Vibration suppression and coordination control of bilateral teleoperation system for flexible-link manipulator with communication delay

Abstract

Bilateral teleoperation (BT) systems can expand the working capabilities of human operators in remote, unstructured, and hazardous environments, and can be applied in a wide range of fields such as nuclear detection, surgery, mining, outer space research, and military operations. At present, there is little research on the control theory of flexible-link BT systems, and communication delay issues have not been considered. However, in practical terms, transmission delay in communication networks is a major problem in BT systems, and even small time delay often reduce system performance and lead to system instability. This paper takes the bilateral teleoperation system composed of master and slave flexible-link manipulators (MSFLM-BT) as the research object, and designs corresponding control scheme based on partial differential equation (PDE) dynamic model in the presence of communication delay, successfully achieving vibration suppression and angle coordination tracking of the system. The stability of the system is proven using the Lyapunov method, and under this control scheme, all closed-loop signals are bounded. Finally, the effectiveness of the proposed control scheme is verified through numerical simulation.

Keywords

Bilateral teleoperation system flexible-link manipulator communication delay vibration suppression angle coordination tracking

1. Introduction

Compared to traditional rigid mechanical systems, flexible mechanical systems have the characteristics of light weight, low energy consumption, fast speed, and high flexibility, and are widely used in service industry, manufacturing industry, aerospace, and other fields. Therefore, the modeling and control of flexible mechanical systems have become one of the key research topics in the field of control in recent years Liu et al. (2023a, 2022a); Zhou et al. (2021); Liu et al. (2022c, 2019, 2022b, 2023b); Zhao et al. (2020); Liu et al. (2023c, 2021), and Ji and Liu (2023). The motion characteristics of flexible mechanical systems are related to both time and position variables, making it a typical distributed parameter system with infinite dimensional characteristics. The establishment of an accurate dynamic model is an important prerequisite for the design of control methods. At present, the models established for flexible systems are divided into ordinary differential equation (ODE) model and partial differential equation (PDE) model. ODE model is a simplified model, which only retains the key modes and ignores the high-order modes Rasouli et al. (2021, 2020), which is easy to cause the observation or control overflow problem Zhao et al. (2020), and then affect the stability of the system. While the PDE model can accurately describe the infinite dimensional characteristics of the system, retain all the modes of the system, and effectively avoid the overflow problem that may occur based on the ODE approximate model Meirovitch and Baruh (1983) and Liu et al. (2022c). In addition, flexible mechanical systems typically use new types of flexible lightweight materials, which are prone to elastic deformation and vibration during motion Liu et al. (2022c); Zhao et al. (2020), and Ji and Liu (2024). These vibrations often interact with the system’s motion, seriously affecting its working state and even damaging its stability, causing the entire system to lose control. This hinders the practical application of flexible mechanical systems in engineering. Therefore, the research on control theory of flexible mechanical systems is of great significance.

However, in reviewing the relevant literature on modeling and control theory of flexible mechanical systems, the bilateral teleoperation (BT) system composed of master-slave flexible mechanical systems has received little attention. A BT system is a master-slave robotic framework characterized by bidirectional energy/information exchange, enabling the operator to control the remote slave while receiving feedback (e.g., position/force) from slave-side environment Tian et al. (2025) and Nahri et al. (2022). BT systems can enable humans to complete tasks without exposure to hazardous environments, and can be applied in a wide range of fields such as nuclear detection, surgery, mining, outer space research, and military operations Jiang et al. (2022); Nguyen et al. (2023); Gu et al. (2024b); Cruz Ortiz et al. (2020), and Chen et al. (2020). In recent decades, research on coordinated control of BT systems has achieved fruitful results Han et al. (2010); Jung and Jeon (2022); Gu et al. (2024a); Cruz-Ortiz et al. (2022); Wang et al. (2022); Peñaloza-Mejía et al. (2017); Wang et al. (2014); Yang et al. (2019), and Wang et al. (2018). Unfortunately, most research is focused on rigid mechanical systems.

The search terms “bilateral teleoperation” and “flexible” were employed to retrieve the 15 most relevant studies from the Web of Science database. After systematic analysis and categorization, the results are summarized in Table 1. Notably, joint flexibility lacks the infinite-dimensional characteristics discussed in the first paragraph, whereas link flexibility inherently exhibits such properties. To avoid terminological ambiguity, the term “flexible-link BT system” is adopted henceforth. As shown in Table 1, there are only three studies on the control theory of BT systems where both the master and slave are flexible-link mechanical systems Li et al. (2021); Wang and Liu (2022), and Cao and Liu (2021). Based on the PDE dynamic model of the bilateral teleoperation system composed of master-slave flexible-link manipulators (MSFLM-BT), reference Li et al. (2021) designs a bilateral coordinated controller to achieve vibration suppression and angle coordinated tracking of the system. In reference Wang and Liu (2022), Wang et al. use a random quantizer to quantize the input signal and propose an adaptive boundary bilateral coordinated quantization control scheme, which achieved adaptive tracking control of the MSFLM-BT system with input signal quantization. Reference Cao and Liu (2021) proposes a novel master-slave system coordination controller based on the nonlinear PDE model of a master-slave two-link rigid-flexible manipulator, which achieves angle coordination tracking and vibration suppression of the system.

Table 1.

Comparison of relevant references on the flexible BT system.

Master system + Slave system	Reference	Upper bound of time-delay
Link flexibility^a (PDE^b)+link flexibility^a (PDE^b)	Li et al. (2021); Wang and Liu (2022); Cao and Liu (2021)	Time-delay is not considered
Rigid + link flexibility^a (PDE^b)	Yaryan et al. (2013); Yagi et al. (2013)	0.5s, 0.2s
Rigid + link flexibility^a (ODE^b)	Rashidinejad et al. (2015); Rasouli et al. (2021); Rasouli et al. (2020); Yaryan et al. (2012)	∖^c ,0.7s, 0.7s,∖^c
Rigid + joint flexibility^a	Nuño et al. (2013); Nuño et al. (2012); Yang et al. (2018); Li et al. (2019); Nuño et al. (2014)	0.42s, 0.55s, 0.3s, 1.06s, 0.55s
Joint flexibility^a + joint flexibility^a	Ye et al. (2014)	0.6s

^aThe flexibility of flexible manipulator is mainly manifested in the flexibility of joint and link. Joint flexibility refers to the twisting deformation of the transmission mechanism and joint shaft of the manipulator, while link flexibility refers to the elastic deformation of the link of the manipulator.

^bPDE/ODE means that the flexible system is modeled as a PDE/ODE model in the reference.

^c∖ means that the upper bound of time-delay is not found in the reference.

However, the above three references do not consider the issue of communication delay. In practical terms, transmission delay in communication networks is a major problem in BT systems. Even small time delays often reduce system performance and lead to system instability, which has become one of the complex problems that researchers urgently need to solve Chen et al. (2020); Forouzantabar et al. (2012), and Li et al. (2013). At present, there have been many studies on the communication delay problem of BT systems Chen et al. (2020); Sun et al. (2016b); Zhang et al. (2022); Sun et al. (2020); Liu and Hu (2019); Sun et al. (2016a); Daly and Wang (2014); Sun et al. (2017); Mohammadi et al. (2017); Gong et al. (2022), and Gong and Ji (2022). In reference Chen et al. (2020), an adaptive robust control scheme based on radial basis function neural network is proposed to ensure the global stability of BT robot system under time-delay. Based on a four channel nonlinear teleoperation system, reference Sun et al. (2016b) proposes a novel time-domain passivity method based on waves, which can ensure system stability in the presence of random time delays. Reference Zhang et al. (2022) applies adaptive control method to estimate the upper bound of time delay for a class of nonlinear BT systems, in order to solve the problem of the significant impact of time delay on system stability. However, these studies are all based on the ODE model for control scheme design, and there is currently no research on the communication delay problem of flexible-link BT systems based on the PDE model. The infinite-dimensional nature of PDE models hinders the direct application of control methods developed for the rigid mechanical systems to flexible mechanical systems. Additionally, it poses significant challenges to the controller design and stability analysis of flexible systems Liu et al. (2023b). This paper focuses on the control theory research for the MSFLM-BT system with the objectives of achieving vibration suppression and angular coordination tracking of master-slave flexible-link manipulators under communication delays. The master manipulator operates in a safe environment to track a desired angle, while the slave manipulator, located in a remote workspace, synchronizes its angular position with the master.

To sum up, this paper is based on the PDE model of MSFLM-BT system, and studies the control scheme in the presence of communication delay between the master and slave flexible-link manipulators. The difficulty of this study can be summarized as follows: (1) The vibration of flexible-link manipulators will affect the angle coordination tracking of the system. (2) At present, there is no research on the communication delay problem of flexible-link BT system based on PDE model. (3) The PDE model of MSFLM-BT system is infinite dimensional, which will bring difficulties to the design and stability analysis of the controller. Therefore, the vibration suppression and angle coordination tracking of MSFLM-BT system in the presence of communication delay is a challenging and meaningful topic, and it is of great significance for the application of flexible-link BT system in practical engineering.

The remainder of this paper is structured as follows: Section 2 presents system modeling and essential preliminaries. Section 3 provides an overview of the control system design and a detailed stability analysis. In Section 4, the effectiveness of the control scheme is verified by numerical simulation. The conclusion of this paper is in Section 5.

2. Modeling and preliminaries

2.1. Modeling

The controller is designed based on the PDE dynamic model of the MSFLM-BT system established in reference Li et al. (2021). The model structure of the system is shown in Figure 1, in which the master manipulator is in a safe environment and tracks the given angle θ_d, and the slave manipulator is located in a remote environment and tracks at the same angle as the master manipulator. In Table 2, the model parameters are elucidated.

Figure 1.

Structure of MSFLM-BT system.

Table 2.

Model parameters.

Variable	Meaning
L	Length of the flexible-link manipulator
ρ	Linear density of the flexible-link manipulator
m	Mass of the end payload
EI	Bending stiffness of the flexible-link manipulator
I _h	Inertia of the hub
θ_i(t)^a	Joint angle
w_i(x_i, t)^a	Elastic deflection
u_i1(t)^a	Control input signal acting on the hub
u_i2(t)^a	Control input signal acting on the end payload

^ai = m, s, m refers to the master flexible-link manipulator and s refers to the slave flexible-link manipulator.

Remark 1

To enhance clarity, the subsequent notations are employed throughout this paper: $*^{'} = \partial * / \partial x_{i}$ , $*^{''} = {\partial *}^{2} / \partial x_{i}^{2}$ , $*^{'''} = {\partial *}^{3} / \partial x_{i}^{3}$ , $*^{''''} = {\partial *}^{4} / \partial x_{i}^{4}$ , $\dot{*} = \partial * / \partial t$ , $\ddot{*} = {\partial *}^{2} / \partial t^{2}$ .

The displacement of the master and slave flexible-link manipulators can be expressed as follows

z_{i} (x_{i}, t) = x_{i} θ_{i} (t) + w_{i} (x_{i}, t)

(1)

It is easy to obtain $\partial^{n} z_{i} (x_{i}, t) / \partial {x_{i}}^{n} = \partial^{n} w_{i} (x_{i}, t) / \partial {x_{i}}^{n}, n \geq 2$ . Considering the dynamics of flexible-link manipulator, the boundary conditions are obtained as follows: $w_{i} (0, t) = w_{i}^{'} (0, t) = 0$ , z_i(0,t) = 0, $z_{i}^{'} (0, t) = θ_{i} (t)$ .

According to the reference Li et al. (2021), we get the PDE model of MSFLM-BT system. It can be seen that vibration and angle are coupled

ρ {\ddot{z}}_{i} (x_{i}, t) = - E I z_{i}^{''''} (x_{i}, t)

(2)

u_{i 1} (t) = I_{h} {\ddot{θ}}_{i} (t) - E I z_{i}^{''} (0, t)

(3)

u_{i 2} (t) = m {\ddot{z}}_{i} (L, t) - E I z_{i}^{‴} (L, t)

(4)

z_{i}^{''} (L, t) = 0

(5)

where i = m, s, m refers to the master flexible-link manipulator and s refers to the slave flexible-link manipulator.

2.2. Preliminaries

Lemma 1

Rahn (2001) For $ϕ_{1} (l, t), ϕ_{2} (l, t) \in R, l \in [0, L], t \in [0, \infty)$ , there exists

ϕ_{1} (l, t) ϕ_{2} (l, t) \leq \frac{1}{2} {ϕ_{1}}^{2} (l, t) + \frac{1}{2} {ϕ_{2}}^{2} (l, t)

(6)

ϕ_{1} (l, t) ϕ_{2} (l, t) \leq \frac{1}{2 v} {ϕ_{1}}^{2} (l, t) + \frac{v}{2} {ϕ_{2}}^{2} (l, t)

(7)

where v > 0

Lemma 2

Hardy et al. (1952) For $ϕ (l, t) \in R, t \in [0, \infty), l \in [0, L]$ , if ∀t ∈ [0, ∞), ϕ(0, t) = 0, then

ϕ^{2} (l, t) \leq L \int_{0}^{L} {ϕ^{'}}^{2} (l, t) d l, \forall l \in [0, L]

(8)

If ϕ′(0, t) = 0, ∀t ∈ [0, ∞), then

\int_{0}^{L} {ϕ^{'}}^{2} (l, t) d l \leq L^{2} \int_{0}^{L} {ϕ^{''}}^{2} (l, t) d l

(9)

Property 1

Queiroz et al. (2000) If the kinetic energy is bounded for ∀t ∈ [0, + ∞), then ${\dot{w}}_{i} (x_{i}, t)$ , ${\dot{w}}_{i}^{'} (x_{i}, t)$ , ${\dot{w}}_{i}^{''} (x_{i}, t)$ and ${\dot{w}}_{i}^{'''} (x_{i}, t)$ are bounded for ∀(x_i, t) ∈ [0, L] × [0, + ∞). The kinetic energy of the system is $E_{k i} = I_{h} {\dot{θ}}_{i}^{2} / 2 + ρ \int_{0}^{L} {\dot{z}}_{i}^{2} (x_{i}, t) d x_{i} / 2 + m {\dot{z}}_{i}^{2} (L, t) / 2$ Li et al. (2021).

Property 2

Queiroz et al. (2000) If the potential energy is bounded for ∀t ∈ [0, + ∞), then $w_{i}^{''} (x_{i}, t)$ , $w_{i}^{'''} (x_{i}, t)$ and $w_{i}^{''''} (x_{i}, t)$ are bounded ∀(x_i, t) ∈ [0, L] × [0, + ∞). The potential energy of the system is $E_{p i} = E I \int_{0}^{L} {w_{i}^{''}}^{2} (x_{i}, t) d x_{i} / 2$ Li et al. (2021).

3. Controller design

Firstly, the angle tracking errors of the master flexible-link manipulator and the slave flexible-link manipulator are defined as ɛ_m(t) = θ_m(t) − θ_d and ɛ_s(t) = θ_s(t) − θ_d, respectively.

In order to reduce the impact of communication delay on the MSFLM-BT system, the following auxiliary systems are introduced

{\dot{ζ}}_{m} (t) = - [θ_{m} (t) - θ_{s} (t - τ_{m s})]

(10)

{\dot{ζ}}_{s} (t) = - [θ_{s} (t) - θ_{m} (t - τ_{s m})]

(11)

{\tilde{ζ}}_{m} (t) = ζ_{m} (t) - θ_{d}

(12)

{\tilde{ζ}}_{s} (t) = ζ_{s} (t) - θ_{d}

(13)

where τ_ms refers to the communication delay from the slave to the master. τ_sm refers to the communication delay from the master to the slave.

To design the control law, we introduce the following auxiliary signals

η_{m} (t) = e_{m} (t) + {\dot{e}}_{m} (t)

(14)

η_{s} (t) = e_{s} (t) + {\dot{e}}_{s} (t)

(15)

μ_{m} (t) = w_{m}^{'} (L, t) + {\dot{z}}_{m} (L, t) - w_{m}^{‴} (L, t)

(16)

μ_{s} (t) = w_{s}^{'} (L, t) + {\dot{z}}_{s} (L, t) - w_{s}^{‴} (L, t)

(17)

where e_m(t) = θ_m(t) − θ_d = ɛ_m(t),

e_{s} (t) = θ_{s} (t) - ζ_{s} (t) = ε_{s} (t) - {\tilde{ζ}}_{s} (t)

The control law is

\begin{matrix} u_{m 1} (t) = - k_{m} η_{m} (t) - r_{m 1} e_{m} (t) - E I z_{m}^{''} (0, t) \\ - (I_{h} + r_{m 2}) {\dot{e}}_{m} (t) + ω_{m} {\dot{ζ}}_{m} (t) \end{matrix}

(18)

\begin{matrix} u_{m 2} (t) = - π_{m} μ_{m} (t) - m {\dot{w}}_{m}^{'} (L, t) \\ - E I z_{m}^{‴} (L, t) + m {\dot{w}}^{‴}_{m} (L, t) \end{matrix}

(19)

\begin{matrix} u_{s 1} (t) = - k_{s} η_{s} (t) - (r_{s 1} + \frac{1}{2}) e_{s} (t) \\ - (I_{h} + r_{s 2} + \frac{1}{2}) {\dot{e}}_{s} (t) - E I z_{s}^{''} (0, t) + I_{h} {\ddot{ζ}}_{s} (t) \end{matrix}

(20)

\begin{matrix} u_{s 2} (t) = - π_{s} μ_{s} (t) - m {\dot{w}}_{s}^{'} (L, t) \\ - E I z_{s}^{'''} (L, t) + m {\dot{w}}^{‴}_{s} (L, t) \end{matrix}

(21)

where k_m, r_m1, r_m2, ω_m, π_m, k_s, r_s1, r_s2, π_s are control parameters.

In Figure 2, the structural diagram for designed controller is illustrated. In this figure, $θ_{m} (t - τ_{s m})$ represents the slave side obtains the angle of the master side from the communication network with a delay of τ_sm, and $θ_{s} (t - τ_{m s})$ represents the master side obtains the angle of the slave side from the communication network with a delay of τ_ms.

Figure 2.

The structure diagram of closed-loop system.

Theorem 1

Considering MSFLM-BT system (2)–(5), the control laws (18)–(21) can successfully realize vibration suppression and angle coordination tracking, that is, $\lim_{t \to \infty} θ_{m} (t) = \lim_{t \to \infty} θ_{s} (t) = θ_{d}$ , $\lim_{t \to \infty} {\dot{θ}}_{i} (t) = 0$ , $\lim_{t \to + \infty} w_{i} (x_{i}, t) = 0$ and $\lim_{t \to + \infty} {\dot{w}}_{i} (x_{i}, t) = 0$ , i = m, s.

Proof of Theorem 1

This proof is established through Lyapunov theory.

Firstly, we design Lyapunov–Krasovskii function:

V (t) = \sum_{m = 1}^{6} V_{m} (t)

(22)

V_{1} (t) = \frac{1}{2} I_{h} η_{m}^{2} (t) + \frac{r_{m 1} + r_{m 2}}{2} e_{m}^{2} (t)

(23)

V_{2} (t) = \frac{1}{2} I_{h} η_{s}^{2} (t) + \frac{r_{s 1} + r_{s 2}}{2} e_{s}^{2} (t) + \frac{1}{2} e_{s}^{2} (t)

(24)

\begin{matrix} V_{3} (t) = \sum_{i = m, s} (\frac{a_{i} E I}{2} \int_{0}^{L} {w_{i}^{''}}^{2} (x_{i}, t) d x_{i} \\ + \frac{a_{i} ρ}{2} \int_{0}^{L} {\dot{z}}_{i}^{2} (x_{i}, t) d x_{i}) \end{matrix}

(25)

V_{4} (t) = \frac{1}{2} m μ_{m}^{2} (t) + \frac{1}{2} m μ_{s}^{2} (t)

(26)

\begin{matrix} V_{5} (t) = \sum_{i = m, s} (b_{i} ρ \int_{0}^{L} x_{i} w_{i}^{'} (x_{i}, t) {\dot{z}}_{i} (x_{i}, t) d x_{i} \\ + c_{i} ρ \int_{0}^{L} (L - x_{i}) w_{i}^{'} (x_{i}, t) {\dot{z}}_{i} (x_{i}, t) d x_{i}) \end{matrix}

(27)

\begin{matrix} V_{6} (t) = \frac{ω_{s}}{2} {\tilde{ζ}}_{s}^{2} + \frac{ω_{m}}{2} \int_{t - τ_{m s}}^{t} ε_{s}^{2} (σ) d σ \\ + \frac{ω_{s}}{2} \int_{t - τ_{s m}}^{t} ε_{m}^{2} (σ) d σ \end{matrix}

(28)

where a_m, b_m, c_m, r_m1, r_m2, ω_m, a_s, b_s, c_s, r_s1, r_s2, ω_s > 0.

Using Lemmas 1–2, we can get

\begin{matrix} |V_{5} (t)| \leq \sum_{i = m, s} (\frac{(b_{i} + c_{i}) ρ L^{3}}{2} \int_{0}^{L} w_{i}^{'' 2} (x_{i}, t) d x_{i} \\ + \frac{(b_{i} + c_{i}) ρ L}{2} \int_{0}^{L} {\dot{z}}_{i}^{2} (x_{i}, t) d x_{i}) \leq β V_{3} (t) \end{matrix}

(29)

where β = L max{(b_m + c_m)ρL²/(a_mEI), (b_m + c_m)/a_m, (b_s + c_s)ρL²/(a_sEI), (b_s + c_s)/a_s}.

Then we have

\begin{matrix} 0 \leq (1 - β) [V_{1} (t) + V_{2} (t) + V_{3} (t) + V_{4} (t) + V_{6} (t)] \leq V (t) \\ \leq (1 + β) [V_{1} (t) + V_{2} (t) + V_{3} (t) + V_{4} (t) + V_{6} (t)] \end{matrix}

(30)

where 1 − β ≥ 0.

${\dot{V}}_{m} (t), m = 1,2, \dots, 5$ is derived (detailed derivation process can be found in equations (A1)–(A5) in the Appendix) as follows:

Combining equations (3), (14), (18), and skillfully using $- η_{m} (t) [r_{m 1} e_{m} (t) + r_{m 2} {\dot{e}}_{m} (t)] + η_{m} (t) [r_{m 1} e_{m} (t) + r_{m 2} {\dot{e}}_{m} (t)]$ and $η_{m} (t) [- ω_{m} {\dot{ζ}}_{m} (t) + ω_{m} {\dot{ζ}}_{m} (t)]$ in the derivation process, we obtain

\begin{aligned} {\dot{V}}_{1} (t) & = - k_{m} η_{m}^{2} (t) - r_{m 1} e_{m}^{2} (t) - r_{m 2} {\dot{e}}^{2}_{m} (t) \\ + ω_{m} e_{m} (t) {\dot{ζ}}_{m} (t) + ω_{m} {\dot{e}}_{m} (t) {\dot{ζ}}_{m} (t) \end{aligned}

(31)

Combining equations (3), (15), (20), and skillfully using $- η_{s} (t) [r_{s 1} e_{s} (t) + r_{s 2} {\dot{e}}_{s} (t)] + η_{s} (t) [r_{s 1} e_{s} (t) + r_{s 2} {\dot{e}}_{s} (t)]$ , $- η_{s} (t) [e_{s} (t) / 2 + {\dot{e}}_{s} (t) / 2] + η_{s} (t) [e_{s} (t) / 2 + {\dot{e}}_{s} (t) / 2]$ and $e_{s} (t) [- ω_{s} {\dot{ζ}}_{s} (t) + ω_{s} {\dot{ζ}}_{s} (t)]$ in the derivation process, we obtain

\begin{aligned} {\dot{V}}_{2} (t) & = - k_{s} η_{s}^{2} (t) - (\frac{1}{2} + r_{s 1}) e_{s}^{2} (t) - (\frac{1}{2} + r_{s 2}) {\dot{e}}_{s}^{2} (t) \\ - ω_{s} e_{s} (t) {\dot{ζ}}_{s} (t) + ω_{s} e_{s} (t) {\dot{ζ}}_{s} (t) \end{aligned}

(32)

By using Lemma 1 and the partial integration method, combining equations (2), (5), (16)–(17), and the boundary conditions $w_{i} (0, t) = w_{i}^{'} (0, t) = 0$ , z_i(0,t) = 0, $z_{i}^{'} (0, t) = θ_{i} (t)$ , we obtain

\begin{aligned} {\dot{V}}_{3} (t) & \leq \sum_{i = m, s} (- a_{i} E I {\dot{θ}}_{i} (t) w_{i}^{''} (0, t) + \frac{a_{i} E I}{2} μ_{i}^{2} (t) \\ - \frac{a_{i} E I}{2} {\dot{z}}_{i}^{2} (L, t) - a_{i} E I w_{i}^{'} (L, t) {\dot{z}}_{i} (L, t)) \end{aligned}

(33)

Combining equations (4), (16), (17), (19), and (21), we obtain

{\dot{V}}_{4} (t) = - π_{m} μ_{m}^{2} (t) - π_{s} μ_{s}^{2} (t)

(34)

\begin{aligned} {\dot{V}}_{5} (t) & \leq \sum_{i = m, s} (- \frac{b_{i} - c_{i}}{2} ρ \int_{0}^{L} {\dot{z}}_{i}^{2} (x_{i}, t) d x_{i} + \frac{b_{i}}{2} E I L μ_{i}^{2} (t) \\ - \frac{3 (b_{i} - c_{i})}{2} E I \int_{0}^{L} z_{i}^{'' 2} (x_{i}, t) d x_{i} + \frac{b_{i}}{2} ρ L {\dot{z}}_{i}^{2} (L, t) \\ - \frac{c_{i} E I L}{2} z_{i}^{'' 2} (0, t) - b_{i} E I L w_{i}^{'} (L, t) {\dot{z}}_{i} (L, t) \\ - \frac{b_{i}}{2} E I L w_{i}^{' 2} (L, t) - b_{i} ρ \int_{0}^{L} x_{i} {\dot{θ}}_{i} (t) \dot{z} (x_{i}, t) d x_{i} \\ - c_{i} ρ \int_{0}^{L} (L - x_{i}) {\dot{θ}}_{i} (t) {\dot{z}}_{i} (x_{i}, t) d x_{i}) \end{aligned}

(35)

Combining equations (10)–(11), we obtain

\begin{aligned} {\dot{V}}_{1} (t) + {\dot{V}}_{2} (t) + {\dot{V}}_{6} (t) \\ = - k_{m} η_{m}^{2} (t) - r_{m 1} e_{m}^{2} (t) - r_{m 2} {\dot{e}}^{2}_{m} (t) \\ - k_{s} η_{s}^{2} (t) - (\frac{1}{2} + r_{s 1}) e_{s}^{2} (t) - (\frac{1}{2} + r_{s 2}) {\dot{e}}_{s}^{2} (t) \\ + ω_{m} {\dot{e}}_{m} (t) {\dot{ζ}}_{m} (t) - ω_{s} e_{s} (t) {\dot{ζ}}_{s} (t) + A + B \end{aligned}

(36)

where

\begin{aligned} A & = ω_{m} e_{m} (t) {\dot{ζ}}_{m} (t) + \frac{ω_{m}}{2} (ε_{s}^{2} (t) - ε_{s}^{2} (t - τ_{m s})) \\ = - ω_{m} e_{m} (t) (θ_{m} (t) - θ_{s} (t - τ_{m s})) \\ + \frac{ω_{m}}{2} (ε_{s}^{2} (t) - ε_{s}^{2} (t - τ_{m s})) \\ = ω_{m} e_{m} (t) (- ε_{m} (t) + ε_{s} (t - τ_{m s})) \\ + \frac{ω_{m}}{2} (ε_{s}^{2} (t) - ε_{s}^{2} (t - τ_{m s})) \\ = - \frac{ω_{m}}{2} [\begin{matrix} 2 ε_{m}^{2} (t) - 2 ε_{m} (t) ε_{s} (t - τ_{m s}) \\ - ε_{s}^{2} (t) + ε_{s}^{2} (t - τ_{m s}) \end{matrix}] \\ = - \frac{ω_{m}}{2} [\begin{matrix} ε_{m}^{2} (t) - 2 ε_{m} (t) ε_{s} (t - τ_{m s}) \\ + ε_{s}^{2} (t - τ_{m s}) + ε_{m}^{2} (t) - ε_{s}^{2} (t) \end{matrix}] \\ = - \frac{ω_{m}}{2} {(ε_{m} (t) - ε_{s} (t - τ_{m s}))}^{2} \\ - \frac{ω_{m}}{2} (ε_{m}^{2} (t) - ε_{s}^{2} (t)) \end{aligned}

(37)

\begin{aligned} B & = ω_{s} e_{s} (t) {\dot{ζ}}_{s} (t) + ω_{s} {\tilde{ζ}}_{s} (t) {\dot{ζ}}_{s} (t) \\ + \frac{ω_{s}}{2} (ε_{m}^{2} (t) - ε_{m}^{2} (t - τ_{s m})) \\ = ω_{s} (ε_{s} (t) - {\tilde{ζ}}_{s} (t)) {\dot{ζ}}_{s} (t) + ω_{s} {\tilde{ζ}}_{s} (t) {\dot{ζ}}_{s} (t) \\ + \frac{ω_{s}}{2} (ε_{m}^{2} (t) - ε_{m}^{2} (t - τ_{s m})) \\ = ω_{s} ε_{s} (t) {\dot{ζ}}_{s} (t) + \frac{ω_{s}}{2} (ε_{m}^{2} (t) - ε_{m}^{2} (t - τ_{s m})) \\ = ω_{s} ε_{s} (t) (- θ_{s} (t) + θ_{m} (t - τ_{s m})) \\ + \frac{ω_{s}}{2} (ε_{m}^{2} (t) - ε_{m}^{2} (t - τ_{s m})) \\ = ω_{s} ε_{s} (t) (- ε_{s} (t) + ε_{m} (t - τ_{s m})) \\ + \frac{ω_{s}}{2} (ε_{m}^{2} (t) - ε_{m}^{2} (t - τ_{s m})) \\ = - \frac{ω_{s}}{2} [\begin{matrix} 2 ε_{s}^{2} (t) - 2 ε_{s} (t) ε_{m} (t - τ_{s m}) \\ - ε_{m}^{2} (t) + ε_{m}^{2} (t - τ_{s m}) \end{matrix}] \\ = - \frac{ω_{s}}{2} [\begin{matrix} ε_{s}^{2} (t) - 2 ε_{s} (t) ε_{m} (t - τ_{s m}) \\ + ε_{m}^{2} (t - τ_{s m}) + ε_{s}^{2} (t) - ε_{m}^{2} (t) \end{matrix}] \\ = - \frac{ω_{s}}{2} {(ε_{s} (t) - ε_{m} (t - τ_{s m}))}^{2} - \frac{ω_{s}}{2} (ε_{s}^{2} (t) - ε_{m}^{2} (t)) \end{aligned}

(38)

Set ω_s equal to ω_m, then we get

\begin{aligned} A + B & = - \frac{ω_{m}}{2} {(ε_{m} (t) - ε_{s} (t - τ_{m s}))}^{2} - \frac{ω_{m}}{2} (ε_{m}^{2} (t) - ε_{s}^{2} (t)) \\ - \frac{ω_{s}}{2} {(ε_{s} (t) - ε_{m} (t - τ_{s m}))}^{2} - \frac{ω_{s}}{2} (ε_{s}^{2} (t) - ε_{m}^{2} (t)) \\ = - \frac{ω_{m}}{2} {(ε_{m} (t) - ε_{s} (t - τ_{m s}))}^{2} \\ - \frac{ω_{s}}{2} {(ε_{s} (t) - ε_{m} (t - τ_{s m}))}^{2} \end{aligned}

(39)

Then ${\dot{V}}_{1} (t) + {\dot{V}}_{2} (t) + {\dot{V}}_{6} (t)$ can be obtained

\begin{aligned} {\dot{V}}_{1} (t) + {\dot{V}}_{2} (t) + {\dot{V}}_{6} (t) \\ = - k_{m} η_{m}^{2} (t) - r_{m 1} e_{m}^{2} (t) - r_{m 2} {\dot{e}}^{2}_{m} (t) - k_{s} η_{s}^{2} (t) \\ - (\frac{1}{2} + r_{s 1}) e_{s}^{2} (t) - (\frac{1}{2} + r_{s 2}) {\dot{e}}_{s}^{2} (t) \\ + ω_{m} {\dot{e}}_{m} (t) {\dot{ζ}}_{m} (t) - \frac{ω_{m}}{2} {(ε_{m} (t) - ε_{s} (t - τ_{m s}))}^{2} \\ - ω_{s} e_{s} (t) {\dot{ζ}}_{s} (t) - \frac{ω_{s}}{2} {(ε_{s} (t) - ε_{m} (t - τ_{s m}))}^{2} \end{aligned}

(40)

Then the derivative of the Lyapunov function can be obtained

\begin{aligned} \dot{V} (t) & = {\dot{V}}_{1} (t) + {\dot{V}}_{2} (t) + {\dot{V}}_{3} (t) + {\dot{V}}_{4} (t) + {\dot{V}}_{5} (t) + {\dot{V}}_{6} (t) \\ \leq - \sum_{i = m, s} ((π_{i} - \frac{a_{i} E I}{2} - \frac{b_{i}}{2} E I L) μ_{i}^{2} (t) \\ + \frac{a_{i} E I - b_{i} ρ L}{2} {\dot{z}}_{i}^{2} (L, t) + \frac{b_{i} - c_{i}}{2} ρ \int_{0}^{L} {\dot{z}}_{i}^{2} (x_{i}, t) d x_{i} \\ + \frac{3 (b_{i} - c_{i})}{2} E I \int_{0}^{L} {z_{i}^{'}}^{2} (x_{i}, t) d x_{i} + \frac{c_{i}}{2} E I L {z_{i}^{'}}^{2} (0, t) \\ + \frac{b_{i}}{2} E I L {w_{i}^{'}}^{2} (L, t) + k_{i} η_{i}^{2} (t)) - r_{m 1} e_{m}^{2} (t) - r_{m 2} {\dot{e}}_{m}^{2} (t) \\ - (\frac{1}{2} + r_{s 1}) e_{s}^{2} (t) - (\frac{1}{2} + r_{s 2}) {\dot{e}}_{s}^{2} (t) \\ - \frac{ω_{m}}{2} {(ε_{m} (t) - ε_{s} (t - τ_{m s}))}^{2} \\ - \frac{ω_{s}}{2} {(ε_{s} (t) - ε_{m} (t - τ_{s m}))}^{2} + \sum_{j = 1}^{10} β_{j} (t) \end{aligned}

(41)

where

- (1 / 2 + r_{s 2}) {\dot{e}}_{s}^{2} (t) = - (1 + 2 r_{s 2}) {\dot{θ}}_{s}^{2} (t) / 2 - (1 + 2 r_{s 2}) {\dot{ζ}}_{s}^{2} (t) / 2 + (1 + 2 r_{s 2}) {\dot{θ}}_{s} (t) {\dot{ζ}}_{s} (t)

, the expression of β_j(t), j = 1, 2, …, 10 is shown in Table 3.

According to Lemmas 1–2 (detailed derivation process can be found in equations (A6)–(A16) in the Appendix), we can further obtain

\begin{aligned} \dot{V} (t) & \leq - \sum_{i = m, s} (χ_{i 1} μ_{i}^{2} (t) + χ_{i 2} {\dot{z}}_{i}^{2} (L, t) \\ + χ_{i 3} \int_{0}^{L} {\dot{z}}_{i}^{2} (x_{i}, t) d x_{i} + χ_{i 4} \int_{0}^{L} {z_{i}^{'}}^{2} (x_{i}, t) d x_{i} \\ + χ_{i 5} {z_{i}^{'}}^{2} (0, t) + \frac{b_{i}}{2} E I L {w_{i}^{'}}^{2} (L, t) + k_{i} η_{i}^{2} (t)) \\ - r_{m 1} e_{m}^{2} (t) - χ_{m 6} {\dot{e}}_{m}^{2} (t) - χ_{m 7} {\dot{ζ}}_{m}^{2} (t) \\ - χ_{s 6} e_{s}^{2} (t) - χ_{s 7} {\dot{θ}}_{s}^{2} (t) - χ_{s 8} {\dot{ζ}}_{s}^{2} (t) \end{aligned}

(42)

where χ_mn > 0, n = 1, 2, …, 7, χ_sk > 0, k = 1, 2, …, 8, and the expressions of χ_mn and χ_sk are shown in Tables 4 and 5, respectively. In Tables 4 and 5, b_j > 0, j = 1, 2, …, 11.

From equation (42), we can get $\dot{V} (t) \leq 0$ . According to Lassalle theorem, we can get $\lim_{t \to \infty} \dot{V} (t) = 0$ . Further, $\lim_{t \to \infty} e_{m} (t) = \lim_{t \to \infty} ε_{m} (t) = \lim_{t \to \infty} {\dot{e}}_{m} (t) = \lim_{t \to \infty} {\dot{θ}}_{m} (t) = \lim_{t \to \infty} e_{s} (t) = \lim_{t \to \infty} {\dot{θ}}_{s} (t) = 0$ , $\lim_{t \to \infty} {\dot{ζ}}_{m} (t) = \lim_{t \to \infty} [θ_{s} (t - τ_{m s}) - θ_{m} (t)] = \lim_{t \to \infty} [ε_{s} (t - τ_{m s}) - ε_{m} (t)] = 0$ and $\lim_{t \to \infty} {\dot{ζ}}_{s} (t) = \lim_{t \to \infty} [θ_{m} (t - τ_{s m}) - θ_{s} (t)] = \lim_{t \to \infty} [ε_{m} (t - τ_{s m}) - ε_{s} (t)] = 0$ can be get. From $\lim_{t \to \infty} ε_{m} (t) = 0$ and $\lim_{t \to \infty} [ε_{s} (t - τ_{m s}) - ε_{m} (t)] = \lim_{t \to \infty} [ε_{m} (t - τ_{s m}) - ε_{s} (t)] = 0$ , $\lim_{t \to \infty} ε_{s} (t) = 0$ can be get, that is, $\lim_{t \to \infty} θ_{m} (t) = \lim_{t \to \infty} θ_{s} (t) = θ_{d}$ .

Integrating both sides of equation (42) at the same time, we get

\begin{aligned} 0 & \leq V (t) + \sum_{i = m, s} (χ_{i 1} \int_{0}^{t} μ_{i}^{2} (τ) d τ + χ_{i 2} \int_{0}^{t} {\dot{z}}_{i}^{2} (L, τ) d τ \\ + χ_{i 3} \int_{0}^{t} \int_{0}^{L} {\dot{z}}_{i}^{2} (x_{i}, τ) d x_{i} d τ + χ_{i 4} \int_{0}^{t} \int_{0}^{L} {z_{i}^{'}}^{2} (x_{i}, τ) d x_{i} d τ \\ + χ_{i 5} \int_{0}^{t} {z_{i}^{'}}^{2} (0, τ) d τ + \frac{b_{i}}{2} E I L \int_{0}^{t} {w_{i}^{'}}^{2} (L, τ) d τ \\ + k_{i} \int_{0}^{t} η_{i}^{2} (τ) d τ) + r_{m 1} \int_{0}^{t} e_{m}^{2} (τ) d τ + χ_{m 6} \int_{0}^{t} {\dot{e}}_{m}^{2} (τ) d τ \\ + χ_{m 7} \int_{0}^{t} {\dot{ζ}}_{m}^{2} (τ) d τ + χ_{s 6} \int_{0}^{t} e_{s}^{2} (τ) d τ + χ_{s 7} \int_{0}^{t} {\dot{θ}}_{s}^{2} (τ) d τ \\ + χ_{s 8} \int_{0}^{t} {\dot{ζ}}_{s}^{2} (τ) d τ \leq V (0) \end{aligned}

(43)

Accordingly, equation (43) follows that $\int_{0}^{L} {\dot{z}}_{m}^{2} (x_{m}, t) d x_{m}$ , $\int_{0}^{t} {\dot{z}}_{m}^{2} (x_{m}, τ) d τ$ , $\int_{0}^{L} {w_{m}^{''}}^{2} (x_{m}, t) d x_{m}$ , $\int_{0}^{t} \int_{0}^{L} {w_{m}^{''}}^{2} (x_{m}, τ) d x_{m} d τ$ , $w_{m}^{''} (0, t)$ , ${\dot{z}}_{m} (L, t)$ , $\int_{0}^{L} {\dot{z}}_{s}^{2} (x_{s}, t) d x_{s}$ , $\int_{0}^{t} {\dot{z}}_{s}^{2} (x_{s}, τ) d τ$ , $\int_{0}^{L} {w_{s}^{''}}^{2} (x_{s}, t) d x_{s}$ , $\int_{0}^{t} \int_{0}^{L} {w_{s}^{''}}^{2} (x_{s}, τ) d x_{s} d τ$ , $w_{s}^{''} (0, t)$ , ${\dot{z}}_{s} (L, t)$ , $\int_{0}^{t} μ_{m}^{2} (τ) d τ$ , $\int_{0}^{t} η_{m}^{2} (τ) d τ$ , $\int_{0}^{t} e_{m}^{2} (τ) d τ$ , $\int_{0}^{t} {\dot{e}}^{2}_{m} (τ) d τ$ , $\int_{0}^{t} {\dot{ζ}}_{m}^{2} (τ) d τ$ , $\int_{0}^{t} μ_{s}^{2} (τ) d τ$ , $\int_{0}^{t} η_{s}^{2} (τ) d τ$ , $\int_{0}^{t} e_{s}^{2} (τ) d τ$ , $\int_{0}^{t} {\dot{θ}}_{s}^{2} (τ) d τ$ , $\int_{0}^{t} {\dot{ζ}}_{s}^{2} (τ) d τ$ , μ_m(t), η_m(t), e_m(t), ${\dot{e}}_{m} (t)$ , ${\dot{ζ}}_{m} (t)$ , μ_s(t), η_s(t), e_s(t), ${\dot{θ}}_{s} (t)$ , ${\dot{ζ}}_{s} (t)$ and V(t) are bounded. Due to ${\dot{θ}}_{s} (t)$ and ${\dot{ζ}}_{s} (t)$ are bounded, ${\dot{e}}_{s} (t)$ is bounded. Due to ${\dot{w}}_{i} (x_{i}, t) = {\dot{z}}_{i} (x_{i}, t) - x_{i} {\dot{θ}}_{i} (t)$ , $\int_{0}^{t} {\dot{w}}_{i}^{2} (x_{i}, τ) d τ$ is bounded. And using Lemma 2, we know that w_i(x_i,t) and $\int_{0}^{t} {w_{i}}^{2} (x_{i}, τ) d τ$ are bounded. Obviously, the kinetic energy and the potential energy of the system are bounded, hence ${\dot{w}}_{i} (x_{i}, t)$ , ${\dot{w}}_{i}^{'} (x_{i}, t)$ , ${\dot{w}}_{i}^{'''} (x_{i}, t)$ , $w_{i}^{'''} (x_{i}, t)$ and $w_{i}^{''''} (x_{i}, t)$ are bounded according to Properties 1–2. Further, we can obtain that ${\ddot{z}}_{i} (x_{i}, t)$ is bounded according equation (2). Then combining equations (18)–(21), it can be obtained that u_m1(t), u_m2(t), u_s1(t), and u_s2(t) are bounded. Therefore, ${\ddot{θ}}_{i} (t), i = m, s$ in equation (3) is bounded. Furthermore, due to ${\ddot{w}}_{i} (x_{i}, t) = {\ddot{z}}_{i} (x_{i}, t) - x_{i} {\ddot{θ}}_{i} (t)$ , ${\ddot{w}}_{i} (x_{i}, t)$ is bounded. To sum up, all closed-loop signals are bounded.

Since w_i(x_i,t), ${\dot{w}}_{i} (x_{i}, t)$ , ${\ddot{w}}_{i} (x_{i}, t)$ , $\int_{0}^{t} {w_{i}}^{2} (x_{i}, τ) d τ$ , and $\int_{0}^{t} {\dot{w}}_{i}^{2} (x_{i}, τ) d τ$ are bounded, then $\lim_{t \to + \infty} w_{i} (x_{i}, t) = 0$ and $\lim_{t \to + \infty} {\dot{w}}_{i} (x_{i}, t) = 0$ can be obtained according to Barbalats lemma.

To sum up, $\lim_{t \to \infty} θ_{m} (t) = \lim_{t \to \infty} θ_{s} (t) = θ_{d}$ , $\lim_{t \to \infty} {\dot{θ}}_{i} (t) = 0$ , $\lim_{t \to + \infty} w_{i} (x_{i}, t) = 0$ and $\lim_{t \to + \infty} {\dot{w}}_{i} (x_{i}, t) = 0$ , The proof of Theorem 1 is completed.

Table 3.

The expression of β_j(t), j = 1, 2, …, 10.

β_j(t)	Expression
β₁(t)	$- a_{m} E I {\dot{θ}}_{m} (t) w_{m}^{''} (0, t)$
β₂(t)	$- (a_{m} + b_{m} L) E I w_{m}^{'} (L, t) {\dot{z}}_{m} (L, t)$
β₃(t)	$- a_{s} E I {\dot{θ}}_{s} (t) w_{s}^{''} (0, t)$
β₄(t)	$- (a_{s} + b_{s} L) E I w_{s}^{'} (L, t) {\dot{z}}_{s} (L, t)$
β₅(t)	$- b_{m} ρ \int_{0}^{L} x_{m} \dot{z} (x_{m}, t) d x_{m} {\dot{θ}}_{m} (t)$
β₆(t)	$- c_{m} ρ \int_{0}^{L} (L - x_{m}) {\dot{θ}}_{m} (t) {\dot{z}}_{m} (x_{m}, t) d x_{m}$
β₇(t)	$- b_{s} ρ \int_{0}^{L} x_{s} {\dot{θ}}_{s} (t) \dot{z} (x_{s}, t) d x_{s}$
β₈(t)	$- c_{s} ρ \int_{0}^{L} (L - x_{s}) {\dot{θ}}_{s} (t) {\dot{z}}_{s} (x_{s}, t) d x_{s}$
β₉(t)	$ω_{m} {\dot{e}}_{m} (t) {\dot{ζ}}_{m} (t)$
β₁₀(t)	$- ω_{s} e_{s} (t) {\dot{ζ}}_{s} (t)$

Table 4.

The expression of χ_mn(t), n = 1, 2, …, 7.

χ_mn(t)	Expression
χ_m1(t)	π_m − a_mEI/2 − b_mEIL/2
χ_m2(t)	$a_{m} E I / 2 - b_{m} ρ L / 2 - b_{2} (a_{m} E I + b_{m} E I L) / 2$
χ_m3(t)	(b_m − c_m)ρ/2 − b₅b_mρL/2 − b₆c_mρL/2
χ_m4(t)	$3 (b_{m} - c_{m}) E I / 2 - (a_{m} E I L + b_{m} E I L^{2}) / (2 b_{2})$
χ_m5(t)	c_mEIL/2 − b₁a_mEI/2
χ_m6(t)	r_m2 − a_mEI/(2b₁) − b_mρL²/(2b₅) − c_mρL²/(2b₆) − ω_m/(2b₉)
χ_m7(t)	ω_m/2 − b₉ω_m/2

Table 5.

The expression of χ_sk(t), k = 1, 2, …, 8.

χ_sk(t)	Expression
χ_s1(t)	π_s − a_sEI/2 − b_sEIL/2
χ_s2(t)	$a_{s} E I / 2 - b_{4} (a_{s} E I + b_{s} E I L) / 2 - b_{s} ρ L / 2$
χ_s3(t)	(b_s − c_s)ρ/2 − b₇b_sρL/2 − b₈c_sρL/2
χ_s4(t)	$3 (b_{s} - c_{s}) E I / 2 - (a_{s} E I L + b_{s} E I L^{2}) / (2 b_{4})$
χ_s5(t)	c_sEIL/2 − b₃a_sEI/2
χ_s6(t)	1/2 − ω_s/(2b₁₀) + r_s1
χ_s7(t)	$- a_{s} E I / (2 b_{3}) - b_{s} ρ L^{2} / (2 b_{7}) - c_{s} ρ L^{2} / (2 b_{8}) + (1 / 2 + r_{s 2}) (1 - 1 / b_{11})$
χ_s8(t)	$ω_{s} / 2 - b_{10} ω_{s} / 2 + (1 / 2 + r_{s 2}) (1 - b_{11})$

4. Simulations

The performance of the control law (18)–(21) for the MSFLM-BT system is verified by applying the finite difference method He et al. (2014) and Zhao et al. (2020). The system parameters determined according to reference Li et al. (2021) are given as: L = 1m, ρ = 0.2 kg/m, m = 0.1 kg, EI = 3Nm², and I_h = 0.1kgm². w_i(x_i, 0) is initialized to $0.01 {x_{i}}^{2}$ and other initial conditions are 0. The control parameters are k_m = 0.5, r_m1 = 0.5, r_m2 = 0.25, ω_m = 0.01, π_m = 0.12, k_s = 0.3, r_s1 = 0.5, r_s2 = 0.25, and π_s = 0.1.

Remark 2

According to equation (42) analysis, increasing the values of k_m, r_m1, r_m2, π_m, k_s, r_s1, r_s2, π_s and appropriately selecting the value of ω_m can ensure that $\dot{V} (t)$ is semi-negative definite. Equation (18) reflects that increasing k_m, r_m1, r_m2 can increase the impact of the angle tracking error e_m(t) and error rate ${\dot{e}}_{m} (t)$ on the control input u_m1(t), and adjusting the ratio of r_m1 and r_m2 can determine the proportion of the impact of error e_m(t) and error rate ${\dot{e}}_{m} (t)$ on u_m1(t), increasing ω_m can increase the impact of the angle θ_s(t − τ_ms) of the slave on u_m1(t). Similarly, equations (19)–(21) can be used to analyze the effects of the values of π_m, k_s, r_s1, r_s2, and π_s on the control inputs u_m2(t), u_s1(t), and u_s2(t), respectively. Increasing the values of ω_m and k_s, r_s1, r_s2 can make the master and slave more sensitive to each other’s angle information, thereby achieving better angle coordination and tracking. Generally speaking, increasing k_m, r_m1, r_m2, π_m, k_s, r_s1, r_s2, π_s help to improve the control performance, but may cause the amplitude of the control signal to be too large. Therefore, the controller parameters should be appropriately adjusted to balance control performance and control signal amplitude in practical applications.

To assess the effect of the control law, simulations of the system are conducted under five scenarios.

Case 1

Without control.

Cases 2–4 are simulated under the proposed control law.

Case 2

τ_sm = 0.5s, τ_ms = 0.5s.

Case 3

τ_sm = 1s, τ_ms = 1s.

Case 4

τ_sm = 1.5s, τ_ms = 1.5s.

Case 5

are simulated under the control law proposed in reference Li et al. (2021). Communication delay is τ_sm = 1s, τ_ms = 1s.

Figure 3 shows the uncontrolled dynamics of the MSFLM-BT system, with the red and blue lines overlapping. It is evident from this figure that the master-slave flexible-link manipulators experience significant vibration in the absence of control. Figures 4 –6 show the vibration suppression and angle coordination tracking results of the MSFLM-BT system under Cases 2–5. Notably, the results of the master flexible-link manipulator overlap under Cases 2–4 in Figures 5 and 6. It is evident from Figures 4 –6 that under different communication delay settings, the proposed control scheme can effectively suppress the vibration of the master-slave flexible-link manipulators and achieve angle coordinated tracking. The output results of the actuators under Cases 2–5 are shown in Figures 7–8. The maximum absolute values of the controllers u_i1(t) and u_i2(t) proposed in this paper are 1.3912Nm and 37.7112N, respectively.

Figure 3.

The simulation results of Case 1 (Without control).

Figure 4.

The deformation and deformation rate results of the MSFLM-BT system under Cases 2–5.

Figure 5.

The middle-point deformation and end-point deformation results of the MSFLM-BT system under Cases 2–5.

Figure 6.

The angle θ_i(t) and angle speed ${\dot{θ}}_{i} (t)$ results of the MSFLM-BT system under Cases 2–5.

Figure 7.

The output of actuator u_i1(t) results of the MSFLM-BT system under Cases 2–5.

Figure 8.

The output of actuator u_i2(t) results of the MSFLM-BT system under Cases 2–5.

The quantitative results of vibration suppression under Cases 2–5 are summarized in Table 6. As evidenced by Figures 4 and 5 and Table 6, the proposed control scheme achieves significantly lower vibration amplitude and frequency while maintaining stable vibration suppression. These results collectively demonstrate the effectiveness of our control strategy in mitigating vibrations.

Table 6.

Quantitative results of vibration suppression under Cases 2–5.

	${\|w_{m} (x_{m}, t)\|}_{\max}$	${\|w_{s} (x_{s}, t)\|}_{\max}$	${\|{\dot{w}}_{m} (x_{m}, t)\|}_{\max}$	${\|{\dot{w}}_{s} (x_{s}, t)\|}_{\max}$
Case 2	0.1556 m	0.0854 m	0.5768 m	0.1918 m
Case 3	0.1556 m	0.0854 m	0.5768 m	0.1918 m
Case 4	0.1556 m	0.0854 m	0.5768 m	0.1918 m
Case 5	0.2620 m	0.1762 m	3.2122 m	1.3469 m

Under identical communication delay condition, the proposed control scheme is compared with the method in reference Li et al. (2021). As shown in Figures 4 –8 and Table 6, the system under Li et al.’s control scheme exhibits increased vibration amplitude, higher oscillation frequency, prolonged settling time, and high-frequency variations in the control signal when subjected to communication delay. In contrast, the proposed control scheme exhibits significantly smoother transient dynamics and faster convergence rates. These results demonstrate that the proposed control scheme can reduce the impact of communication delay on system stability.

Therefore, for MSFLM-BT system with communication delay, the proposed control scheme can achieve vibration suppression and angle coordination tracking. To sum up, the simulation results validate the efficacy of the control scheme.

5. Conclusion

This paper investigates the vibration suppression and coordinated control problem of a bilateral teleoperation system composed of master and slave flexible-link manipulators in the presence of communication delay. Using the PDE dynamics model of MSFLM-BT system, a corresponding control scheme is designed based on Lyapunov–Krasovskii function. Under this control scheme, system stability is ensured in the presence of communication delay, and vibration suppression and angle coordinated tracking of the system are achieved. Numerical simulation verifies the effectiveness of the control scheme. This study has the following limitations:

The model assumes fixed communication delays, whereas real-world networks typically experience time-varying or random delays. Future work could extend the proposed control scheme to handle time-varying delays via robust or adaptive strategies. In addition, the maximum limit of communication delay cannot be calculated quantitatively, we will strive to solve this problem in future research.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work was supported by the National Natural Science Foundation of China under Grant No. 62273020 and 62473009.

ORCID iD

Jinkun Liu

Appendix

(A1)

\begin{aligned} {\dot{V}}_{1} (t) & = η_{m} (t) [I_{h} {\dot{e}}_{m} (t) + I_{h} {\ddot{e}}_{m} (t) + r_{m 1} e_{m} (t) + r_{m 2} {\dot{e}}_{m} (t)] \\ + (r_{m 1} + r_{m 2}) e_{m} (t) {\dot{e}}_{m} (t) \\ - η_{m} (t) [r_{m 1} e_{m} (t) + r_{m 2} {\dot{e}}_{m} (t)] \\ = η_{m} (t) [\begin{matrix} I_{h} {\dot{e}}_{m} (t) + u_{m 1} (t) + E I z_{m}^{''} (0, t) \\ + r_{m 1} e_{m} (t) + r_{m 2} {\dot{e}}_{m} (t) \\ - ω_{m} {\dot{ζ}}_{m} (t) + ω_{m} {\dot{ζ}}_{m} (t) \end{matrix}] \\ - r_{m 1} e_{m}^{2} (t) - r_{m 2} {\dot{e}}^{2}_{m} (t) \\ = - k_{m} η_{m}^{2} (t) - r_{m 1} e_{m}^{2} (t) - r_{m 2} {\dot{e}}^{2}_{m} (t) \\ + ω_{m} e_{m} (t) {\dot{ζ}}_{m} (t) + ω_{m} {\dot{e}}_{m} (t) {\dot{ζ}}_{m} (t) \end{aligned}

(A2)

\begin{aligned} {\dot{V}}_{2} (t) & = η_{s} (t) [\begin{matrix} I_{h} {\dot{e}}_{s} (t) + I_{h} {\ddot{e}}_{s} (t) + r_{s 1} e_{s} (t) \\ + r_{s 2} {\dot{e}}_{s} (t) + \frac{1}{2} e_{s} (t) + \frac{1}{2} {\dot{e}}_{s} (t) \end{matrix}] \\ + (r_{s 1} + r_{s 2}) e_{s} (t) {\dot{e}}_{s} (t) \\ + e_{s} (t) [{\dot{e}}_{s} (t) - ω_{s} {\dot{ζ}}_{s} (t) + ω_{s} {\dot{ζ}}_{s} (t)] \\ - η_{s} (t) [r_{s 1} e_{s} (t) + r_{s 2} {\dot{e}}_{s} (t)] - η_{s} (t) [\frac{1}{2} e_{s} (t) + \frac{1}{2} {\dot{e}}_{s} (t)] \\ = - k_{s} η_{s}^{2} (t) - (\frac{1}{2} + r_{s 1}) e_{s}^{2} (t) - (\frac{1}{2} + r_{s 2}) {\dot{e}}_{s}^{2} (t) \\ - ω_{s} e_{s} (t) {\dot{ζ}}_{s} (t) + ω_{s} e_{s} (t) {\dot{ζ}}_{s} (t) \end{aligned}

(A3)

\begin{aligned} {\dot{V}}_{3} (t) & = \sum_{i = m, s} (a_{i} E I \int_{0}^{L} w_{i}^{''} (x_{i}, t) {\dot{w}}^{''}_{i} (x_{i}, t) d x_{i} \\ - a_{i} E I \int_{0}^{L} {\dot{z}}_{i} (x, t) z_{i}^{''''} (x_{i}, t) d x_{i}) \\ = \sum_{i = m, s} a_{m} E I [- {\dot{z}}_{m} (L, t) z_{m} ‴ (L, t) - {\dot{θ}}_{m} (t) w_{m}^{''} (0, t)] \\ \leq \sum_{i = m, s} (- a_{i} E I {\dot{θ}}_{i} (t) w_{i}^{''} (0, t) + \frac{a_{i} E I}{2} μ_{i}^{2} (t) \\ - \frac{a_{i} E I}{2} {\dot{z}}_{i}^{2} (L, t) - a_{i} E I w_{i}^{'} (L, t) {\dot{z}}_{i} (L, t)) \end{aligned}

(A4)

\begin{aligned} {\dot{V}}_{4} (t) & = μ_{m} (t) [m {\dot{w}}^{'}_{m} (L, t) + u_{m 2} (t) + E I z_{m} ‴ (L, t) \\ - m {\dot{w}}^{‴}_{m} (L, t)] + μ_{s} (t) [m {\dot{w}}^{'}_{s} (L, t) + u_{s 2} (t) \\ + E I z_{s} ‴ (L, t) - m {\dot{w}}^{‴}_{s} (L, t)] \\ = - π_{m} μ_{m}^{2} (t) - π_{s} μ_{s}^{2} (t) \end{aligned}

(A5)

\begin{aligned} {\dot{V}}_{5} (t) & = \sum_{i = m, s} (b_{i} ρ \int_{0}^{L} x_{i} {\dot{w}}^{'}_{i} (x_{i}, t) {\dot{z}}_{i} (x_{i}, t) d x_{i} \\ + b_{i} ρ \int_{0}^{L} x_{i} w_{i}^{'} (x_{i}, t) {\ddot{z}}_{i} (x_{i}, t) d x_{i} \\ + c_{i} ρ \int_{0}^{L} (L - x_{i}) {\dot{w}}^{'}_{i} (x_{i}, t) {\dot{z}}_{i} (x_{i}, t) d x_{i} \\ + c_{i} ρ \int_{0}^{L} (L - x_{i}) w_{i}^{'} (x_{i}, t) {\ddot{z}}_{i} (x_{i}, t) d x_{i}) \\ \leq \sum_{i = m, s} (- \frac{b_{i} - c_{i}}{2} ρ \int_{0}^{L} {\dot{z}}_{i}^{2} (x_{i}, t) d x_{i} + \frac{b_{i}}{2} E I L μ_{i}^{2} (t) \\ - \frac{3 (b_{i} - c_{i})}{2} E I \int_{0}^{L} z_{i}^{'' 2} (x_{i}, t) d x_{i} + \frac{b_{i}}{2} ρ L {\dot{z}}_{i}^{2} (L, t) \\ - \frac{c_{i} E I L}{2} z_{i}^{'' 2} (0, t) - b_{i} E I L w_{i}^{'} (L, t) {\dot{z}}_{i} (L, t) \\ - \frac{b_{i}}{2} E I L w_{i}^{' 2} (L, t) - b_{i} ρ \int_{0}^{L} x_{i} {\dot{θ}}_{i} (t) \dot{z} (x_{i}, t) d x_{i} \\ - c_{i} ρ \int_{0}^{L} (L - x_{i}) {\dot{θ}}_{i} (t) {\dot{z}}_{i} (x_{i}, t) d x_{i}) \end{aligned}

According to Lemma 1–2, we can obtain (A6)

β_{1} (t) \leq \frac{a_{m} E I}{2 b_{1}} {\dot{θ}}_{m}^{2} (t) + \frac{a_{m} E I b_{1}}{2} {w_{m}^{''}}^{2} (0, t)

(A7)

\begin{aligned} β_{2} (t) & \leq \frac{a_{m} E I L + b_{m} E I L^{2}}{2 b_{2}} \int_{0}^{L} {w_{m}^{''}}^{2} (x_{m}, t) d x_{m} \\ + \frac{(a_{m} E I + b_{m} E I L) b_{2}}{2} {\dot{z}}_{m}^{2} (L, t) \end{aligned}

(A8)

β_{3} (t) \leq \frac{a_{s} E I}{2 b_{3}} {\dot{θ}}_{s}^{2} (t) + \frac{a_{s} E I b_{3}}{2} {w_{s}^{''}}^{2} (0, t)

(A9)

\begin{aligned} β_{4} (t) & \leq \frac{a_{s} E I L + b_{s} E I L^{2}}{2 b_{4}} \int_{0}^{L} {w_{s}^{''}}^{2} (x_{s}, t) d x_{s} \\ + \frac{(a_{s} E I + b_{s} E I L) b_{4}}{2} {\dot{z}}_{s}^{2} (L, t) \end{aligned}

(A10)

β_{5} (t) \leq \frac{b_{m} ρ L^{2}}{2 b_{5}} {\dot{θ}}_{m}^{2} (t) + \frac{b_{m} ρ L b_{5}}{2} \int_{0}^{L} {\dot{z}}_{m}^{2} (x_{m}, t) d x_{m}

(A11)

β_{6} (t) \leq \frac{c_{m} ρ L^{2}}{2 b_{6}} {\dot{θ}}_{m}^{2} (t) + \frac{c_{m} ρ L b_{6}}{2} \int_{0}^{L} {\dot{z}}_{m}^{2} (x_{m}, t) d x_{m}

(A12)

β_{7} (t) \leq \frac{b_{s} ρ L^{2}}{2 b_{7}} {\dot{θ}}_{s}^{2} (t) + \frac{b_{s} ρ L b_{7}}{2} \int_{0}^{L} {\dot{z}}_{s}^{2} (x_{s}, t) d x_{s}

(A13)

β_{8} (t) \leq \frac{c_{s} ρ L^{2}}{2 b_{8}} {\dot{θ}}_{s}^{2} (t) + \frac{c_{s} ρ L b_{8}}{2} \int_{0}^{L} {\dot{z}}_{s}^{2} (x_{s}, t) d x_{s}

(A14)

β_{9} (t) \leq \frac{ω_{m}}{2 b_{9}} {\dot{θ}}_{m}^{2} (t) + \frac{ω_{m} b_{9}}{2} {\dot{ζ}}_{m}^{2} (t)

(A15)

β_{10} (t) \leq \frac{ω_{s}}{2 b_{10}} {e_{s}}^{2} (t) + \frac{ω_{s} b_{10}}{2} {\dot{ζ}}_{s}^{2} (t)

(A16)

\begin{aligned} (1 + 2 r_{s 2}) {\dot{θ}}_{s} (t) {\dot{ζ}}_{s} (t) & \leq \frac{(\frac{1}{2} + r_{s 2})}{b_{11}} {\dot{θ}}_{s}^{2} (t) \\ + (\frac{1}{2} + r_{s 2}) b_{11} {\dot{ζ}}_{s}^{2} (t) \end{aligned}

where b_i > 0, i = 1, 2, …, 11.

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