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Abstract

A new adaptive tracking control problem of a nonlinear teleoperation system is considered in this article, which differs from the previous ones in that both kinematics and dynamics of manipulators are uncertain and perturbation of the actuator parameters exists due to overheating of the actuator. To overcome the first which is also the key problem, we propose a new adaptive tracking control framework which additionally contains a neural network mechanism and a robust adaptive mechanism. Based on the framework, an adaptive controller is further established to compensate for the perturbation of the actuator parameters online. By employing the neural networks and adaption method in control design, an extended teleoperation control system is effectively accommodated. With the Lyapunov theory, the tracking error is shown to converge to a residual around zero as the time goes to infinity. The effectiveness of the proposed scheme has been verified in the physical platform.

Keywords

Adaptive control teleoperation system kinematic uncertainty neural networks

Introduction

In recent years, research of bilateral control for nonlinear teleoperation system has collected much attention over the last decades due to its various applications such as disaster rescue, undersea exploration, surveillance, and data acquisition, as observed in the literature.^1–8 Lee and Spong¹ defined a control framework and proposed a position feedback approach for bilateral teleoperation with a proportional–derivative (PD) controller. Chopra et al.² presented a passivity-based control algorithm such that a good position tracking performance can be guaranteed using additional position control on both master and slave robots. Note that the scattering-based methods cannot guarantee position tracking in the presence of time delay, which is in fact a tedious task. Chopra et al.³ raised a coordination architecture to ensure state synchronization of a teleoperation system without using the scattering transformation, which was only applicable to systems without gravity. Subsequently, Nuno et al.⁴ proposed an adaptive technique to remove this restriction. In Nuno et al.,⁵ a globally stable PD-like controller was newly designed to achieve the synchronization of bilateral teleoperation without the delayed derivative action. Moreover, Hashemzadeh et al.⁶ developed a new algorithm, which was capable of handling actuator saturation of a bilateral teleoperation system. However, all the mentioned control schemes above are only applicable to the state synchronization of the joint space. Actually, in the control of a teleoperation system, tasks to be executed are usually given in the Cartesian space, which are more convenient in practical application.

Compared with the joint-space tracking control problem of a teleoperation system,^1–6 task-space tracking control of a teleoperation system is more challenging due to the coupling of kinematic and dynamic uncertainties, which has been an active research topic in recent years. Kawada et al.⁹ proposed a task-space controller to guarantee the delay-independent asymptotic stability of a teleoperation system. In Nath et al.,¹⁰ by combining a nonlinear observer and the invertible transformation technique, a new controller was designed for bilateral synchronization of a kinematically redundant teleoperation system. Wang and Xie¹¹ presented a delay-robust control scheme to achieve position/orientation synchronization based on the task-space passivity framework, in which the Jacobian matrix was assumed to be known. To deal with the uncertain kinematics and dynamics preceded by robotic systems, some interesting results have been presented in the literature.^12–15 Cheah et al.¹² developed an adaptive Jacobian controller for the tracking control of a manipulator with kinematic and dynamic uncertainties. Moreover, Wang and Xie¹³ further proposed an adaptive inverse dynamics controller to ensure the uniform performance of robot over the entire workspace. Talebi and Khorasani¹⁴ presented a neural network (NN)-based method to counteract the effect caused by uncertainties in the model and sensory measurements. Clearly, the control for robot with uncertain kinematics and dynamics has also received much attention. However, the aforementioned works only consider a single robot and their proposed schemes may not be applied to the teleoperation system.

To cope with the control problem of the teleoperation system with uncertain kinematics and dynamics, much effort has been made in previous studies.^16–22 Especially, in Liu and Chopra,¹⁶ an adaptive control algorithm was designed to guarantee the task-space synchronization of a teleoperation system. As an extended research, Liu and Khong¹⁷ further proposed an alternative control algorithm for nonlinear teleoperators with uncertain kinematics and dynamics, in which the requirement of task-space velocity information was removed. In Zhao et al.,^18,19 fuzzy adaptive control approaches were proposed to handle the uncertainties of nonlinear systems. Furthermore, by incorporating a switching-based approach, a unified framework was presented to handle the kinematic uncertainties in Zhai and Xia.²⁰ However, all the aforementioned control schemes ignore actuator uncertainty. In fact, actuator parameters usually vary due to overheating of the actuator or changing ambient temperature, which may cause the degradation of control performance. As a matter of fact, it is still a challenging and unsolved problem to ensure the performance of the teleoperation system with uncertain kinematics, dynamics, and actuator model.

Motivated by such an observation, in this article, we propose an adaptive controller for tracking control of a teleoperation system. One of the major difficulties that obstruct the implementation of the proposed scheme is how to ensure the control performance of a teleoperation system with uncertain kinematics, dynamics, and actuator model. To remove this obstacle, NNs are designed so that the kinematic and dynamic uncertainties can be handled. Moreover, an adaptive controller is established to compensate for the perturbation of actuator parameters online. In summary, this article has the following contributions:

Being different from the previous task-space tracking controller,^16,17,20 the newly proposed one additionally contains a NN mechanism and an adaptive mechanism. Specifically, the proposed NN mechanism is designed to deal with the unknown dynamics, which successfully eliminate the restriction in previous studies^16,17,20 that robot’s mechanical structure is supposed to be known. Subsequently, the adaptive mechanism is employed to cancel the overlarge NNs compensation error. As a result, the proposed adaptive tracking controller is more feasible than those traditionally presented.

Compared with the schemes developed in previous studies,^16,17,20 the actuator parameter transform matrix and parameter adaption law are designed to counteract the effect of overheating of the actuator on tracking control performance. Furthermore, the proposed adaptive tracking controller ensures that the tracking error converges to a small neighborhood of zero, without the need for known kinematic and dynamic parameters.

The remainder of this article is organized as follows. In section “System description and problem statement,” the problem formulation and preliminaries are presented. An adaptive tracking control approach is proposed in section “Adaptive tracking control design” and the experimental results are given in section “Experimental results and analysis.” Finally, this article is concluded in section “Conclusion.”

System description and problem statement

Overview of the teleoperation system

We have implemented a teleopeartion system in our laboratory for the testification of the proposed tracking control approach (Figure 1). The experimental testbed is composed of two heterogeneous robots. The master robot is a Dobot manipulator with 3 degrees of freedom (DOFs), and the slave robot is an AnnoRobot manipulator with 6 DOFs. Specifically, the Dobot manipulator is equipped with an electronic control board and a touch screen. The rigid AnnoRobot manipulator consists of six links and six rotational joints. Moreover, network adapter is installed in the two heterogeneous robots, respectively, and they can communicate with each other through Transmission Control Protocol (TCP) sockets.

Figure 1.

Block diagram of the proposed control scheme.

A closed-loop control architecture for a teleoperation system is constructed. Two major features of teleoperation have been implemented as follows:

Using the network adapter, the master robot can communicate with the slave robot. Thus, remote control could be completed.

The master robot is equipped with touch screen. The slave robot can be remotely controlled to accomplish specific tasks.

Problem statement

As illustrated in Figure 1, the teleoperation system consists of the master manipulator and slave manipulator. Let $i = {m, s}$ be the master and slave manipulators, respectively. Then the kinematics are governed by

\begin{matrix} x_{i} = f_{i} (q_{i}) \end{matrix}

(1)

\begin{matrix} {\overset{\cdot}{x}}_{i} = J_{i} {\overset{\cdot}{q}}_{i} \end{matrix}

(2)

where $q_{i} \in R^{n}$ is the vector of the ith joint position, $f_{i} (q_{i})$ denotes the mapping function between $x_{i}$ and $q_{i}$ , and $J_{i}$ represents the Jacobian matrix describing the nonlinear mapping between task velocity ${\overset{\cdot}{x}}_{i}$ and joint velocity ${\overset{\cdot}{q}}_{i}$ . In this work, it is postulated that the Jacobian matrix $J_{i}$ is not measurable for controller design, which relaxes the restriction in previous studies.^16,17,20

It is well known that the dynamic differentiation equation of the teleoperation system can be expressed as follows

{\begin{matrix} \begin{matrix} u_{i} = H_{i} (q_{i}) {\overset{\cdot\cdot}{q}}_{i} + (\frac{1}{2} {\overset{\cdot}{H}}_{i} (q_{i}) + C_{i} (q_{i}, {\overset{\cdot}{q}}_{i})) {\overset{\cdot}{q}}_{i} + G_{i} (q_{i}) \\ u_{i} = T_{i} τ_{i} \end{matrix} \end{matrix}

(3)

where $H_{i} (q_{i}) \in R^{n \times n}$ denotes the positive inertial matrix, $(1 / 2) {\overset{\cdot}{H}}_{i} (q_{i}) + C (q_{i}, {\overset{\cdot}{q}}_{i}) \in R^{n \times n}$ is the Coriolis and centrifugal matrix, $G_{i} (q_{i})$ represents the gravitational force vector, and $τ_{i}$ is the designed control torque that acted on the ith joint. $T_{i} \in R^{n \times n}$ is the actuator parameter transmission matrix which is a diagonal matrix and relates the designed torque to the actual torque $u_{i}$ . It is worthy to point out that the $H_{i} (q_{i})$ and $C_{i} (q_{i}, {\overset{\cdot}{q}}_{i})$ matrices are both hard to be precisely obtained, and such uncertain dynamics problem imposes a challenging difficulty on the control design.

Note that the dynamic model (3) has the following properties:

Property 1. The inertial matrix $H_{i} (q_{i})$ is uniformly positive definite.

Property 2. $C (q_{i}, {\overset{\cdot}{q}}_{i})$ is a skew-symmetric matrix, which implies that $\forall ζ \in R^{n}$ satisfies

\begin{matrix} ζ^{T} C_{i} (q_{i}, {\overset{\cdot}{q}}_{i}) ζ = 0 \end{matrix}

Property 3. For any differentiable vector, the following equation holds

J (q) ξ = Y_{k} (q) α_{k} \in R^{n}

where $Y_{k} (q)$ is the kinematic matrix and $α_{k}$ is a constant vector containing kinematic parameters.

To deal with the unknown dynamic parameters, the radial basis function neural networks (RBFNNs)^23–26 are employed. In general, the RBFNN contains three layers including the input layer, hidden layer, and the output layer. The input vector x is in the input layer, and each node in the hidden layer includes a Gaussian basis function, which can be formulated as follows

\begin{matrix} Ψ_{i} (x) = \exp (\frac{- {(x - b_{i})}^{T} (x - b_{i})}{σ_{i}^{2}}) \end{matrix}

(4)

where $σ_{i} \in R$ is the width of the Gaussian function and $b_{i}$ is the center of the receptive field. It is well known that RBFNNs can approximate any nonlinear continuous function $f (x)$ over a compact set $x \in Ω$ with any prescribed accuracy $ϵ > 0$ . Thus, the following lemma holds.

Lemma 1

Let $f (x)$ be any given continuous function defined on a compact set $Ω$ . Then, there exists an RBFNN such that^27–29

sup_{x \in Ω} | f (x) - \sum_{i = 1}^{N} W_{i}^{*} Ψ_{i} (x) | \leq ϵ

where $W_{i}^{*}$ is the ideal weight vector, N is the number of neural nodes in the hidden layer, and $ϵ$ denotes an arbitrary positive constant.

Adaptive tracking control design

Clearly, the Jacobian matrix J is unknown. We design $\hat{J}$ to approximate the constant matrix J, and the estimated error is represented by

\begin{matrix} \tilde{J} = \hat{J} - J \end{matrix}

(5)

For stabilizing the system in equation (3), the nominal reference of joint-space velocity is defined as

\begin{matrix} {\overset{\cdot}{q}}_{ri} = {\hat{J}}_{i}^{- 1} (q_{i}) ({\hat{\overset{\cdot}{x}}}_{i} + e_{i}) \end{matrix}

(6)

where the subscript $i = m, s$ denotes the master and slave robots, ${\hat{J}}_{i}^{- 1}$ represents the inverse of ${\hat{J}}_{i}$ , and $e_{i}$ represents the tracking errors, which are defined as $e_{m} = x_{s} - x_{m}$ and $e_{s} = x_{m} - x_{s}$ .

Subsequently, an adaptive sliding vector $s_{i}$ is defined as

s_{i} = {\overset{\cdot}{q}}_{i} - {\overset{\cdot}{q}}_{ri}

(7)

From the definitions of $Δ x_{i} = {\hat{x}}_{i} - x_{i}$ and equation (5), we can rewrite equation (7) as

\begin{matrix} \begin{matrix} s_{i} = J_{i}^{- 1} ({\overset{\cdot}{x}}_{i} - J_{i} {\overset{\cdot}{q}}_{ri}) \\ = J^{- 1} ({\overset{\cdot}{x}}_{i} - {\hat{\overset{\cdot}{x}}}_{i} - e_{i} + {\tilde{J}}_{i} {\overset{\cdot}{q}}_{ri}) \end{matrix} \\ = J_{i}^{- 1} (- Δ {\overset{\cdot}{x}}_{i} - e_{i} + {\tilde{J}}_{i} {\overset{\cdot}{q}}_{ri}) \end{matrix}

(8)

Adaptive controller design

In this article, NNs are used to approximate the dynamics (equation (3)) for its excellent approximation ability. To alleviate the computational burden, the dynamics (equation (3)) are divided into two parts. Specifically, let $N_{i}^{1} (X_{i}^{1}) = H_{i} (q_{i}) {\overset{\cdot\cdot}{q}}_{ri}$ and $N_{i}^{2} (X_{i}^{2}) = ((1 / 2) {\overset{\cdot}{H}}_{i} (q_{i}) + C_{i} (q_{i}, {\overset{\cdot}{q}}_{i})) {\overset{\cdot}{q}}_{ri} + G_{i} (q_{i})$ , where $X_{i}^{1} = [{\overset{\cdot\cdot}{q}}_{ri}^{T}, q_{i}^{T}]^{T}$ and $X_{i}^{2} = [{\overset{\cdot}{q}}_{ri}^{T}, {\overset{\cdot}{q}}_{i}^{T}, q_{i}^{T}]$ . NNs are designed to approximate $N_{i}^{1} (X_{i}^{1})$ and $N_{i}^{2} (X_{i}^{2})$ , respectively.

From Lemma 1, we have

\begin{matrix} N_{i}^{1} (X_{i}^{1}) = W_{i}^{1} Ψ_{i}^{1} (X_{i}^{1}) + ϵ_{i}^{1} \end{matrix}

(9)

\begin{matrix} N_{i}^{2} (X_{i}^{2}) = W_{i}^{2} Ψ_{i}^{2} (X_{i}^{2}) + ϵ_{i}^{2} \end{matrix}

(10)

where $ϵ_{i}^{1}$ and $ϵ_{i}^{2}$ denote the minimums of modeling errors and $W_{i}^{1}$ and $W_{i}^{2}$ are the vectors of the weights of NNs, respectively.

Now, we present the proposed tracking controller as follows

u_{i} (t) = - {\hat{T}}_{i}^{- 1} τ_{i} + {\hat{T}}_{i}^{- 1} Y_{a, i} (τ_{i}) {\hat{α}}_{a, i}

(11)

where ${\hat{T}}_{i} \in R_{n \times n}$ denotes the approximate transmission matrix and the second is an estimated dynamic compensation term. In particular, $Y_{a, i} (τ_{i}) = diag {- τ_{i 1}, - τ_{i 2}, \dots, - τ_{in}}$ is a diagonal matrix and ${\hat{α}}_{a, i}$ is the estimate of the actuator adaption parameter $α_{a, i}$ .

Specifically

τ_{i} = - {\hat{N}}_{i} + L_{si} s_{i} - L_{di} {\hat{J}}_{i}^{T} {\overset{\cdot}{e}}_{i} - ψ_{i}

(12)

where ${\hat{N}}_{i} = {\hat{N}}_{i}^{1} (X_{i}^{1}) + {\hat{N}}_{i}^{2} (X_{i}^{2})$ , $L_{si}$ and $L_{di}$ are the positive definite matrices for $i = m, s$ , and $ψ_{i}$ represents the robust adaptive compensator as defined in equation (13)

ψ_{i} = - {\hat{R}}_{i} ϕ (s_{i})

(13)

where $R_{i} \geq sup | ϵ_{i} |$ , ${\hat{R}}_{i}$ represents the estimate of R_i’s upper bound, and $ϕ (•)$ is a sign function.

By substituting the controller equation (12) into the dynamics equation (3), the closed-loop teleoperation system can be formulated as

\begin{matrix} H_{i} (q_{i}) {\overset{\cdot}{s}}_{i} + (\frac{1}{2} {\overset{\cdot}{H}}_{i} (q_{i}) + C_{i} (q_{i}, {\overset{\cdot}{q}}_{i})) s \\ - {\tilde{N}}_{i} + B_{i} + (T_{i} {\hat{T}}_{i}^{- 1} - I) τ_{i} - T_{i} {\hat{T}}_{i}^{- 1} Y_{a, i} (τ_{i}) {\hat{α}}_{a, i} = 0 \end{matrix}

(14)

where $B_{i} = L_{si} s_{i} - L_{di} {\hat{J}}_{i}^{T} {\overset{\cdot}{e}}_{i} - Φ_{i}$ , ${\tilde{N}}_{i} = {\hat{N}}_{i} - N_{i} = ({\tilde{W}}_{i}^{1})^{T} Ψ_{i}^{1} (X_{i}^{1}) + ({\tilde{W}}_{i}^{2})^{T} Ψ_{i}^{2} (X_{i}^{2})$ is the estimate error, and $Φ_{i} = ψ_{i} - ϵ_{i}$ .

In combination with the definition of $Y_{a, i}$ , the last two terms of equation (14) can be reformulated as

\begin{matrix} (T_{i} {\hat{T}}_{i}^{- 1} - I) τ_{i} - T_{i} {\hat{T}}_{i}^{- 1} Y_{a, i} (τ_{i}) {\hat{α}}_{a, i} \\ = Y_{a, i} (τ_{i}) ({\bar{α}}_{a, i} - T_{i} {\hat{T}}_{i}^{- 1} {\hat{α}}_{a, i}) \end{matrix}

(15)

where ${\bar{α}}_{a, i} = 1 - (T_{ij} / {\hat{T}}_{ij})$ and $T_{ij}$ and ${\hat{T}}_{ij}$ are the jth diagonal elements of $T_{i}$ and ${\hat{T}}_{i}$ .

By substituting equation (15) into equation (14), the closed-loop dynamics can be further rewritten as

\begin{matrix} H_{i} (q_{i}) {\overset{\cdot}{s}}_{i} + (\frac{1}{2} {\overset{\cdot}{H}}_{i} (q_{i}) + C_{i} (q_{i})) s - {\tilde{N}}_{i} \\ + B_{i} + Y_{a, i} (τ_{i}) Δ {\bar{α}}_{a, i} = 0 \end{matrix}

(16)

where $Δ {\bar{α}}_{a, i} = {\bar{α}}_{a, i} - T_{i} {\hat{T}}_{i}^{- 1} {\hat{α}}_{a, i}$ . Thus, the following equation holds

\begin{matrix} Δ {\overset{\cdot}{\bar{α}}}_{a, i} = - T_{i} {\hat{T}}_{i}^{- 1} {\overset{\cdot}{\hat{α}}}_{a, i} \end{matrix}

(17)

Multiplying $s_{i}^{T}$ on both sides of the closed-loop teleoperation system (16), we can get

\begin{matrix} \frac{1}{2} s_{i}^{T} {\overset{\cdot}{H}}_{i} (q_{i}) s_{i} = s^{T} (+ {\tilde{N}}_{i} - B_{i} - Y_{a, i} (τ_{i}) Δ {\bar{α}}_{a, i} \\ - H_{i} (q_{i}) {\overset{\cdot}{s}}_{i} - C_{i} (q_{i}, {\overset{\cdot}{q}}_{i}) s) \end{matrix}

(18)

The parameter adaption law for the actuator is defined as

\begin{matrix} {\overset{\cdot}{\hat{α}}}_{a, i} = - ϒ_{ai} Y_{a, i}^{T} s_{i} \end{matrix}

(19)

The parameter adaption law is defined as

\begin{matrix} {\overset{\cdot}{\hat{α}}}_{k, i} = - ϒ_{ki} L_{si} Y_{k, i}^{T} Δ x_{i} \end{matrix}

(20)

The robust adaptive compensator is defined as

\begin{matrix} {\overset{\cdot}{\hat{R}}}_{i} = Γ_{i} ∥ s_{i}^{T} ∥ - Ω_{i} {\hat{R}}_{i} \end{matrix}

(21)

The parameter adaption laws for NN are given as

\begin{matrix} {\overset{\cdot}{\hat{W}}}_{i}^{1} = - Ξ_{1 i} Ψ_{i}^{1} (X_{i}^{1}) s_{i}^{T} - Λ_{1 i} {\hat{W}}_{i}^{1} \end{matrix}

(22)

\begin{matrix} {\overset{\cdot}{\hat{W}}}_{i}^{2} = - Ξ_{2 i} Ψ_{i}^{2} (X_{i}^{2}) s_{i}^{T} - Λ_{2 i} {\hat{W}}_{i}^{2} \end{matrix}

(23)

where $Ξ_{1 i}, Ξ_{2 i}, Λ_{1 i}, Λ_{2 i}$ are the positive definite diagonal matrices.

Remark 1

In contrast to the existing works dealing with kinematic uncertainty, the difference should be clarified. The feedback part in equation (12) can be reformulated as $L_{si} J_{i}^{- 1} (- Δ {\overset{\cdot}{x}}_{i} - e_{i} + {\tilde{J}}_{i} {\overset{\cdot}{q}}_{ri}) - L_{di} {\hat{J}}_{i}^{T} {\overset{\cdot}{e}}_{i}$ , which can be intuitively interpreted as the inverse Jacobian feedback of the task-space tracking errors and the joint reference velocity. It is worthy to point out that the inverse Jacobian feedback of the kinematic parameter estimation error supplies the control law (equation (12)) with the ability of handling the kinematic uncertainties. To clearly illustrate the overall closed-loop control architecture, a graphical result is shown in Figure 2.

Figure 2.

Closed-loop control architecture.

Stability analysis

Theorem 1

Consider a teleoperation system with uncertain kinematics and dynamics controlled by the proposed controller (equation (12)) with the corresponding adaption rules given by equations (19)–(23). It is proven that all physical variables in the closed-loop system remain semi-globally uniformly ultimately bounded (SGUUB). Moreover, the tracking errors $e_{m}$ and $e_{s}$ are asymptotically convergent to a small neighborhood of zero.

Proof

The designed Lyapunov function candidate consists of two parts, which are $V_{1}$ for the control tracking part and the second one $V_{2}$ for the adaption part, respectively. In particular

\begin{matrix} V_{1} = \frac{1}{2} s_{i}^{T} H_{i} (q_{i}) s_{i} + \frac{1}{2} L_{di} e_{i}^{T} e_{i} + \frac{1}{2} Δ α_{k, i}^{T} ϒ_{ki}^{- 1} Δ α_{k, i} \\ + \frac{1}{2} L_{si} Δ x_{i}^{T} Δ x + \frac{1}{2} Δ {\bar{α}}_{a, i}^{T} ϒ_{ai}^{- 1} {\hat{T}}_{i} T_{i}^{- 1} Δ {\bar{α}}_{a, i} \end{matrix}

(24)

\begin{matrix} V_{2} = \frac{1}{2} \sum_{j = 1}^{Ni} ({\tilde{W}}_{i}^{1})_{j}^{T} Ξ_{1 i}^{- 1} ({\tilde{W}}_{i}^{1})_{j} + \frac{1}{2} \sum_{j = 1}^{Ni} ({\tilde{W}}_{i}^{2})_{j}^{T} Ξ_{2 i}^{- 1} ({\tilde{W}}_{i}^{2})_{j} \\ + \frac{1}{2} \sum_{j = 1}^{Ni} ({\tilde{R}}_{i})_{j}^{T} Γ_{i}^{- 1} ({\tilde{R}}_{i})_{j} \end{matrix}

(25)

where $L_{di}$ and $L_{si}$ are the appropriate positive constants and $N_{i}$ denotes the total number of neural nodes.

By differentiating $V_{1}$ with respect to the time variable, one has

\begin{matrix} {\overset{\cdot}{V}}_{1} = s_{i}^{T} H_{i} (q_{i}) {\overset{\cdot}{s}}_{i} + \frac{1}{2} s_{i}^{T} {\overset{\cdot}{H}}_{i} (q_{i}) s_{i} + L_{di} e_{i}^{T} {\overset{\cdot}{e}}_{i} \\ + Δ α_{k, i}^{T} ϒ_{ki}^{- 1} Δ {\overset{\cdot}{α}}_{k, i} + L_{si} Δ x_{i}^{T} Δ \overset{\cdot}{x} \\ + Δ {\bar{α}}_{a, i}^{T} ϒ_{ai}^{- 1} {\hat{T}}_{i} T_{i}^{- 1} Δ \overset{\cdot}{\bar{α_{a, i}}} \end{matrix}

(26)

Substituting equations (17)–(19) into equation (26), we have

\begin{matrix} {\overset{\cdot}{V}}_{1} = s_{i}^{T} H_{i} (q_{i}) {\overset{\cdot}{s}}_{i} + s^{T} (+ {\tilde{N}}_{i} - B_{i} - Y_{a, i} (τ_{i}) Δ {\bar{α}}_{a, i} \\ - H_{i} (q_{i}) {\overset{\cdot}{s}}_{i} - C_{i} (q_{i}, {\overset{\cdot}{q}}_{i}) s_{i}) + L_{di} e_{i}^{T} {\overset{\cdot}{e}}_{i} \\ + Δ α_{k, i}^{T} ϒ_{ki}^{- 1} Δ {\overset{\cdot}{α}}_{k, i} + L_{si} Δ x_{i}^{T} Δ {\overset{\cdot}{x}}_{i} \\ + Δ {\bar{α}}_{a, i}^{T} ϒ_{ai}^{T} {\hat{T}}_{i} T_{i}^{- 1} Δ \overset{\cdot}{\bar{α_{a, i}}} \\ = s^{T} (+ {\tilde{N}}_{i} - B_{i} - C_{i} (q_{i}, {\overset{\cdot}{q}}_{i}) s) + L_{di} e_{i}^{T} {\overset{\cdot}{e}}_{i} \\ + Δ α_{k, i}^{T} ϒ_{ki}^{- 1} Δ {\overset{\cdot}{α}}_{k, i} + L_{si} Δ x_{i}^{T} Δ {\overset{\cdot}{x}}_{i} \end{matrix}

(27)

Using Property 2, equation (27) can be formulated as

\begin{matrix} {\overset{\cdot}{V}}_{1} = s_{i}^{T} (+ {\tilde{N}}_{i} - B_{i}) + L_{di} e_{i}^{T} {\overset{\cdot}{e}}_{i} \\ + Δ α_{k, i}^{T} ϒ_{ki}^{- 1} Δ {\overset{\cdot}{α}}_{k, i} + L_{si} Δ x_{i}^{T} Δ {\overset{\cdot}{x}}_{i} \end{matrix}

(28)

By combining the definition of $Δ x_{i} = {\hat{x}}_{i} - x_{i}$ with Property 3 and equation (20), the last term in equation (28) can be rewritten as

\begin{matrix} L_{si} Δ x_{i}^{T} Δ {\overset{\cdot}{x}}_{i} = L_{si} Δ x_{i}^{T} (\overset{\cdot}{\hat{x_{i}}} - {\overset{\cdot}{x}}_{i}) \\ = L_{si} Δ x_{i}^{T} (Y_{ki} {\hat{α}}_{ki} - Y_{ki} α_{ki}) \\ = L_{si} Δ x_{i}^{T} Y_{ki} Δ α_{ki} \end{matrix}

(29)

Invoking equation (20) yields

\begin{matrix} Δ α_{k, i}^{T} ϒ_{ki}^{- 1} Δ {\overset{\cdot}{α}}_{k, i} = Δ α_{k, i}^{T} ϒ_{ki}^{- 1} \overset{\cdot}{\hat{α_{k, i}}} \\ = - L_{si} Δ α_{k, i}^{T} Y_{k, i}^{T} Δ x_{i} \end{matrix}

(30)

By recalling $\hat{\overset{\cdot}{x_{i}}} = {\hat{J}}_{i} {\overset{\cdot}{q}}_{i}$ , we have

\begin{matrix} {\hat{J}}_{i} s_{i} = {\hat{J}}_{i} {\overset{\cdot}{q}}_{i} - {\hat{J}}_{i} {\overset{\cdot}{q}}_{ri} \\ = {\hat{J}}_{i} {\overset{\cdot}{q}}_{i} - e_{i} - \hat{\overset{\cdot}{x_{i}}} \\ = - e_{i} \end{matrix}

(31)

From the definition of $Φ_{i}$ and equation (13), the following inequality holds

\begin{matrix} s_{i}^{T} Φ_{i} = s_{i}^{T} (ψ_{i} - ϵ_{i}) \leq | s_{i}^{T} | R_{i} + s_{i}^{T} ψ_{i} \\ = - | s_{i}^{T} | {\tilde{R}}_{i} \end{matrix}

(32)

In combination with the definition of variable B, substituting equations (29)–(32) into equation (28) yields

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq s_{i}^{T} {\tilde{N}}_{i} - L_{si} s_{i}^{T} s_{i} + L_{di} ({\hat{J}}_{i} s_{i})^{T} {\overset{\cdot}{e}}_{i} - | s_{i}^{T} | {\tilde{R}}_{i} + L_{di} e_{i}^{T} {\overset{\cdot}{e}}_{i} \\ \leq s_{i}^{T} {\tilde{N}}_{i} - L_{si} s_{i}^{T} s_{i} - | s_{i}^{T} | {\tilde{R}}_{i} \end{matrix}

(33)

Subsequently, differentiating $V_{2}$ yields

\begin{matrix} {\overset{\cdot}{V}}_{2} = - \sum_{j = 1}^{N_{i}} ({\tilde{W}}_{i}^{1})_{j}^{T} Ξ_{i}^{- 1} (\overset{\cdot}{\tilde{W_{i}^{1}}})_{j} - \sum_{j = 1}^{N_{i}} ({\tilde{W}}_{i}^{2})_{j}^{T} Ξ_{i}^{- 1} (\overset{\cdot}{\tilde{W_{i}^{2}}})_{j} \\ + \sum_{j = 1}^{N_{i}} ({\tilde{R}}_{i})_{j}^{T} Γ_{i}^{- 1} (\overset{\cdot}{\tilde{R_{i}}})_{j} \end{matrix}

(34)

Using the parameter adaption laws (equations (21)–(23)), ${\overset{\cdot}{V}}_{2}$ is rewritten as

\begin{matrix} {\overset{\cdot}{V}}_{2} = - s_{i}^{T} {\tilde{N}}_{i} - \sum_{j = 1}^{N_{i}} ({\tilde{W}}_{i}^{1})_{j}^{T} Ξ_{1 i}^{- 1} Λ_{1 i} ({\hat{W}}_{i}^{1})_{j} \\ - \sum_{j = 1}^{N_{i}} ({\tilde{W}}_{i}^{2})_{j}^{T} Ξ_{2 i}^{- 1} Λ_{2 i} ({\hat{W}}_{i}^{2})_{j} + \sum_{j = 1}^{N_{i}} ({\tilde{R}}_{i})_{j}^{T} Γ_{i}^{- 1} (\overset{\cdot}{\tilde{R_{i}}})_{j} \\ = - s_{i}^{T} {\tilde{N}}_{i} - \sum_{j = 1}^{N_{i}} ({\tilde{W}}_{i}^{1})_{j}^{T} Ξ_{1 i}^{- 1} Λ_{1 i} ({\hat{W}}_{i}^{1})_{j} + {\tilde{R}}_{i} ∥ s_{i}^{T} ∥ \\ - \sum_{j = 1}^{N_{i}} ({\tilde{W}}_{i}^{2})_{j}^{T} Ξ_{2 i}^{- 1} Λ_{2 i} ({\hat{W}}_{i}^{2})_{j} - \sum_{j = 1}^{N_{i}} ({\tilde{R}}_{i})_{j}^{T} Γ_{i}^{- 1} Ω_{i} ({\hat{R}}_{i})_{j} \end{matrix}

(35)

Invoking Yong’s inequality, we have

\begin{matrix} - ({\tilde{W}}_{i}^{1})_{j}^{T} Ξ_{1 i}^{- 1} Λ_{1 i} ({\hat{W}}_{i}^{1})_{j} \leq - \frac{1}{2} ({\tilde{W}}_{i}^{1})_{j}^{T} Ξ_{1 i}^{- 1} Λ_{1 i} ({\tilde{W}}_{i}^{1})_{j} \\ + \frac{1}{2} (W_{i}^{1})_{j}^{T} Ξ_{1 i}^{- 1} Λ_{1 i} (W_{i}^{1})_{j} \end{matrix}

(36)

\begin{matrix} - ({\tilde{W}}_{i}^{2})_{j}^{T} Ξ_{2 i}^{- 1} Λ_{2 i} ({\hat{W}}_{i}^{2})_{j} \leq - \frac{1}{2} ({\tilde{W}}_{i}^{2})_{j}^{T} Ξ_{2 i}^{- 1} Λ_{2 i} ({\tilde{W}}_{i}^{2})_{j} \\ + \frac{1}{2} (W_{i}^{2})_{j}^{T} Ξ_{2 i}^{- 1} Λ_{2 i} (W_{i}^{2})_{j} \end{matrix}

(37)

\begin{matrix} - ({\tilde{R}}_{i})_{j}^{T} Γ_{i}^{- 1} Ω_{i} ({\tilde{R}}_{i})_{j} \leq - \frac{1}{2} ({\tilde{R}}_{i})_{j}^{T} Γ_{i}^{- 1} Ω_{i} ({\tilde{R}}_{i})_{j} \\ + \frac{1}{2} (R_{i})_{j}^{T} Γ_{i}^{- 1} Ω_{i} (R_{i})_{j} \end{matrix}

(38)

By differentiating $V (t) = V_{1} (t) + V_{2} (t)$ with respect to time, one has

\begin{matrix} \overset{\cdot}{V} (t) = {\overset{\cdot}{V}}_{1} (t) + {\overset{\cdot}{V}}_{2} (t) \\ \leq - L_{si} s_{i}^{T} s_{i} - \frac{1}{2} \sum_{j = 1}^{N_{i}} ({\tilde{W}}_{i}^{1})_{j}^{T} Ξ_{1 i}^{- 1} Λ_{1 i} ({\tilde{W}}_{i}^{1})_{j} \\ - \frac{1}{2} \sum_{j = 1}^{N_{i}} ({\tilde{W}}_{i}^{2})_{j}^{T} Ξ_{2 i}^{- 1} Λ_{2 i} ({\tilde{W}}_{i}^{2})_{j} \\ - \frac{1}{2} \sum_{j = 1}^{N_{i}} ({\tilde{R}}_{i})_{j}^{T} Γ_{i}^{- 1} Ω_{i} ({\tilde{R}}_{i})_{j} + ϱ \end{matrix}

(39)

where

\begin{matrix} ϱ_{i} = \frac{1}{2} \sum_{j = 1}^{N_{i}} (W_{i}^{1})_{j}^{T} Ξ_{1 i}^{- 1} Λ_{1 i} (W_{i}^{1})_{j} + \frac{1}{2} \sum_{j = 1}^{N_{i}} (W_{i}^{2})_{j}^{T} Ξ_{2 i}^{- 1} Λ_{2 i} (W_{i}^{2})_{j} \\ + \frac{1}{2} \sum_{j = 1}^{N_{i}} (R_{i})_{j}^{T} Γ_{i}^{- 1} Ω_{i} (R_{i})_{j} \end{matrix}

(40)

Supposing that the gain matrices $L_{si} > 0$ , a positive constant h is further defined as follows

\begin{matrix} h = \min {2 L_{si} ξ_{\min} (H (q_{i})), ξ_{\min} (Λ_{1 i}), \\ ξ_{\min} (Λ_{2 i}), ξ_{\min} (Ω_{i}); i = s, m} \end{matrix}

(41)

where $ξ_{\min} (•)$ represents the minimum eigenvalue of the matrix (•).

Thus, $\overset{\cdot}{V}$ can be reformulated as

\begin{matrix} \overset{\cdot}{V} (t) \leq - hV + ϱ \end{matrix}

(42)

where $ϱ = ϱ_{s} + ϱ_{m}$ . By integrating equation (42) over a given time period [0, t], we have

\begin{matrix} V (t) \leq (V (0) - \frac{ϱ}{h}) e^{- ht} + \frac{ϱ}{h} \end{matrix}

(43)

As indicated in equation (43), the Lyapunov function candidate V is nonincreasing. Subsequently, $s_{i}$ , $e_{i}$ , $Δ x_{i}$ , ${\tilde{W}}_{i}$ , and ${\tilde{R}}_{i}$ are bounded for $i = m, s$ . According to equation (43), the closed-loop signals are SGUUB. It can be concluded that the tracking errors $e_{m}$ and $e_{s}$ can be made arbitrarily small by choosing the parameters appropriately.

Experimental results and analysis

System setup

To validate the proposed control scheme, we have established the testbed (Figure 3). The experimental setup and system implementation are described in this section. The D-H parameters of Dobot and AnnoRobot manipulators are presented in Tables 1 and 2. In addition, two scenarios involving bilateral synchronization and effectiveness verification are carried out.

Figure 3.

Schematic diagram of the established testbed.

Table 1.

The D-H parameters of Dobot manipulator.

Joint	$θ$	d (mm)	a (°)	$α (°)$
1	$θ_{1}$	80	0	90
2	$θ_{2}$	0	135	0
3	$θ_{3}$	0	160	0

Table 2.

The D-H parameters of AnnoRobot manipulator.

Joint	$θ$	d (mm)	a (°)	$α (°)$
1	$θ_{1}$	264	0	90
2	$θ_{2}$	0	225	0
3	$θ_{3}$	70	0	90
4	$θ_{4}$	147.3	0	90
5	$θ_{5}$	40	0	90
6	$θ_{6}$	0	150	0

The proposed control system consists of two parts including the main control board and the driving board. Specifically, the ATmega64 acts as the chipset of the driving board, and it has three important functions: first, it collects the real-time data through a set of sensors, for example, the angle velocity ${\overset{\cdot}{q}}_{i}$ from MPU-6050 via an serial peripheral interface (SPI) bus; second, it receives the control commands from the main control board through an RS232 interface with 115,200 bps; finally, the driving board runs the proportion–differentiation (PD) control algorithm such that the motor speed converges to the desired control speed.

Experimental results

In the first case, the objective is to control the end effector of the slave manipulator to track one of the master manipulators. Both the master and slave robots are Dobot manipulators. The robotic manipulators are initialized as $x_{m} (0) = (0.24 m, 0.01 m, 0.095 m)$ , $x_{s} (0) = (0.15 m, 0.015 m, 0.095 m)$ , ${\overset{\cdot}{x}}_{m} = {\overset{\cdot}{x}}_{s} = (0, 0, 0)$ . Recalling Theorem 1, to construct the presented controller, the designed parameters are selected as $L_{si} = diag (15)$ , $L_{di} = diag (9)$ , $Ξ_{1 i} = Ξ_{2 i} = diag (1)$ , $Λ_{1 i} = Λ_{2 i} = diag (1)$ . The initializations for adaptive parameters are given by $Γ_{i} = diag (0.5)$ and $Ω_{i} = diag (0.1)$ . It is worthy to point out that, in practical implementation, the joint velocity ${\overset{\cdot}{q}}_{i}$ and acceleration ${\overset{\cdot\cdot}{q}}_{i}$ are physically restricted to [−3.1 rad/s, 3.1 rad/s] and [–16.3 rad/s, 16.3 rad/s], respectively. For this reason, $N_{i}^{1} (X_{i}^{1})$ totally contains 20 centers evenly spaced in $[- 16.3, 16.3] \times [- 3.14, 3.14]$ . Then, the widths are chosen as $σ_{1} = σ_{2} = d_{\min} / \sqrt{2 N}$ , where $d_{\min}$ is the minimal distance between any two centers. Similarly, the centers for $N_{i}^{2} (X_{i}^{2})$ are evenly spaced in $[- 3.1, 3.1] \times [- 3.1, 3.1] \times [- 3.14, 3.14]$ . Then, the widths are also calculated correspondingly. The actuator parameter transmission matrix $T_{i}$ is estimated as ${\hat{T}}_{i} = {1.1, 1}$ . It should be pointed out that the actuator parameter transmission matrix would change due to overheating of the actuator. Thus, the adaptation law (equation (19)) is designed to compensate for the parameters’ perturbation. Moreover, in the first experiment, the sampling time is set at $Δ t = 0.01 s$ . With the above initializations, the tracking control algorithm in Theorem 1 is applied to the teleoperation system (3).

To show the effectiveness of the proposed scheme in terms of tracking performance, the motion trajectory of the master’s end effector is expected to be a right triangle, and the three vertices of the right triangle are defined as $(0.244, 0, 0.095)$ , $(0.124, 0, 0.095)$ , $(0.124, - 0.05, 0.095)$ . The human operator sends control commands to the master robot by dragging the components of the robot on touch screen, and the experimental results are shown in Figures 4 and 5. As shown in Figures 4 and 5, there are some peaks in the evolutions of tracking errors, which coincide with the ones of x and y positions. It is noticeable that the teleoperation system is stable and the tracking errors between the master and slave manipulators are steered to a small neighborhood of zero asymptotically.

Figure 4.

Case 1: x position trajectories and tracking errors.

Figure 5.

Case 1: y position trajectories and tracking errors.

To further demonstrate the effectiveness and the general applicability of the proposed algorithm, comparative experiments with the one in Liu and Khong¹⁷ are carried out. In the second case, the master and slave robots are, respectively, Dobot manipulator and AnnoRobot manipulator. To simulate the working condition of overheating of the actuator, both the master and slave robots are required to perform 20,000 reciprocating motions and each duration lasts 5 s. Subsequently, the master robot is required to draw a rectangle and the corresponding coordinates of the four vertices are defined as $(0.27 m, 0 m, 0.48 m)$ , $(0.19 m, 0 m, 0.48 m)$ , $(0.19 m, - 0.05 m, 0.48 m)$ , and $(0.27 m, - 0.05 m, 0.48 m)$ . The initial task-space positions are given by $x_{m} (0) = (0.26 m, 0.001 m, 0.48 m)$ , $x_{s} (0) = (0.25 m, 0.012 m, 0.48 m)$ , ${\overset{\cdot}{x}}_{m} = {\overset{\cdot}{x}}_{s} = (0, 0, 0)$ . To facilitate the analysis, the control parameters and initial conditions are kept the same as the first experiment. With the above parameters, the proposed controller in Theorem 1 is implemented, and the experimental results are shown in Figures 6 and 7. As shown in Figures 6 and 7, the tracking errors $e_{m}$ and $e_{s}$ are both convergent to a small neighborhood of zero under the control of the presented adaptive controller. It should be pointed out that, unlike the proposed method, the one in Liu and Khong¹⁷ did not take into account the parameters’ perturbation in the actuator so that it cannot guarantee a satisfactory convergence speed. On the other hand, the proposed scheme ensures a better control performance due to the fact that the parameter adaption law is additionally incorporated in the proposed controller.

Figure 6.

Case 2: x position trajectories and tracking errors.

Figure 7.

Case 2: y position trajectories and tracking errors.

Conclusion

In this article, we consider the adaptive tracking control problem of a teleoperation system with uncertain kinematics and dynamics. A novel actuator update law is proposed to compensate for the actuator parameters’ perturbation. By resorting to the adaptive control technique and in combination with the NN technique, an adaptive tracking control approach is established such that the tracking errors are steered into an adjustable neighborhood of zero. Finally, the effectiveness of the proposed control approach was confirmed by experimental results.

Footnotes

Handling Editor: Dan Zhang

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 work was supported by the National Natural Science Foundation of China under Grant Nos 61573108, U1613223, U1501251, and U1613223, in part by the Natural Science Foundation of Guangdong Province under Grant Nos 2016A030313715 and 2016A030313018, in part by the China Postdoctoral Science Foundation under Grant No. 2018M633353, in part by the Fundamental Research Funds for the Central Universities under Grant No. ZYGX2016J140, in part by the Scientific and Technical Supporting Programs of Sichuan Province under Grant Nos 2016GZ0395, 2017GZ0391, and 2017GZ0392, in part by the Science and Technology Project of Guangdong Province under Grant No. 2016A020220003, and in part by the Science and Technology Planning Program of Zhongshan City under Grant Nos 2017SF0603 and 2017A1024.

ORCID iD

Liang Yang

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Adaptive tracking control of a nonlinear teleoperation system with uncertainties in kinematics and dynamics

Abstract

Keywords

Introduction

System description and problem statement

Overview of the teleoperation system

Problem statement

Lemma 1

Adaptive tracking control design

Adaptive controller design

Remark 1

Stability analysis

Theorem 1

Proof

Experimental results and analysis

System setup

Experimental results

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References