Bilateral coordination control of flexible master–slave manipulators using a partial differential equation model

Abstract

In this study, we evaluate the coordination tracking control problem of a flexible master–slave teleoperation system. The system under consideration is based on a dynamic model described by a set of partial differential equations. Existing research on bilateral controllers is based on teleoperation systems composed of rigid master robots and rigid or flexible slave robots. In this work, we consider teleoperation systems with flexible master and slave robots. We dynamically model flexible master–slave manipulators using partial differential equations. Based on the dynamic model, a bilateral coordination controller is developed to realize the coordination angle tracking and vibration suppression of flexible master–slave manipulators. The teleoperation system is proven to be asymptotically stable under the control scheme. Numerical simulation results illustrate that the proposed controller is effective.

Keywords

Bilateral teleoperation system flexible manipulator coordination control vibration control partial differential equation model

1. Introduction

Bilateral teleoperation systems (BTSs) have received extensive attention (Chen et al., 2016; Islam et al., 2015; Sun et al., 2017; Yang et al., 2019a; Yuan et al., 2019) because they can be used for remote or dangerous tasks such as minimally invasive surgery (Dong et al., 2018), nuclear cleanup (Manocha et al., 2001), and outer space research (Penin et al., 2000). A BTS consists of a human operator and remote master–slave robots that communicate through a control and communication medium (see Figure 1). The human operator and master robot are in a control room, whereas the slave robot is in a remote location. The operator sends motion commands to the master robot. The master robot then uses a control and communication medium to exchange information (positions, speeds, and/or forces) between master and slave robots so that the slave robot can follow the trajectory of the master robot from a distance (Hokayem and Spong, 2006). The advantages of remote manipulation and the requirements of system stability and robot operation accuracy have promoted the research on the control of the BTS.

Many control schemes have been proposed for the BTS (Yang et al., 2016, 2017, 2018). Lee and Spong (2006) proposed a novel control scheme for a multi-degree-of-freedom (DOF) nonlinear system, where the position coordination of master and slave robots is achieved under a proposed proportional differential (PD) controller. To compensate for disturbances and coordinate positions between the master and slave, an improved proportional integral-differential (PID) controller was designed by Forouzantabar et al. (2012); this controller demonstrated improved performance over the controller designed by Lee and Spong (2006). To solve a dynamic nonlinear BTS problem, Forouzantabar et al. (2017) considered a nonlinear sliding mode controller, which improves the position tracking and force tracking performance. To deal with the dynamic uncertainties of the BTS, an extended state observer-based sliding mode control method was designed (Yang et al., 2019b). Yang et al. (2015) concentered on mitigating the input saturation, external disturbances, and uncertainties of the BTS, and proposed a neural network-based adaptive prescribed performance controller to handle these problems. Zhao et al. (2019a) also investigated actuator saturation and system uncertainty problems of the nonlinear BTS and developed an active disturbance rejection control scheme.

Most studies in the literature use rigid robots as the research objects. However, with the rapid rate of development of technology in the industrial, aerospace, medical, and other fields, robots must be flexible. Li et al. (2019) considered a BTS with a rigid master manipulator and a flexible-joint slave manipulator; a composite adaptive controller was designed to guarantee the flexible slave manipulator follows the master robot. A novel control framework was proposed for the synchronization of the nonlinear BTS with a rigid master and a flexible-joint slave (Yang et al., 2018). A PD controller was proposed to synchronize the motion of the master and slave robots with underactuated flexible joints (Ye et al., 2014). Yaryan et al. (2012) developed an input-shaping PD control scheme to address the input tracking and vibrational elimination problem of a BTS-based surgical robot with flexible links. Rashidinejad et al. (2015) developed a BTS with a flexible-link slave manipulator, where the tip position tracking problem was addressed by the output redefinition method. Rasouli et al. (2020) considered a BTS composed of a rigid master robot and a flexible-link slave robot; they proposed a simple PD control structure to handle the control aspect of the system in the presence of actuator faults and disturbances.

In many recent studies, BTSs are lumped parameter systems described by ordinary differential equations (ODEs). In this study, a BTS composed of master-slave manipulators with link flexibility is considered (see Figure 2). Given that the deformation and vibration of the flexible manipulators are related to position and time, the BTS is essentially a distributed parameter system (DPS). Common ODE models ignore the higher-order modes of the DPS, which may initiate a control spillover problem. However, a partial differential equation (PDE)-based dynamic model can retain all modes to restrain spillover and accurately describe the DPS. Xing et al. (2019) established a PDE model for a flexible string system. A flexible aerial refueling house was modeled as a DPS expressed by PDEs (Liu et al., 2017). Zhao et al. (2019b) developed a PDE model-based vibration control scheme of a flexible manipulator. A BTS with flexible-link slave robot that was modeled using a set of PDEs was developed in Yaryan et al. (2013). In this work, they proposed a modified PD controller to achieve the trajectory tracking and end tip vibration reduction.

The vibrations caused by the link flexibility of master–slave manipulators make the remote operation more complicated. The boundary control method can effectively suppress the vibration of flexible systems (He et al., 2018; Zhao et al., 2019c). A boundary control scheme was concerned to achieve the angle tracking and vibration elimination of a flexible manipulator (He et al., 2019). Xing et al. (2019) constructed a boundary constrained controller to restrain the vibrations of flexible string systems. Xing and Liu (2019) designed a boundary controller to address position tracking and vibration problems of an overhead crane with a flexible cable.

In this study, we design a bilateral coordination control scheme for a BTS. Our model solves the problem of angle tracking and vibration elimination. The contributions of this study are reflected as follows:

The BTS studied in this article is composed of a flexible master manipulator and a flexible slave manipulator. First, we modeled the DPS using a set of PDEs. The established PDE model can accurately characterize nonlinear dynamic characteristics of the BTS.

A bilateral coordination control scheme is proposed for the BTS. Under the action of the designed controller, the flexible slave manipulator can follow the flexible master manipulator to reach an ideal angle.

Deformations and vibrations of the flexible master and slave manipulators can be suppressed by a boundary control method. Therefore, the operational complexity of a BTS, caused by vibrations, can be reduced.

The rest of this article is arranged as follows: The PDE modeling process of the BTS is given in Section 2. In Section 3, a bilateral coordination controller is constructed to restrain vibration and ensure that the slave follows the master. The system’s asymptotic stability is validated by using LaSalle’s invariance principle. In Section 4, numerical simulations illustrate the effectiveness of the designed control scheme. Section 5 gives the conclusion of this article.

Notations

Throughout this article, ${(*)}_{m} and {(*)}_{s}$ denote the flexible master and slave manipulator, respectively. ${(*)}_{t} = \partial (*) / \partial t and {(*)}_{t t} = \partial^{2} (*) / \partial t^{2}$ denote the partial derivatives of $(*)$ with respect to time. ${(*)}^{'} = \partial (*) / \partial x, {(*)}^{''} = \partial^{2} (*) / \partial x^{2}$ , ${(*)}^{'''} = \partial^{3} (*) / \partial x^{3}$ , and ${(*)}^{(4)} = \partial^{4} (*) / \partial x^{4}$ represent the partial derivatives of $(*)$ with respect to space.

2. Problem formulation

2.1. System description

We consider the BTS shown in Figure 1. The human operator and the flexible master manipulator are in the control room, whereas the flexible slave manipulator is in the remote environment. The trace information and control inputs are transmitted through the control and communication medium. We assume that there is no communication delay in the BTS. As shown in Figure 2, the master and slave robots use the same flexible manipulator with one DOF. In Figure 2, $X O Y$ is the global coordinate system and $x O y$ represents the body-fixed coordinate system. System parameters are defined as follows: each flexible master manipulator and slave manipulator has length $L$ , hub inertia $I_{h}$ , bending stiffness $E I$ , mass per unit length $ρ,$ and payload mass $m$ . F_m(t) and F_s(t) denote the force inputs generated by actuators. M_m(t) and M_s(t) represent the torque inputs generated by motors. $θ_{m} (t) and θ_{s} (t)$ are joint angles, $y_{m} (x_{m}, t) and y_{s} (x_{s}, t)$ denote the elastic deflections, $w_{m} (x_{m}, t) = x_{m} θ_{m} + y_{m} (x_{m}, t) and w_{s} (x_{s}, t) = x_{s} θ_{s} + y_{s} (x_{s}, t)$ represent the displacements of the flexible master–slave manipulators, and it easily follows that $w_{m}^{(υ)} (x_{m}, t) = y_{m}^{(υ)} (x_{m}, t) and w_{s}^{(υ)} (x_{s}, t) = y_{s}^{(υ)} (x_{s}, t)$ hold when $υ \geq 2$ .

Remark 1

For simplicity, variable $t$ is omitted without causing any confusion. For example, $θ_{m}$ is used to represent $θ_{m} (t)$ .

We consider a master robot as an example to derive the dynamic model. The potential energy, kinetic energy, and virtual work of the flexible master manipulator can be expressed as

{\begin{cases} E_{k m} = \frac{1}{2} I_{h} θ_{m t}^{2} + \frac{ρ}{2} \int_{0}^{L} w_{m t}^{2} (x_{m}, t) d x_{m} + \frac{1}{2} m w_{m t}^{2} (L, t) \\ E_{p m} = \frac{1}{2} \int_{0}^{L} E I y_{m}^{″ 2} (x_{m}, t) d x_{m} \\ δ W_{v m} = M_{m} δ θ_{m} + F_{m} δ w_{m} (L, t) \end{cases}

(1)

Next, we apply Hamilton’s principle to derive the PDE dynamic model, which is characterized by

\int_{t_{1}}^{t_{2}} (δ E_{k} - δ E_{p} + δ W_{v}) d t = 0

(2)

where

t \in [t_{1}, t_{2}], t_{1} and t_{2}

are time constants and

δ (*)

is the vibration of

(*)

By substituting equation (1) into (2), the PDE model of the master can be obtained by solving the system

{\begin{cases} ρ w_{m t t} (x_{m}, t) = - E I y_{m}^{(4)} (x_{m}, t) \\ F_{m} = m w_{m t t} (L, t) - E I y_{m}^{‴} (L, t) \\ M_{m} = I_{h} θ_{m t t} - E I y_{m}^{″} (0, t) \\ y_{m} (0, t) = y_{m}^{'} (0, t) = y_{m}^{″} (L, t) = 0 \end{cases}

(3)

Similarly, the PDE model of the flexible slave manipulator is

{\begin{cases} ρ w_{s t t} (x_{s}, t) = - E I y_{s}^{(4)} (x_{s}, t) \\ F_{s} = m w_{s t t} (L, t) - E I y_{s}^{‴} (L, t) \\ M_{s} = I_{h} θ_{s t t} - E I y_{s}^{″} (0, t) \\ y_{s} (0, t) = y_{s}^{'} (0, t) = y_{s}^{″} (L, t) = 0 \end{cases}

(4)

2.2. Preliminaries

In this subsection, we introduce some necessary assumptions and lemmas, which we introduce as a precursor to the subsequent controller design and stability analysis.

Assumption 1

The flexible master and slave manipulators are all single-link mechanisms with one DOF; they only move in the horizontal plane without considering gravity.

Assumption 2

The physical parameters $I_{h}$ , $E I$ , $L$ , $ρ$ , and payload mass $m$ of flexible master and slave manipulators are assumed to be the same.

Assumption 3

The flexible manipulator studied in this article is a delay-free manipulator, which means that the communication delay on this BTS is not considered.

3. Control design and analysis

In this section, a bilateral coordination controller is designed for the BTS. The control objectives are (1) the angle coordination and tracking between master and slave can be realized. The flexible slave manipulator can follow the master to attain an ideal angle, namely, $θ_{s} \to θ_{m}, θ_{m} \to θ_{d}, θ_{m t} \to 0, and θ_{s t} \to 0$ ; (2) the deflection and vibration of both the flexible master and slave manipulators can be eliminated, namely, $y_{m} (x_{m}, t) \to 0, y_{s} (x_{s}, t) \to 0, y_{m t} (x_{m}, t) \to 0, and y_{s t} (x_{s}, t) \to 0$ .

For the above control objectives, the coordination control law is designed as

{\begin{cases} M_{m} = - k_{p 1} e_{m} + k_{p 2} e_{s} - k_{d 1} θ_{m t} \\ F_{m} = - k_{1} w_{m t} (L, t) \end{cases}

(5)

{\begin{cases} M_{s} = - k_{p 2} e_{s} - k_{d 2} θ_{s t} \\ F_{s} = - k_{2} w_{s t} (L, t) \end{cases}

(6)

where

k_{p 1}, k_{d 1}, k_{1}, k_{p 2}, k_{d 2}, and k_{2}

are positive constants.

e_{m} = θ_{m} - θ_{d} and e_{s} = θ_{s} - θ_{m}

are the angle errors, and

θ_{d}

is a constant that denotes the ideal angle.

Remark 2

The signals used for the designed coordination control law (equations (5) and (6)) are all available. Angles $θ_{m}$ and $θ_{s}$ can be measured by photoelectrical encoders, $θ_{m t}$ and $θ_{s t}$ can be obtained by tachometers, and then $e_{m}$ and $e_{s}$ can be calculated directly. The deformations $w_{m} (L, t)$ and $w_{s} (L, t)$ can be measured by laser displacement sensors, and the vibrations $w_{m t} (L, t)$ and $w_{s t} (L, t)$ can be calculated via a backward difference algorithm.

Theorem 1

The flexible master-slave manipulator (equations (3) and (4)) is asymptotically stable with Assumptions 1–3 and the bilateral coordination control law (equations (5) and (6)). Meanwhile, the control goal can be accomplished, namely, $θ_{s} \to θ_{m}$ , $θ_{m} \to θ_{d}$ , $θ_{m t} \to 0$ , $θ_{s t} \to 0$ , $y_{m} (x_{m}, t) \to 0$ , $y_{s} (x_{s}, t) \to 0$ , $y_{m t} (x_{m}, t) \to 0$ , and $y_{s t} (x_{s}, t) \to 0$ , $\forall x_{m}, x_{s} \in [0, L]$ as $t \to \infty$ .

Proof

The Lyapunov function candidate is

V = E_{1} + E_{2} + E_{3}

(7)

where

E_{1} = \frac{ρ}{2} \int_{0}^{L} w_{m t}^{2} (x_{m}, t) d x_{m} + \frac{1}{2} \int_{0}^{L} E I y_{m}^{″ 2} (x_{m}, t) d x_{m} + \frac{ρ}{2} \int_{0}^{L} w_{s t}^{2} (x_{s}, t) d x_{s} + \frac{1}{2} \int_{0}^{L} E I y_{s}^{″ 2} (x_{s}, t) d x_{s}

(8)

E_{2} = \frac{1}{2} I_{h} θ_{m t}^{2} + \frac{1}{2} I_{h} θ_{s t}^{2} + \frac{1}{2} k_{p 1} e_{m}^{2} + \frac{1}{2} k_{p 2} e_{s}^{2}

(9)

E_{3} = \frac{1}{2} m w_{m t}^{2} (L, t) + \frac{1}{2} m w_{s t}^{2} (L, t)

(10)

and

V

is positive definite from equations (8)–(10).

The derivative of equation (7) is

V_{t} = E_{1 t} + E_{2 t} + E_{3 t}

(11)

where

\begin{array}{l} E_{1 t} = \int_{0}^{L} - E I w_{m}^{(4)} (x_{m}, t) w_{m t} (x_{m}, t) d x_{m} + \int_{0}^{L} E I y_{m}^{″} (x_{m}, t) y_{m t}^{″} (x_{m}, t) d x_{m} \\ + \int_{0}^{L} - E I w_{s}^{(4)} (x_{s}, t) w_{s t} (x_{s}, t) d x_{s} + \int_{0}^{L} E I y_{s}^{″} (x_{s}, t) y_{s t}^{″} (x_{s}, t) d x_{s} \\ = \int_{0}^{L} - E I w_{m t} (x_{m}, t) d [w_{m}^{‴} (x_{m}, t)] + \int_{0}^{L} E I w_{m}^{″} (x_{m}, t) d [w_{m t}^{'} (x_{m}, t)] \\ + \int_{0}^{L} - E I w_{s t} (x_{s}, t) d [w_{s}^{‴} (x_{s}, t)] + \int_{0}^{L} E I w_{s}^{″} (x_{s}, t) d [w_{s t}^{'} (x_{s}, t)] \\ = {- [E I w_{m t} (x_{m}, t) w_{m}^{‴} (x_{m}, t)] |}_{0}^{L} + {E I [w_{m}^{″} (x_{m}, t) w_{m t}^{'} (x_{m}, t)] |}_{0}^{L} \\ {- [E I w_{s t} (x_{s}, t) w_{s}^{″} (x_{s}, t)] |}_{0}^{L} + {E I [w_{s}^{″} (x_{s}, t) w_{s t}^{'} (x_{s}, t)] |}_{0}^{L} \\ = - E I [w_{m t} (L, t) y_{m}^{‴} (L, t) + w_{s t} (L, t) y_{s}^{‴} (L, t)] \\ - E I [y_{m}^{″} (0, t) θ_{m t} + y_{s}^{″} (0, t) θ_{s t}] \end{array}

(12)

\begin{array}{l} E_{2 t} = θ_{m t} I_{h} θ_{m t t} + θ_{s t} I_{h} θ_{s t t} + k_{p 1} e_{m} e_{m t} + k_{p 2} e_{s} e_{s t} \\ = θ_{m t} [M_{m} + E I y_{m}^{″} (0, t)] + θ_{s t} [M_{s} + E I y_{s}^{″} (0, t)] \\ + k_{p 1} e_{m} θ_{m t} + k_{p2} e_{s} (θ_{s t} - θ_{m t}) \\ = θ_{m t} [M_{m} + E I y_{m}^{″} (0, t) + k_{p 1} e_{m} - k_{p 2} e_{s}] \\ + θ_{s t} [M_{s} + E I y_{s}^{″} (0, t) + k_{p 2} e_{s}] \end{array}

(13)

E_{3 t} = w_{m t} (L, t) m w_{m t t} (L, t) + w_{s t} (L, t) m w_{s t t} (L, t) = w_{m t} (L, t) [F_{m} + E I y_{m}^{‴} (L, t)] + w_{s t} (L, t) [F_{s} + E I y_{s}^{‴} (L, t)]

(14)

Substituting equations (12)–(14) and the coordination control law (5) and (6) into (11) yields

\begin{array}{l} V_{t} = - E I [w_{m t} (L, t) y_{m}^{‴} (L, t) + w_{s t} (L, t) y_{s}^{‴} (L, t)] \\ - E I [y_{m}^{″} (0, t) θ_{m t} + y_{s}^{″} (0, t) θ_{s t}] \\ + θ_{m t} [M_{m} + E I y_{m}^{″} (0, t) + k_{p 1} e_{m} - k_{p 2} e_{s}] \\ + θ_{s t} [M_{s} + E I y_{s}^{″} (0, t) + k_{p 2} e_{s}] \\ + w_{m t} (L, t) [F_{m} + E I y_{m}^{‴} (L, t)] \\ + w_{s t} (L, t) [F_{s} + E I y_{s}^{‴} (L, t)] \\ = - k_{d 1} θ_{m t}^{2} - k_{d 2} θ_{s t}^{2} - k_{1} w_{m t}^{2} (L, t) - k_{2} w_{s t}^{2} (L, t) \\ \leq 0 \end{array}

(15)

Next, the asymptotic stability of the closed-loop system is proven by using the LaSalle invariance principle extended to infinite dimensional systems (Rahn, 2001). To use the extended LaSalle invariance principle, the BTS must have the desired necessary conditions. The proof to develop the necessary conditions is lengthy and is shown in Appendix 1.

Let $V_{t} \equiv 0$ , from equation (15), we have

θ_{m t} = θ_{s t} = w_{m t} (L, t) = w_{s t} (L, t) \equiv 0

(16)

which indicates

θ_{m t t} = θ_{s t t} = w_{m t t} (L, t) = w_{s t t} (L, t) \equiv 0

(17)

Next, we analyze the flexible master manipulator. Substituting equations (16) and (17) into (3) yields

{\begin{matrix} ρ y_{m t t} (x_{m}, t) = - E I y_{m}^{(4)} (x_{m}, t) \\ y_{m}^{(4)} (L, t) = 0 \\ y_{m} (0, t) = y_{m}^{'} (0, t) = y_{m}^{″} (L, t) = 0 \end{matrix}

(18)

By using the method of separation of variables (Krstic and Smyshlyaev, 2008), we have

y_{m} (x_{m}, t) = χ_{m} (x_{m}) φ_{m} (t)

(19)

where $χ_{m} (x_{m})$ depends on the position on the manipulator, $φ_{m} (t)$ depends on time.

Equations (18) and (19) show that

ρ χ_{m} (x_{m}) φ_{m t t} (t) = - E I χ_{m}^{(4)} (x_{m}) φ_{m} (t)

(20)

Arranging equation (20) and introducing $η^{4}$ yields

\frac{χ_{m}^{(4)} (x_{m})}{χ_{m} (x_{m})} = \frac{ρ φ_{m t t} (t)}{- E I φ_{m} (t)} = η^{4}

(21)

which means

χ_{m}^{(4)} (x_{m}) - η^{4} χ_{m} (x_{m}) = 0

(22)

Solving equation (22) yields

χ_{m} (x_{m}) = b_{1} e^{η x_{m}} + b_{2} e^{- η x_{m}} + b_{3} cos (η x_{m}) + b_{4} sin (η x_{m})

(23)

where

b_{1}, \dots, b_{4}

are unknown constants.

Equations (18) and (19) indicate

χ_{m} (0) = χ_{m}^{'} (0) = χ_{m}^{″} (L) = χ_{m}^{(4)} (L) = 0

(24)

Under the initial condition (equation (24)), equation (23) has a unique solution: $b_{1} = b_{2} = b_{3} = b_{4} = 0$ . That is, $χ_{m} (x_{m}) = 0$ . Similarly, $χ_{s} (x_{s}) = 0$ can be obtained. Then, we have $y_{m} (x_{m}, t) = y_{m t} (x_{m}, t) = y_{m}^{″} (0, t) = y_{s} (x_{s}, t) = y_{s t} (x_{s}, t) = y_{s}^{″} (0, t) = 0$ . Then, $M_{m} = M_{s} = 0$ can be further obtained from equations (3) and (4). Combining equations (5) and (6), we arrive at $e_{m} = e_{s} = 0$ , which yields $θ_{s} = θ_{m} = θ_{d}$ . Therefore, we conclude that $e_{m} \to 0$ , $e_{s} \to 0$ , $θ_{m} \to θ_{d}$ , $θ_{s} \to θ_{m}$ , $θ_{m t} \to 0$ , $θ_{s t} \to 0$ , $y_{m} (x_{m}, t) \to 0$ , $y_{s} (x_{s}, t) \to 0$ , $y_{m t} (x_{m}, t) \to 0$ and $y_{s t} (x_{s}, t) \to 0$ hold when $t \to \infty$ . Therefore, we show that the system is asymptotically stable under the designed controller (equations (5) and (6)) using the extended LaSalle invariance principle. Thus, Theorem 1 is proven.

4. Numerical simulations

In this section, the finite difference method is applied to simulate the PDE dynamic model and the designed controller. The system’s physical parameters are $I_{h} = 0.1 {kgm}^{2}$ , $E I = 3 {Nm}^{2}$ , $L = 1 m$ , $ρ = 0.2 {kgm}^{- 1}$ , and $m = 0.1 kg$ . The ideal angle is $θ_{d} = 1 rad$ . The initial conditions are $θ_{m} (0) = 0.4 rad$ , $θ_{s} (0) = 0 rad$ , $y_{m} (x_{m}, 0) = 0.01 x_{m}^{2}$ , $y_{m t} (x_{m}, 0) = 0$ , $y_{s} (x_{s}, 0) = 0.01 x_{s}^{2}$ , and $y_{s t} (x_{s}, 0) = 0$ . To analyze and verify the control performance of the controller, we simulate the BTS for the following cases:

Case 1

Without control. $M_{m} = M_{s} = F_{m} = F_{s} = 0$ .

Case 2

With the developed controller (equations (5) and (6)). Controller parameters are designed as: $k_{p 1} = 14$ , $k_{p 2} = 20$ , $k_{d 1} = 3$ , $k_{d 2} = 2$ , $k_{1} = 2$ and $k_{2} = 1.5$ .

Case 3

With PD control (equations (25) and (26)). Controller parameters are chosen as: $κ_{p 1} = 60$ , $κ_{d 1} = 7$ , $κ_{1} = 1$ , $κ_{2} = 2$ , $κ_{p 2} = 40$ , $κ_{d 2} = 6$ , $κ_{3} = 1$ and $κ_{4} = 2$ .

{\begin{cases} M_{m} = - κ_{p 1} e_{m} - κ_{d 1} e_{m t} \\ F_{m} = - κ_{1} w_{m} (L, t) - κ_{2} w_{m t} (L, t) \end{cases}

(25)

{\begin{cases} M_{s} = - κ_{p 2} e_{s} - κ_{d 2} e_{s t} \\ F_{s} = - κ_{3} w_{s} (L, t) - κ_{4} w_{s t} (L, t) \end{cases}

(26)

The simulation result for case 1 is given in Figure 3. The flexible manipulator vibrates a lot, which is unacceptable.

The simulation results for case 2 and case 3 are shown in Figures 4–9. Figure 4 depicts the angle tracking responses of flexible master–slave manipulators. In Figure 4(a), the slave robot can follow the master robot to reach the ideal angle signal under the proposed controller within 2 s, whereas in Figure 4(b), there are still tracking errors. Figure 5 shows the angle speed of flexible master-slave manipulators. As can be seen from Figure 5, the angular velocities of master–slave robots eventually converge to zero in cases 2–3. Figures 6 and 7 give the deflections of flexible master–slave manipulators. As shown in Figures 6(a) and 7(a), the proposed control scheme can suppress the deformations and vibrations of the flexible master and slave arms. However, in Figures 6(b) and 7(b), PD control cannot inhibit the deformation of the master–slave robot. The deformation of the flexible master and slave manipulators at $L$ and $L / 2$ , as shown in Figure 8, can be suppressed under the designed controller, but it still exists under the PD control strategy. The control inputs of flexible master-slave manipulators are depicted in Figure 9.

In conclusion, the proposed control scheme is effective and has superior control performance than the PD control strategy to coordinate angle tracking and vibration suppression.

5. Conclusion

In this study, we evaluate coordination tracking and vibration elimination of a BTS with flexible master and slave manipulators. The BTS under consideration is modeled as a DPS described by PDEs, and a coordinate controller is developed for this PDE dynamic model. The flexible slave manipulator can follow the master to reach the ideal angle under the designed controller, and vibration suppression can be eliminated. Numerical simulations successfully verify the effectiveness of the proposed control scheme. In the future, we will consider the coordination of tracking and vibration suppression of BTSs with time delay.

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: This study was supported by the Research Fund for the National Natural Science Foundation of China [grant number 61873296]; and the CAS Prospective Deployment Project [grant number ZDRW-KT-2019-1-010402].

ORCID iD

Jinkun Liu

Appendix 1

To use the extended LaSalle invariance principle to verify the asymptotic stability of the BTS, a BTS should satisfy some necessary conditions, as detailed below. Let

\begin{array}{l} z = {[\begin{matrix} z_{1} & z_{2} & z_{3} & z_{4} & z_{5} & z_{6} & z_{7} & z_{8} & z_{9} & z_{10} \end{matrix}]}^{T} \\ = {[\begin{matrix} e_{1} & e_{1 t} & e_{2} & e_{2 t} & w_{1 t} (L, t) & w_{2 t} (L, t) & y_{1} (x_{1}, t) & y_{1 t} (x_{1}, t) & y_{2} (x_{1}, t) & y_{2 t} (x_{1}, t) \end{matrix}]}^{T} \end{array}

The closed-loop system can be rewritten as follows (27)

z_{t} = A z, z (0) \in H

where

A z = {[{\begin{matrix} z_{2} & ϑ_{1} & z_{4} & ϑ_{2} & ϑ_{3} & ϑ_{4} & z_{8} & ϑ_{5} & z_{10} & ϑ \end{matrix}}_{6}]}^{T}

\forall z \in D (A)

and

\begin{array}{l} ϑ_{1} = \frac{1}{I_{h}} (- k_{p 1} z_{1} + k_{p 2} z_{3} - k_{d 1} z_{2} + E I z_{7}^{″} (0)) \\ ϑ_{2} = \frac{1}{I_{h}} (k_{p 1} z_{1} + (k_{d 1} - k_{d 2}) z_{2} - 2 k_{p 2} z_{3} - k_{d 2} z_{4} - E I z_{7}^{″} (0) + E I z_{9}^{″} (0)) \\ ϑ_{3} = \frac{1}{m} (- k_{1} z_{5} + E I z_{7}^{‴} (L)) \\ ϑ_{4} = \frac{1}{m} (- k_{2} z_{6} + E I z_{9}^{‴} (L)) \\ ϑ_{5} = - \frac{E I}{ρ} z_{7}^{(4)} - \frac{x_{1}}{I_{h}} (- k_{p 1} z_{1} + k_{p 2} z_{3} - k_{d 1} z_{2} + E I z_{7}^{″} (0)) \\ ϑ_{6} = - \frac{E I}{ρ} z_{9}^{(4)} - \frac{x_{2}}{I_{h}} (- k_{d 2} z_{2} - k_{p 2} z_{3} - k_{d 2} z_{4} + E I z_{9}^{″} (0)) \end{array}

The spaces related to $z$ mentioned in equation (27) are defined as (28)

ℋ = ℝ^{6} \times H^{2} \times L^{2} \times H^{2} \times L^{2}

where

L^{2} (Ω) = {f | \int_{Ω} {| f (x) |}^{2} d x < \infty}

H^{k} (Ω) = {f | f, f^{'}, f^{″}, \dots, f^{(k)} \in L^{2} (Ω)}

, and

Ω = [0, L]

;

ℝ^{6} (z_{1}, z_{2}, z_{3}, z_{4}, z_{5}, z_{6})

denotes a real value, the first

H^{2}

signifies that

z_{7}^{'}

and

z_{7}^{″}

are bounded, the first

L^{2}

signifies that

z_{8}

and

z_{8 t}

are bounded, the second

H^{2}

signifies that

z_{9}^{'}

and

z_{9}^{″}

are bounded, and the second

L^{2}

means that

z_{10}

and

z_{10 t}

are bounded.

The domain of $A$ is (29)

D (A) = {z \in ℝ^{6} \times H^{4} \times H^{2} \times H^{4} \times H^{2} | \begin{array}{l} z_{7} (0) = 0, z_{7}^{'} (0) = 0, z_{7}^{″} (L) = 0, \\ z_{9} (0) = 0, z_{9}^{'} (0) = 0, z_{9}^{″} (L) = 0 \end{array}}

where

ℝ^{6} (z_{1}, z_{2}, z_{3}, z_{4}, z_{5}, z_{6})

denotes the real value, the first

H^{4}

suggests that

z_{7}^{'}, z_{7}^{″}, z_{7}^{‴}, and z_{7}^{(4)}

are bounded, whereas the second

H^{4}

suggests that

z_{9}^{'}, z_{9}^{″}, z_{9}^{‴}, and z_{9}^{(4)}

are bounded. The first

H^{2}

suggests that

z_{8}^{'} and z_{8}^{″}

are bounded, whereas the second

H^{2}

suggests that

z_{10}^{'} and z_{10}^{″}

are bounded.

The Lyapunov candidate (7) can be designated as the energy function of system (27). Therefore, the closed system is negative semidefinite. Equation (15) yields $V_{t} \leq 0$ , which implies that the operator $A$ is dissipative.

Next, we provide the proof of the following conditions:

The operator $A$ generates a $C_{0}$ -semigroup of contraction $T (t)$ on $ℋ$ .

The solution trajectories of the system, for $t > 0$ , are precompact in $ℋ$ .

Proof

(1) Because $V_{t} \leq 0$ , the operator $A$ is dissipative. We consider the expression $(λ I - A) z = g$ , which has a unique solution for any $λ > 0$ and $g \in ℋ$ according to Lax–Milgram (Pazy, 2012). Therefore, the range of operator $(λ I - A)$ is onto $ℋ$ . According to the Lumer–Phillips theorem, operator $A$ generates a $C_{0}$ -semigroup of contractions $T (t)$ on $ℋ$ (Rahn, 2001).

(2) We define $g = {[\begin{matrix} g_{1} & g_{2} & g_{3} & g_{4} & g_{5} & g_{6} & g_{7} & g_{8} & g_{9} & g_{10} \end{matrix}]}^{T} \in H$ and consider the equation (30)

A z = g

where

\begin{array}{l} g_{1} = z_{2} \\ g_{2} = \frac{1}{I_{h}} (- k_{p 1} z_{1} + k_{p 2} z_{3} - k_{d 1} z_{2} + E I z_{7}^{″} (0)) \\ g_{3} = z_{4} \\ g_{4} = \frac{1}{I_{h}} (k_{p 1} z_{1} + (k_{d 1} - k_{d 2}) z_{2} - 2 k_{p 2} z_{3} - k_{d 2} z_{4} - E I z_{7}^{″} (0) + E I z_{9}^{″} (0)) \\ g_{5} = \frac{1}{m} (- k_{1} z_{5} + E I z_{7}^{‴} (L)) \\ g_{6} = \frac{1}{m} (- k_{2} z_{6} + E I z_{9}^{‴} (L)) \\ g_{7} = z_{8} \\ g_{8} = - \frac{E I}{ρ} z_{7}^{(4)} - \frac{x_{1}}{I_{h}} (- k_{p 1} z_{1} + k_{p 2} z_{3} - k_{d 1} z_{2} + E I z_{7}^{″} (0)) \\ g_{9} = z_{10} \\ g_{10} = - \frac{E I}{ρ} z_{9}^{(4)} - \frac{x_{2}}{I_{h}} (- k_{d 2} z_{2} - k_{p 2} z_{3} - k_{d 2} z_{4} + E I z_{9}^{″} (0)) \end{array}

Solving the above equations yields

\begin{array}{l} z_{1} = \frac{1}{k_{p 1}} (- (k_{d 1} + k_{d 2}) g_{1} - 2 I_{h} g_{2} - k_{d 2} g_{3} - I_{h} g_{4} + E I z_{7}^{″} (0) + E I z_{9}^{″} (0)) \\ z_{2} = g_{1} \\ z_{3} = \frac{1}{k_{p 2}} (- k_{d 2} g_{1} - I_{h} g_{2} - k_{d 2} g_{3} - I_{h} g_{4} + E I z_{9}^{″} (0)) \\ z_{4} = g_{3} \\ z_{5} = \frac{1}{k_{1}} (- m g_{5} + E I z_{7}^{‴} (L)) \\ z_{6} = \frac{1}{k_{2}} (- m g_{6} + E I z_{9}^{‴} (L)) \\ z_{7} = - \frac{ρ}{E I} \int_{0}^{x_{1}} \int_{0}^{ς_{1}} \int_{0}^{ς_{2}} \int_{0}^{ς_{3}} g_{8} (ς_{4}) d ς_{4} d ς_{3} d ς_{2} d ς_{1} - \frac{x_{1}^{5}}{5!} \frac{ρ}{E I} g_{2} + \sum_{j = 0}^{3} σ_{j} x_{1}^{j} \\ z_{8} = g_{7} \\ z_{9} = - \frac{ρ}{E I} \int_{0}^{x_{2}} \int_{0}^{ς_{1}} \int_{0}^{ς_{2}} \int_{0}^{ς_{3}} g_{10} (ς_{4}) d ς_{4} d ς_{3} d ς_{2} d ς_{1} - \frac{x_{2}^{5}}{5!} \frac{ρ}{E I} (g_{2} + g_{4}) + \sum_{j = 0}^{3} υ_{j} x_{2}^{j} \\ z_{10} = g_{9} \end{array}

where

σ_{j} and υ_{j}, j = 0,1,2,3

are uniquely determined by boundary conditions in equations (3) and (4).

Hence, equation (30) has a unique solution $z \in D (A)$ , which suggests that $z = A^{- 1} g$ . Therefore, there exists $A^{- 1}$ that maps $ℋ$ onto $D (A)$ . Furthermore, because $A^{- 1}$ maps every bounded set of $ℋ$ into a bounded set of $D (A)$ , the embedding of the latter space into $ℋ$ is compact. This implies that $A^{- 1}$ is a compact operator (Pazy, 2012).

The spectrum of $A$ consists entirely of isolated eigenvalues. It also proves that for any $λ > 0$ in the resolvent set of $A$ , the operator ${(λ I - A)}^{- 1}$ is a compact operator. Hence, the solution trajectories of system (27), for $t > 0$ , are precompact in $ℋ$ (Morgul, 1998).

The following three conditions can be obtained from the analysis in this study: (1) the operator $A$ generates a $C_{0}$ -semigroup of contraction $T (t)$ on $ℋ$ , (2) the energy of the system $V > 0$ has a negative time derivative $V_{t} \leq 0$ , and (3) all solution trajectories of the system, for $t > 0$ , are precompact in $ℋ$ . Thus, we can apply the extended LaSalle invariance principle to verify the asymptotic stability of the BTS.

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