

    As we pave the path forward, we envision delving deeper into enhancing the nuances of our approach. While we have not directly compared our method with industry benchmarks in this study, future endeavors might involve such comparative analyses. Nonetheless, our primary goal remains: to further elucidate, validate, and promote the unique potential and adaptability of our teleoperation strategy across diverse operational scenarios.

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

    This paper is supported by National Natural Science Foundation of China under grant No. 52305066.

    The authors declare there is no conflict of interest.



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