Adaptive bilateral control of teleoperators with actuator uncertainty and quantized input

Abstract

This article addresses the control and stabilization problem of bilateral teleoperator system with input quantized by the so-called hysteresis-type quantizer, where the joint actuator model, robot dynamics, and kinematics are uncertain. In order to overcome the control issue that only discrete and finite control values can be applied to the robot dynamics, a decomposition-based technique is adopted to cope with the quantized control signals. By incorporating the Nussbaum-gain function to the proposed controller, an adaptive telecontrol scheme together with the parameter updating laws is developed to achieve position and velocity tracking of teleoperator system in the presence time delays without a priori knowledge of actuator and quantizer model parameters, upon which, the robot dynamics and kinematics uncertainties are also effectively accommodated. It is proven by the Lyapunov method that the closed-loop dynamics are bounded, and the position as well as velocity tracking errors converges to zero. Simulation results verify the proposed adaptive telecontrol scheme.

Keywords

Adaptive control robot control bilateral teleoperators quantizer actuator uncertainties

Introduction

The past two decades have witnessed a rapid development on bilateral control for teleoperator system due to its powerful capabilities on accomplishing complicated tasks in remote or hazardous environments, including outer space exploration, surgical operations, nuclear plant handling, and so on.^1–6 Bilateral teleoperation system, commonly comprising remote and local robots, extends the human operation capability and self-perception to execute the specific task from local robots to remote robots, giving humans the high degree of adaptation to handle unexpectable changes. To achieve such objectives, many results have been devoted to the bilateral control for teleoperation systems via the scattering theory or the wave variables.^1–3,7 However, this kind of classic scattering transformation scheme cannot ensure position coordination in the sense that the position errors between remote and local robot asymptotically converge to zero. Thus, the study on teleoperator control without using the scattering theory becomes a mostly control theoretic arena.

To cope with the position and velocity synchronization for bilateral teleoperator system subject to dynamic and kinematical uncertainty without using the scattering theory, some outstanding works can be found in the previous literature.^8–19 The results in literature^8–12 demonstrate that adaptive P-type or PD-type controllers can globally stabilize the teleoperation system under constant or variable time delays, simultaneously providing favorable performance of joint-space position and velocity synchronization. Specially, the approaches proposed in Wang and Xie^17,20 extend joint-space synchronization to task-space position/orientation coordination by exploiting the passivity property of the quaternion-based orientation kinematics. In the recent notable publications,^13,14 adaptive delay-based control schemes are presented to achieve task-space position and velocity tracking for teleoperator system with uncertain dynamics and kinematics. However, one strict requirement of the above results^8–17,20,21 is that the control signal generator must be physically equipped in the actuating mechanism, inevitably reducing the flexibility of robot operation in the hazardous and remote environment with access restrictions. An intuitive perception for teleoperator system is that the control signals should be transferred by the wireless devices rather than the physical connection (e.g. control umbilical cord). To economize limited communication bandwidth, the control signals are required to be processed by the so-called quantizer such that a continuous signal is mapped to discrete finite sets.

Along the line of quantized control input, some novel results are given in literature.^22–28 By utilizing uniform quantizer,²² the control signals are divided into equivalent and average quantization blocks, yet, this kind of quantizer is generally not applicable to the asymptotically tracking control since only fixed quantization level is available even if exact quantization precision is required. To overcome this issue, logarithmic quantizers are adopted in the works by Fu and Xie²³ and Xing et al.,²⁴ and the adaptive control schemes are based upon the sector-bounded property of quantization errors to design appropriate control laws. In previous literature,^25–28 hysteretic quantizer, which can be regarded as a combination of two logarithmic quantizers with identical gain but opposite direction, is used to avoid chattering phenomenon. However, the above results assume that the structural property and exact parametric knowledge of the adopted quantizer are known in the controller design, which seems unpractical since the model parameters are user-defined and adjusted to obtain optimal control performances. Moreover, it is still unclear how to design the control algorithms for the bilateral teleoperators with hysteresis-like quantized input, simultaneously accommodating the quantizer model uncertainties as well as dynamic and kinematical uncertainties. Thus, quantized input control for teleoperators remains an open problem.

Besides, another possible restriction of the traditional teleoperator control^8–17,20,21 is the assumption of perfect and accurate actuator output during the whole execution period. It is well known that the actuator is actually uncalibrated and varies with the temperature of electronic component, yet, only few results are reported in the works by Cheah and colleagues^29–31 to aim at this control topic. These solutions cope with the actuator uncertainties by establishing an adaptive model parameter updating law to online estimate the uncalibrated actuator model. Unfortunately, such adaptive techniques are only robust to the constant or slow time-varying parameter case, and the results in Wang et al.^30,31 focus on joint servoing loop control rather than torque-based control. In particular, the nonlinear couplings between the actuator mechanism and quantizer operations inherently exist in the robot dynamics, making the control design for teleoperator system more challenging.

Motivated by the above-mentioned analysis, we shall specifically attack the telecontrol problem for robotic teleoperator system with quantized input from a new perspective that the quantizer model parameter, actuator models, robot dynamics, and kinematics are not exactly obtainable. The contributions of this article is mainly threefold:

For the sake of avoiding chattering phenomenon and obtaining variable quantization levels along with the control input, hysteresis quantizer is adopted, and the robotic dynamics preceded by quantized inputs distinguish our work from the traditional teleoperator control schemes.^8–17,20,21 Via fully exploiting the structural feature of hysteresis quantizer and utilizing the sector-bound property, a novel decomposition-based method, which is inspired by the works by Zhou et al.²⁵ and Wang et al.,²⁸ is subsequently developed to cope with the control issue that only limited and quantized control inputs are available (see Theorem 1), for which the control design and stability analysis for the bilateral teleoperator system turn out to be feasible.

To achieve precise position and velocity tracking control, actuator uncertainties are also considered. By incorporating the Nussbaum-gain function to the proposed controller, the control difficulties resulting from the nonlinear couplings between actuator mechanism and quantized operations are then conquered without an exact knowledge of actuator model and quantizer parameters, in contrast to the existing related works on actuator uncertainties^29–31 that require additionally computational resources to estimate the unknown actuator parameters.

With full considerations of the uncertain robot dynamics and kinematics, an adaptive telecontroller integrated with Nussbaum-type gains is established for the teleoperator system in the presence of constant time delays such that the position tracking errors and velocity errors converge to zero, extending the results in literatures^8–17,20,21 to a more practical application.

Preliminary and problem statement

Robot kinematics and dynamics with actuator uncertainty and quantized input

In the teleoperation system, it is assumed that the bilateral manipulators(i.e. local and remote robots) are nonredundant. Let $i = {l, r}$ be the local manipulator and remote manipulator, respectively. The forward kinematic of manipulator can be represented as follows^13,32

{\begin{matrix} x_{r} = h_{r} (q_{r}) & {\overset{\cdot}{x}}_{r} = J_{r} (q_{r}) {\overset{\cdot}{q}}_{r} \\ x_{l} = h_{l} (q_{l}) & {\overset{\cdot}{x}}_{l} = J_{l} (q_{l}) {\overset{\cdot}{q}}_{l} \end{matrix}

(1)

where $q_{i} \in R^{n}$ denotes the vector of joint positions, $x_{i}$ represents the task space coordinate, $h_{i} (q_{i}) : R^{n} \to R^{n}$ is the nonlinear mapping describing the relation between the joint space and task space, $n$ is the degree of freedom (DOF) of manipulators, ${\overset{\cdot}{q}}_{i}$ and ${\overset{\cdot}{x}}_{i}$ are the angle velocities and translation velocity, and $J_{i} (q_{i})$ is the standard Jacobian matrix. Note that the kinematics models have the following properties.²⁹

Property 1

The robotic kinematics is linear in a set of kinematic parameters $a_{k, i} \in R^{w}$ such that

{\overset{\cdot}{x}}_{i} = J_{i} (q_{i}), ξ = Y_{k, i} (q_{i}, ξ) a_{k, i}

where $Y_{k_{i}} (q_{i}, ξ) \in R^{n \times w}$ is the so-called kinematic regressor matrix. Following the works by Liu and Chopra¹³ and Cheah et al.,²⁹ the dynamic equation of n-DOFs teleoperator has the following form

{\begin{matrix} H_{r} (q_{r}) {\overset{\cdot\cdot}{q}}_{r} + S_{r} (q_{r}, {\overset{\cdot}{q}}_{r}) {\overset{\cdot}{q}}_{r} + G_{r} (q_{r}) = K_{r} Q_{r} (τ_{r}) + F_{r} \\ H_{l} (q_{l}) {\overset{\cdot\cdot}{q}}_{l} + S_{l} (q_{l}, {\overset{\cdot}{q}}_{l}) {\overset{\cdot}{q}}_{l} + G_{l} (q_{l}) = K_{l} Q_{l} (τ_{l}) - F_{l} \end{matrix}

(2)

where for $i = {l, r}$ , $H_{i} (q_{i}) \in R^{n \times n}$ are the symmetric manipulator inertia matrices, $S_{i} (q_{i}, {\overset{\cdot}{q}}_{i}) \in R^{n \times n}$ are the centripetal and Coriolis matrices, $G_{i} (q_{i}) \in R^{n}$ are the gravitational torques, $F_{i} \in R^{n}$ are the forces exerted on the local and remote manipulators, $K_{i} Q_{i} (τ_{i}) \in R^{n}$ represent the vectors of applied torques with $K_{i} = diag (k_{i, 1}, \dots, k_{i, n}) \in R^{n \times n}$ being the proportional actuator gains and $Q_{i} (\cdot)$ being the quantitative operations. Several basic properties of robot dynamics are listed as follows.^32–35

Property 2

The symmetric inertia matrixes $H_{i} (q_{i})$ are uniformly positive definite.

Property 3

The Coriolis matrices $S_{i} (q_{i}, {\overset{\cdot}{q}}_{i})$ can be appropriately defined such that $S_{i} (q_{i}, {\overset{\cdot}{q}}_{i}) - (1 / 2) {\overset{\cdot}{H}}_{i} (q_{i})$ is a skew symmetric. Thus, we have $η^{T} {S_{i} (q_{i}, {\overset{\cdot}{q}}_{i}) - (1 / 2) {\overset{\cdot}{H}}_{i} (q_{i})} η = 0$ for $η \in R^{n \times 1}$ being any vector.

Property 4

The Lagrangian robot dynamics (2) is linear in a set of dynamics parameters $a_{d, i} \in R^{v}$ such that

H_{i} (q_{i}) \overset{\cdot}{ζ} + S_{i} (q_{i}, {\overset{\cdot}{q}}_{i}) ζ + G_{i} (q_{i}) = Y_{d, i} (q_{i}, {\overset{\cdot}{q}}_{i}, ζ, \overset{\cdot}{ζ}) a_{d, i}

where $Y_{d, i} (q_{i}, {\overset{\cdot}{q}}_{i}, ζ, \overset{\cdot}{ζ}) \in R^{n \times v}$ are the so-called dynamics regressor matrix and $ζ \in R^{n}$ is a differentiable vector.

Remark 1

In the most recent studies on bilateral control for the teleoperation system,^13,14,17 the proportional actuator gains $K_{i}$ are set as identity matrix or implicitly assumed to be known. Nevertheless, it is explicitly pointed out in Cheah et al.²⁹ that this proportional actuator gains $K_{i}$ will change with varying temperature due to the motor motion and actuator status. Note that such variation will inevitably give rise to time-varying proportional actuator gains in the robot dynamics, thus leading to imprecise control torque or even system instability. Hence, it is a challenging but significative work to consider the bilateral telecontrol in the presence of the actuator uncertainty or calibration error; that is, the proportional actuator gains $K_{i}$ are positive but unknown and time-varying within $[k_{min}, k_{max}]$ where $k_{min}$ and $k_{max}$ are the lower and upper bounds of $k_{i, j}$ , respectively.

Remark 2

In this article, it is assumed that the physical connection between controller and actuating mechanism is unavailable, in which case that a more practical and more rational requirement is considered in the bilateral teleoperation system. To economize limited bandwidth, the controller outputs $τ_{i}$ are processed by quantizer $Q_{i} (\cdot)$ such that the control signal to the plant is divided into piece-wise constant function of time. This is in contrast to the traditionally bilateral telecontrol studies that continuous controller outputs are utilized. Obviously, limited and partitioned control signals lead to complicated control design and stability analysis. Thus, it is interesting and meaningful to investigate telecontrol for manipulator system with quantized input.

Problem formulation

In this work, the time delays $T_{i}$ in communication channel, as discussed by Liu and colleagues,^13,14 are considered in the teleoperation system. Define the tracking errors and velocity errors with time delays as follows

{\begin{matrix} e_{r} = x_{l} (t - T_{l}) - x_{r} (t) & {\overset{\cdot}{e}}_{r} = {\overset{\cdot}{x}}_{l} (t - T_{l}) - {\overset{\cdot}{x}}_{r} (t) \\ e_{l} = x_{r} (t - T_{r}) - x_{l} (t) & {\overset{\cdot}{e}}_{l} = {\overset{\cdot}{x}}_{r} (t - T_{r}) - {\overset{\cdot}{x}}_{l} (t) \end{matrix}

(3)

where $x_{l} (t - T_{l})$ and $x_{r} (t - T_{r})$ are the task space coordinates with communication delays. To simplify the problem and discussion, we assume that $x_{i} \equiv x_{i} (t)$ . The control objectives addressed are defined as follows:

Problem 1

Suppose that the proportional actuator gains $K_{i}$ are unknown and time-varying, the controller outputs are processed by quantizer $Q_{i} (\cdot)$ , and the constant kinematic parameters $a_{k, i}$ and dynamic parameters $a_{d, i}$ are unknown and design an adaptive motion controller for the teleoperation system such that the position tracking errors and velocity errors asymptotically converge to zero, that is, $\lim_{t \to \infty} e_{r} = e_{l} = 0$ and $\lim_{t \to \infty} {\overset{\cdot}{e}}_{r} = {\overset{\cdot}{e}}_{l} = 0$ .

Instrumental lemma

In this article, Nussbaum gain technique is adopted to cope with the unknown gain control problem. Any even function $N (χ)$ is called the Nussbaum function if the following holds

\lim_{κ \to \infty} \sup \int_{κ_{0}}^{κ} N (χ) d χ = + \infty

(4)

\lim_{κ \to \infty} \inf \int_{κ_{0}}^{κ} N (χ) d χ = - \infty

(5)

where $κ_{0}$ is the initial value of $κ$ . Throughout this article, the Nussbaum-type function $N (χ) = χ^{2} \cos (χ)$ is utilized.

Lemma 1

Suppose that $ω_{i, j} (\cdot)$ are smooth and time-varying functions defined within $[{\underline{ω}}_{i, j}, {\bar{ω}}_{i, j}]$ and $0 \notin [{\underline{ω}}_{i, j}, {\bar{ω}}_{i, j}]$ , $\forall t \in [0, t)$ , and $i = {r, l}, j = {1, \dots, n}$ . Let $V (\cdot) \geq 0$ and $N (χ_{i, j})$ be an even smooth Nussbaum-type function defined in $[0, t)$ with $χ_{i, j} (t)$ being a smooth function for $i = {r, l}, j = {1, \dots, n}$ .^36,37 If the following inequality holds

V (t) \leq Θ + \sum_{i \in {r, l}} \sum_{j = 1}^{n} \int_{0}^{t} [ω_{i, j} (δ) N (χ_{i, j} (δ)) + 1] {\overset{\cdot}{χ}}_{i, j} (δ) d δ

where $Θ$ is positive constant.

Then, according to Chen et al.³⁶ and Liu and Tong,³⁷ the following conclusions can be drawn: $V (t)$ , $χ_{i, j} (t)$ , $ω_{i, j} (t) N (χ_{i, j} (t)) {\overset{\cdot}{χ}}_{i, j} (t)$ must be bounded on $[0, t_{f})$ for $i = {r, l}, j = {1, \dots, n}$ .

Adaptive bilateral task-space controller for teleoperation system with quantized input

In this section, we investigate the adaptive task-space telecontroller design for the bilateral teleoperation system in the presence of time-varying proportional actuator gains, limited and partitioned control signals, and unknown kinematic parameters as well as dynamic parameters.

The hysteretic quantizer

As illustrated above, the control signals are transferred by wireless device such that the physical connections between controller and actuating mechanism can be removed. For the sake of reducing excessive information exchange, the control signals corresponding to the jth joint of ith manipulator are processed by the hysteretic quantizer^25,28

Q (τ_{i, j}) = {\begin{matrix} u_{k} sgn (τ_{i, j}) & \begin{matrix} \frac{u_{k}}{1 + φ} < | τ_{i, j} | \leq u_{k}, {\overset{\cdot}{τ}}_{i, j} < 0, or \\ u_{k} < | τ_{i, j} | \leq \frac{u_{k}}{1 - φ}, {\overset{\cdot}{τ}}_{i, j} > 0 \end{matrix} \\ {\bar{u}}_{k} sgn (τ_{i, j}) & \begin{matrix} u_{k} < | τ_{i, j} | \leq \frac{u_{k}}{1 - φ}, {\overset{\cdot}{τ}}_{i, j} < 0, or \\ \frac{u_{k}}{1 - φ} < | τ_{i, j} | \leq \frac{u_{k} (1 + φ)}{1 - φ}, {\overset{\cdot}{τ}}_{i, j} > 0 \end{matrix} \\ 0 & \begin{matrix} 0 \leq | τ_{i, j} | < \frac{u_{min}}{1 + φ}, {\overset{\cdot}{τ}}_{i, j} < 0, or \\ \frac{u_{min}}{1 + φ} \leq | τ_{i, j} | \leq u_{min}, {\overset{\cdot}{τ}}_{i, j} > 0 \end{matrix} \\ Q (τ_{i, j} (t_{-})) & otherwise \end{matrix}

(6)

where ${\bar{u}}_{k} = u_{k} (1 + φ)$ , $u_{k} = ϵ^{(1 - k)} u_{min}$ with $k = 1, 2, \dots$ , $u_{min} > 0$ , $0 < ϵ < 1$ , $φ = 1 - ϵ / 1 + ϵ$ , and $Q (τ_{i, j} (t_{-}))$ is the quantized value in the last sampling time.

Remark 3

It should be noted that another kind of frequently encountered quantizer is called the uniform quantizer,²⁶ whose characteristic lies in the fact that the quantization density is fixed. However, one possible weakness of this quantizer may be the phenomenon that limited quantization precision is available, since each of the quantization level is changeless due to the fixed quantization density, especially when the control signals approach to zero. Thus, hysteresis quantizer, as an extension of the logarithmic quantizer²⁶ and with the advantage of factitiously adjusting quantization density and avoiding signal chattering, is adopted in the bilateral teleoperation system.

Remark 4

It is worth mentioning that the quantization density coefficient $φ$ of hysteresis quantizer and the quantization level $u_{k} = ϵ^{(1 - k)} u_{min}$ are manually defined, which literally implies that more different quantized levels can be obtained with the changes of quantizer input. Moreover, the hysteresis-type quantizer (equation (6)) can be treated as a switch combination of two logarithmic quantizers whose coarseness and quantized levels depend on coefficient $φ$ , making the quantizer (equation (6)) having the capability of avoiding chattering. Note that a large value of $ϵ$ means a high-density quantized level, and $u_{min} > 0$ can be considered as the size of the dead zone for $Q (\cdot)$ . Figure 1 plots the hysteresis characteristics of quantizer (equation (6)) when $τ_{i, j} > 0$ . By exploring (equation (6)), the output of quantizer can be mathematically divided into discrete value $Q (τ_{i}) \in Ω_{Q} = {0, \pm u_{k}, \pm u_{k} (1 + φ), k = 1, 2, \dots}$ . Hence, only limited quantization levels correspond to a segment of the measured quantizer input; in other words, only limited quantization values of control torques can be applied to the bilateral teleoperation system, giving rise to challenging controller design and complex stability analysis. To overcome this issue, a decomposition-based method for quantized input is proposed.

Figure 1.

Map of hysteresis quantizer for $τ_{i, j} > 0$ .

Theorem 1

The output of hysteresis quantizer $Q (τ_{i, j})$ can be decomposed into a linear part and a nonlinear disturbance-like part as follows

Q (τ_{i, j}) = ρ_{i, j} τ_{i, j} + d_{i, j}

(7)

where $ρ_{i, j} \in [1 - φ, 1 + φ]$ is called the linear gain and $d_{i, j} \in [- u_{min}, u_{min}]$ is called the disturbance-like part.

Proof

Actually, this theorem is visually reflected by the slopes of $1 - φ$ and $1 + φ$ in Figure 1. The proof is divided into the following two parts:

P1. When $| τ_{i, j} | \leq u_{min}$ holds, the quantizer output can be denoted as follows

Q (τ_{i, j}) = 0 = τ_{i, j} + d_{i, j}

where $ρ_{i, j} = 1 \in [1 - φ, 1 + φ]$ and $d_{i, j} = - τ_{i, j} \in [- u_{min}, u_{min}]$ .

P2. When $| τ_{i, j} | > u_{min}$ holds, the quantizer output can be subsequently denoted as follows

Q (τ_{i, j}) = \frac{Q (τ_{i, j})}{τ_{i, j}} τ_{i, j}

where $ρ_{i, j} = Q (τ_{i, j}) / τ_{i, j} \in [1 - φ, 1 + φ]$ due to the sector bound property and $d_{i, j} = 0 \in [- u_{min}, u_{min}]$ . Therefore, the linear gain $ρ_{i, j}$ and the disturbance-like part $d_{i, j}$ are defined as follows

ρ_{i, j} = {\begin{matrix} \frac{Q (τ_{i, j})}{τ_{i, j}} & τ_{i, j} \in (- \infty, - u_{min}) \cup (u_{min}, + \infty) \\ 1 & τ_{i, j} \in [- u_{min}, u_{min}] \end{matrix}

(8)

and

d_{i, j} = {\begin{matrix} 0 & τ_{i, j} \in (- \infty, - u_{min}) \cup (u_{min}, + \infty) \\ - τ_{i, j} & τ_{i, j} \in [- u_{min}, u_{min}] \end{matrix}

(9)

The proof is completed.▪

Then, the teleoperator dynamics can be thus represented as follows

{\begin{matrix} H_{r} (q_{r}) {\overset{\cdot\cdot}{q}}_{r} + S_{r} (q_{r}, {\overset{\cdot}{q}}_{r}) {\overset{\cdot}{q}}_{r} + G_{r} (q_{r}) = Ω_{r} τ_{r} + Φ_{r} + F_{r} \\ H_{l} (q_{l}) {\overset{\cdot\cdot}{q}}_{l} + S_{l} (q_{l}, {\overset{\cdot}{q}}_{l}) {\overset{\cdot}{q}}_{l} + G_{l} (q_{l}) = Ω_{l} τ_{l} + Φ_{l} - F_{l} \end{matrix}

(10)

where $Ω_{i} = diag (ω_{i, 1}, \dots, ω_{i, n}) \in R^{n \times n}$ with $ω_{i, j} = k_{i, j} ρ_{i, j}$ and $Φ_{i} = [ϕ_{i, 1}, \dots, ϕ_{i, n}]^{T} \in R^{n}$ with $ϕ_{i, j} = k_{i, j} d_{i, j}$ . According to Theorem 1, one can obtain that $| ω_{i, j} | \leq k_{max} (1 + φ) = {\bar{ω}}_{i, j}, ω_{i, j} \in [{\underline{ω}}_{i, j}, {\bar{ω}}_{i, j}]$ and $| ϕ_{i, j} | \leq k_{\max} u_{\min} = {\bar{ϕ}}_{i, j}$ ; that is, $ω_{i, j}$ and $ϕ_{i, j}$ are both bounded due to the boundedness of $k_{i, j}$ , $ρ_{i, j}$ , and $d_{i, j}$ .

Remark 5

It is important to be noted that the modified gain matrixes $Ω_{i}$ and the modified disturbance-like term $Φ_{i}$ are unknown and time-varying but bounded. Observing equations (2), (6), and (10), the couplings between quantized function and the actuator gain are decomposed into two bounded terms, providing an improved controllability and simplifying the stability analysis. As shown later, the proposed controller design does not require the exact knowledge of quantizer model parameters and actuator parameters; that is, the restrictive conditions of known actuator gains in Liu and colleagues^13,14 and known quantizer density $ϵ$ in Xing et al.²⁶ are subsequently removed.

Controller design for teleoperator with quantized input in free motion

In this section, it is assumed that the teleoperators move in a external force–free environment such that $F_{e} = F_{l} = 0$ . To achieve the control objectives, an intuitionistic idea is to employ the position tracking errors and velocity errors feedback in controller design. Following the works by Liu and colleagues,^13,14 we define the joint-space reference velocity

{\overset{\cdot}{q}}_{r, i} = {\hat{J}}_{i}^{- 1} (q_{i}) Λ e_{i}

(11)

where for $i = {l, r}$ , ${\hat{J}}_{i} (q_{i})$ is the estimated Jacobian matrix obtained by replacing the true parameters $a_{k, i}$ with the estimations ${\hat{a}}_{k, i}$ , $Λ$ is a positive constant. Then, by differentiating equation (11) with respect to time, the nominal acceleration can be represented as follows

{\overset{\cdot\cdot}{q}}_{r, i} = {\overset{\cdot}{\hat{J}}}_{i}^{- 1} (q_{i}) Λ e_{i} + {\hat{J}}_{i}^{- 1} (q_{i}) Λ {\overset{\cdot}{e}}_{i}

(12)

where ${\overset{\cdot}{\hat{J}}}_{i}^{- 1} (q_{i}) = - {\hat{J}}_{i}^{- 1} (q_{i}) {\overset{\cdot}{\hat{J}}}_{i}^{- 1} (q_{i}) {\hat{J}}_{i}^{- 1} (q_{i})$ is the derivative of the ${\hat{J}}_{i}^{- 1} (q_{i})$ . The error vector in the joint space is given as follows

s_{i} = {\overset{\cdot}{q}}_{i} - {\overset{\cdot}{q}}_{r, i}

(13)

Then, the task-space sliding vector is defined as follows

r_{i} = {\hat{J}}_{i} (q_{i}) s_{i} = {\hat{\overset{\cdot}{x}}}_{i} - Λ e_{i}

(14)

where ${\hat{\overset{\cdot}{x}}}_{i} = {\hat{J}}_{i} (q_{i}) {\overset{\cdot}{q}}_{i}$ is the estimated task-space velocity. Based on the nominal references defined, the adaptive Nussbaum-type controller is given as follows

τ_{i} = - N_{i}^{*} (χ_{i}) {\bar{τ}}_{i}

(15)

with the virtual controller being

\begin{matrix} {\bar{τ}}_{i} = & Y_{d, i} {\hat{a}}_{d, i} - Ξ_{s, i} s_{i} - Ξ_{r, i} {\hat{J}}_{i}^{T} (q_{i}) r_{i} \\ + Ξ_{e, i} {\hat{J}}_{i}^{T} {\overset{\cdot}{e}}_{i} - \tanh (s_{i} / ν) {\hat{Φ}}_{i} \end{matrix}

(16)

where the Nussbaum gain matrices $N_{i}^{*} (χ_{i})$ are defined as $N_{i}^{*} (χ_{i}) = diag (N (χ_{i, 1}), \dots, N (χ_{i, n})) \in R^{n \times n}$ , $χ_{i} = [χ_{i, 1}, \dots, χ_{i, n}]^{T} \in R^{n}$ , ${\hat{Φ}}_{i} = [{\hat{ϕ}}_{i, 1}, \dots, {\hat{ϕ}}_{i, n}]^{T}$ are the estimates of ${\bar{Φ}}_{i} = [{\bar{ϕ}}_{i, 1}, \dots, {\bar{ϕ}}_{i, n}]^{T} \in R^{n}$ with ${\bar{ϕ}}_{i, j}$ being the bound of $ϕ_{i, j}$ defined in equation (10), the hyperbolic tangent matrices are defined as $\tanh (s_{i} / ν) = diag (\tanh (s_{i, 1} / ν), \dots, \tanh (s_{i, n} / ν))$ with $ν = 1 / (1 + t^{2})$ . $Ξ_{s, i}$ , $Ξ_{r, i}$ , and $Ξ_{e, i} \in R^{n \times n}$ are all diagonal positive definite. From Property 4, we have

\begin{array}{l} Y_{d, i} {\hat{a}}_{d, i} \equiv Y_{d, i} (q_{i}, {\dot{q}}_{i}, {\dot{q}}_{r, i}, {\overset{\cdot\cdot}{q}}_{r, i}) {\hat{a}}_{d, i} \\ \equiv {\hat{H}}_{i} (q_{i}) {\overset{\cdot\cdot}{q}}_{r, i} + {\hat{S}}_{i} (q_{i}, {\dot{q}}_{i}) q_{r, i} + {\hat{G}}_{i} (q_{i}) \end{array}

(17)

where ${\hat{H}}_{i} (q_{i})$ , ${\hat{S}}_{i}$ , and ${\hat{G}}_{i} (q_{i})$ are the estimates of $H_{i} (q_{i})$ , $S_{i}$ , and $G_{i} (q_{i})$ by replacing the true parameters $a_{d, i}$ with the estimates ${\hat{a}}_{d, i}$ . By substituting equations (15) and (16) into equation (10), the closed-loop dynamics can be denoted as follows

\begin{array}{l} H_{i} (q_{i}) {\dot{s}}_{i} + S_{i} (q_{i}, {\dot{q}}_{i}) s_{i} = - Ω_{i} N_{i}^{*} (χ_{i}) {\bar{τ}}_{i} + Φ_{i} - {\bar{τ}}_{i} \\ + Y_{d, i} Δ a_{d, i} - Ξ_{s, i} s_{i} - Ξ_{r, i} {\hat{J}}_{i}^{T} r_{i} + Ξ_{e, i} {\hat{J}}_{i}^{T} \dot{e} \\ - \tanh (s_{i} / ν) {\hat{Φ}}_{i} \end{array}

(18)

where $Δ a_{d, i} = {\hat{a}}_{d, i} - a_{d, i}$ . It can be seen that the controller design given in equations (15) and (16) utilizes the estimated values rather than the true values of dynamic parameters and kinematic parameters, which directly implies that proposed scheme has the capacity of accommodating the dynamic and kinematic uncertainties for the robotic teleoperation system. Moreover, a priori knowledge of the actuator and quantizer model parameters is not required to be known. The parameter update laws for ${\hat{a}}_{d, i}$ are given as follows

{\overset{\cdot}{\hat{a}}}_{d, i} = - \underset{d, i}{Π} Y_{d, i}^{T} s_{i}

(19)

where $Π_{d, i}$ is the positive diagonal matrix. For the sake of obtaining the smooth task-space velocity signal, a filtered-based method³⁸ is introduced as follows

{\overset{\cdot}{y}}_{i} + ψ y_{i} = ψ {\overset{\cdot}{x}}_{i}

(20)

where $ψ$ is a positive constant. Let $p$ be the Laplace variable; equation (20) can be formulated as follows

y_{i} = \frac{ψ p}{p + ψ}, x_{i} = W_{k, i} a_{k, i}

(21)

where $W_{k, i} = (ψ p / p + ψ) Y_{k, i}$ with $Y_{k, i} \equiv Y_{k, i} (q_{i}, {\overset{\cdot}{q}}_{i})$ from Property 1. Then, the update laws for ${\hat{a}}_{k, i}$ are designed as follows

{\overset{\cdot}{\hat{a}}}_{k, i} = - \underset{k, i}{Π} [W_{k, i}^{T} Π_{w, i} (W_{k, i} {\hat{a}}_{k, i} - y_{i}) + Y_{k, i}^{T} Δ x_{i} + Ξ_{e, i} Y_{k, i}^{T} {\overset{\cdot}{e}}_{i}]

(22)

where $Δ x_{i} = {\hat{x}}_{i} - x_{i}$ and $Π_{k, i}, Π_{w, i}$ are positive diagonal matrices. In addition, the Nussbaum variables are updated as follows

{\overset{\cdot}{χ}}_{i} = - diag (ε_{i}) diag (s_{i}) {\bar{τ}}_{i}

(23)

where $ε_{i} = [ε_{i, 1}, \dots, ε_{i, n}]^{T}$ with $ε_{i, j}$ being positive constant. Finally, ${\hat{Φ}}_{i}$ is updated as follows

{\overset{\cdot}{\hat{Φ}}}_{i} = \underset{T, i}{Π} \tanh (s_{i} / ν) s_{i}

(24)

where $Π_{T, i} = diag (π_{T, i, 1}, \dots, π_{T, i, n})$ are positive diagonal matrices. Now, we are ready to present the following result.

Theorem 2

For a external force–free task space such that $F_{r} = F_{l} = 0$ , the bilateral adaptive controllers (equations (15) and (16)) together with the parameter updating laws (equations (19), (22)–(24)) guarantee the position tracking and velocity tracking for the teleoperation system (equations (1) and (2)) with input modulated by quantizer (equation (6)). That is, $x_{r} - x_{l} \to 0$ and ${\overset{\cdot}{x}}_{r} - {\overset{\cdot}{x}}_{l} \to 0$ as $t \to \infty$ , and in addition, ${\overset{\cdot}{x}}_{r} \to 0$ and ${\overset{\cdot}{x}}_{l} \to 0$ .

Proof

Let us define a Lyapunov-like function candidate as follows

\begin{matrix} V_{1} (t) = & \frac{1}{2} \sum_{i \in {r, l}} {s_{i}^{T} H_{i} (q_{i}) s_{i} + Δ a_{d, i}^{T} Π_{d, j}^{- 1} Δ a_{d, i} + Δ a_{k, i}^{T} Π_{k, i}^{- 1} Δ a_{k, i} \\ + e_{i}^{T} Λ Ξ_{e, i} e_{i} + Δ x_{i}^{T} Δ x_{i} + Δ Φ_{i}^{T} Π_{T, i}^{- 1} Δ Φ_{i} \\ + Ξ_{e, i} \int_{t - T_{i}}^{t} {\overset{\cdot}{x}}_{i}^{T} (δ) {\overset{\cdot}{x}}_{i} (δ) d δ} \end{matrix}

(25)

where $Δ a_{d, i} = {\hat{a}}_{d, i} - a_{d, i}$ , $Δ a_{k, i} = {\hat{a}}_{k, i} - a_{k, i}$ , and $Δ Φ_{i} = {\hat{Φ}}_{i} - {\bar{Φ}}_{i}$ . Premultiplying both sides of equation (18) by $s_{i}^{T}$ and using the skew symmetry property of $S_{i} (q_{i}, {\overset{\cdot}{q}}_{i}) - (1 / 2) {\overset{\cdot}{H}}_{i} (q_{i})$ and substituting equation (19) into (25), the time derivative of $V_{1}$ can be written as follows

\begin{matrix} {\overset{\cdot}{V}}_{1} (t) = & \sum_{i \in {r, l}} {- s_{i}^{T} Ω_{i} N_{i}^{*} (χ_{i}) {\bar{τ}}_{i} + s_{i}^{T} Φ_{i} - s_{i}^{T} {\bar{τ}}_{i} - s_{i}^{T} Ξ_{s, i} s_{i} \\ - r_{i}^{T} Ξ_{r, i} r_{i} + Ξ_{e, i} s_{i}^{T} {\hat{J}}_{i}^{T} {\overset{\cdot}{e}}_{i} - s_{i}^{T} \tanh (s_{i} / ν) {\hat{Φ}}_{i} \\ + Δ a_{k, i}^{T} Π_{k, i}^{- 1} {\overset{\cdot}{\hat{a}}}_{k, i} + e_{i}^{T} Λ Ξ_{e, i} {\overset{\cdot}{e}}_{i} + Δ x_{i}^{T} Δ {\overset{\cdot}{x}}_{i} + Δ Φ_{i}^{T} Π_{T, i}^{- 1} {\overset{\cdot}{\hat{Φ}}}_{i} \\ + \frac{1}{2} Ξ_{e, i} {\overset{\cdot}{x}}_{i}^{T} {\overset{\cdot}{x}}_{i} - \frac{1}{2} Ξ_{e, i} {\overset{\cdot}{x}}_{i}^{T} (t - T_{i}) {\overset{\cdot}{x}}_{i} (t - T_{i})} \end{matrix}

(26)

From equation (14), we have

Ξ_{e, i} {\hat{J}}_{i}^{T} s_{i}^{T} (q_{i}) {\overset{\cdot}{e}}_{i} = Ξ_{e, i} r_{i}^{T} {\overset{\cdot}{e}}_{i} = Ξ_{e, i} {\hat{\overset{\cdot}{x}}}_{i}^{T} {\overset{\cdot}{e}}_{i} - Ξ_{e, i} Λ e_{i}^{T} {\overset{\cdot}{e}}_{i}

(27)

Also, note that

\begin{matrix} Δ {\overset{\cdot}{x}}_{i} & = {\hat{\overset{\cdot}{x}}}_{i} - {\overset{\cdot}{x}}_{i} = {\hat{J}}_{i} (q_{i}) {\overset{\cdot}{q}}_{i} - J_{i} (q_{i}) {\overset{\cdot}{q}}_{i} \\ = Y_{k, i} {\hat{a}}_{k, i} - Y_{k, i} a_{k, i} = Y_{k, i} Δ a_{k, i} \end{matrix}

(28)

Ξ_{e, i} {\hat{\overset{\cdot}{x}}}_{i}^{T} {\overset{\cdot}{e}}_{i} = Ξ_{e, i} Δ a_{k, i}^{T} Y_{k, i}^{T} {\overset{\cdot}{e}}_{i} + Ξ_{e, i} {\overset{\cdot}{x}}_{i}^{T} {\overset{\cdot}{e}}_{i}

(29)

Combining equations (27)–(29) and substituting (22) into (26), it can be obtained that

\begin{matrix} V_{1} (t) = & \sum_{i \in {r, l}} {- s_{i}^{T} Ω_{i} N_{i}^{*} (χ_{i}) {\bar{τ}}_{i} + s_{i}^{T} Φ_{i} - s_{i}^{T} {\bar{τ}}_{i} - s_{i}^{T} Ξ_{s, i} s_{i} \\ - r_{i}^{T} Ξ_{r, i} r_{i} - s_{i}^{T} \tanh (s_{i} / ν) {\hat{Φ}}_{i} + Δ Φ_{i}^{T} Π_{T, i}^{- 1} {\overset{\cdot}{\hat{Φ}}}_{i} \\ - Δ a_{k, i}^{T} W_{k, i}^{T} \underset{w, i}{Π} W_{k, i} Δ a_{k, i}^{T} + Ξ_{e, i} {\overset{\cdot}{x}}_{i}^{T} {\overset{\cdot}{e}}_{i} + \frac{1}{2} Ξ_{e, i} {\overset{\cdot}{x}}_{i}^{T} {\overset{\cdot}{x}}_{i} \\ - \frac{1}{2} Ξ_{e, i} {\overset{\cdot}{x}}_{i}^{T} (t - T_{i}) {\overset{\cdot}{x}}_{i} (t - T_{i})} \end{matrix}

(30)

Using the relationships ${\overset{\cdot}{e}}_{r} = {\overset{\cdot}{x}}_{l} (t - T_{l}) - {\overset{\cdot}{x}}_{r}$ and ${\overset{\cdot}{e}}_{l} = {\overset{\cdot}{x}}_{r} (t - T_{r}) - {\overset{\cdot}{x}}_{l}$ , we have

\begin{matrix} \sum_{i \in {r, l}} Ξ_{e, i} {\overset{\cdot}{x}}_{i}^{T} {\overset{\cdot}{e}}_{i} + \frac{1}{2} Ξ_{e, i} {\overset{\cdot}{x}}_{i}^{T} {\overset{\cdot}{x}}_{i} - \frac{1}{2} Ξ_{e, i} {\overset{\cdot}{x}}_{i}^{T} (t - T_{i}) {\overset{\cdot}{x}}_{i} (t - T_{i}) \\ = - \frac{1}{2} Ξ_{e, i} {\overset{\cdot}{e}}_{r}^{T} {\overset{\cdot}{e}}_{r} - \frac{1}{2} Ξ_{e, i} {\overset{\cdot}{e}}_{l}^{T} {\overset{\cdot}{e}}_{l} \end{matrix}

(31)

Moreover, the following inequation holds

\begin{matrix} \sum_{i \in {r, l}} s_{i}^{T} Φ_{i} - s_{i}^{T} \tanh (s_{i} / ν) {\hat{Φ}}_{i} \\ + Δ Φ_{i}^{T} Π_{T, i}^{- 1} {\overset{\cdot}{\hat{Φ}}}_{i} \leq \sum_{i \in {r, l}} {| s_{i} |^{T} {\bar{Φ}}_{i} - s_{i}^{T} \tanh (s_{i} / ν) {\bar{Φ}}_{i} \\ - s_{i}^{T} \tanh (s_{i} / ν) Δ Φ_{i} + Δ Φ_{i}^{T} Π_{T, i}^{- 1} {\overset{\cdot}{\hat{Φ}}}_{i}} \\ \leq \sum_{i \in {r, l}} {\sum_{j = 1}^{n} 0.2785 ν {\bar{Φ}}_{i, j} - s_{i}^{T} \tanh (s_{i} / ν) Δ Φ_{i} + Δ Φ_{i}^{T} Π_{T, i}^{- 1} {\overset{\cdot}{\hat{Φ}}}_{i}} \end{matrix}

(32)

Thus, substituting the parameter updating laws (equations (22) and (24)) into equation (30) yields

\begin{matrix} {\overset{\cdot}{V}}_{1} (t) & \leq \sum_{i \in {r, l}} {- s_{i}^{T} Ξ_{s, i} s_{i} - r_{i}^{T} Ξ_{r, i} r_{i} - Δ a_{k, i}^{T} W_{k, i}^{T} \underset{w, i}{Π} W_{k, i} Δ a_{k, i}^{T} \\ - \frac{1}{2} Ξ_{e, i} {\overset{\cdot}{e}}_{i}^{T} {\overset{\cdot}{e}}_{i} + \sum_{j = 1}^{n} 0.2785 ν {\bar{Φ}}_{i, j} + \sum_{j = 1}^{n} \frac{1}{π_{T, i, j}} {\overset{\cdot}{χ}}_{i, j} \\ + \sum_{j = 1}^{n} \frac{1}{π_{T, i, j}} ω_{i, j} N (χ_{i, j}) {\overset{\cdot}{χ}}_{i, j}} \\ \leq \sum_{i \in {r, l}} {- s_{i}^{T} Ξ_{s, i} s_{i} - r_{i}^{T} Ξ_{r, i} r_{i} - Δ a_{k, i}^{T} W_{k, i}^{T} \underset{w, i}{Π} W_{k, i} Δ a_{k, i}^{T} \\ - \frac{1}{2} Ξ_{e, i} {\overset{\cdot}{e}}_{i}^{T} {\overset{\cdot}{e}}_{i} + \sum_{j = 1}^{n} 0.2785 ν {\bar{Φ}}_{i, j} \\ + \sum_{j = 1}^{n} \frac{1}{π_{T, i, j}} [ω_{i, j} N (χ_{i, j}) + 1] {\overset{\cdot}{χ}}_{i, j}} \end{matrix}

(33)

Taking the time integration of equation (33) over $[0, t]$ , we have

\begin{matrix} {\overset{\cdot}{V}}_{1} (t) \leq \sum_{i \in {r, l}} {\int_{0}^{t} (- s_{i}^{T} Ξ_{s, i} s_{i} - r_{i}^{T} Ξ_{r, i} r_{i} - \frac{1}{2} Ξ_{e, i} {\overset{\cdot}{e}}_{i}^{T} {\overset{\cdot}{e}}_{i} \\ - Δ a_{k, i}^{T} W_{k, i}^{T} \underset{w, i}{Π} W_{k, i} Δ a_{k, i}) d δ} + \sum_{i \in {r, l}} \sum_{j = 1}^{n} \int_{0}^{t} 0.2785 ν {\bar{Φ}}_{i, j} d δ \\ + \sum_{i \in {r, l}} \sum_{j = 1}^{n} \frac{1}{π_{T, i, j}} \int_{0}^{δ} [ω_{i, j} (δ) N (χ_{i, j} (δ)) + 1] {\overset{\cdot}{χ}}_{i, j} (δ) d δ + Θ_{1} \\ \leq \sum_{i \in {r, l}} \sum_{j = 1}^{n} \frac{1}{π_{T, i, j}} \int_{0}^{δ} [ω_{i, j} (δ) N (χ_{i, j} (δ)) + 1] {\overset{\cdot}{χ}}_{i, j} (δ) d δ + Θ \end{matrix}

(34)

where $Θ_{1}$ and $Θ = Θ_{1} + \sum_{i \in {r, l}} \sum_{j = 1}^{n} \int_{0}^{t} 0.2785 ν {\bar{Φ}}_{i, j} d δ$ are both bounded.

Hence, according to Lemma 1, we can obtain that $V_{1} (t)$ , $χ_{i, j} (t)$ , $ω_{i, j} (t) N (χ_{i, j} (t)) {\overset{\cdot}{χ}}_{i, j} (t)$ are bounded for $i = {r, l}$ , $j = {1, \dots, n}$ . With some simple deductions, the boundedness of $V_{1} (t)$ directly implies that all the signals in the closed-loop system are bounded. Furthermore, from equation (34), it can be obtained that $s_{i}, r_{i}, {\overset{\cdot}{e}}_{i}$ are bounded and square integrable. By applying Barbalat’s Lemma, the boundedness of ${\overset{\cdot}{s}}_{i}, {\overset{\cdot}{r}}_{i}$ indicates that $\lim_{t \to \infty} s_{i} \to 0$ , $\lim_{t \to \infty} r_{i} \to 0$ . Combining the definitions of $s_{i}, r_{i}$ in equations (11), (13), and (14), it can be concluded that $\lim_{t \to \infty} e_{i} \to 0$ , $\lim_{t \to \infty} {\overset{\cdot}{e}}_{i} \to 0$ and $\lim_{t \to \infty} {\overset{\cdot}{x}}_{i} \to 0$ for $i = {r, l}$ .▪

In this article, the external forces exerted on the remote and local robots are assumed to be passive maps as follows^7,39

\int_{0}^{t} s_{r}^{T} F_{r} d δ \leq Θ_{m}, - \int_{0}^{t} s_{l}^{T} F_{l} d δ \leq Θ_{l}

(35)

where $Θ_{r}$ and $Θ_{l}$ are nonnegative constants. Then, we can obtain the following result.

Theorem 3

If external forces exerted on the remote and local robots satisfy (equation (35)), then, the bilateral adaptive controllers (equations (15) and (16)) and the parameter updating laws (equations (19), (22)–(24)) ensure the position tracking and velocity tracking for the teleoperation system (equations (1) and (2)) with input modulated by quantizer (equation (6)), that is, $\lim_{t \to \infty} e_{i} \to 0$ , $\lim_{t \to \infty} {\overset{\cdot}{e}}_{i} \to 0$ , and $\lim_{t \to \infty} {\overset{\cdot}{x}}_{i} \to 0$ for $i = {r, l}$ .

Proof

This proof can be briefly derived from the proof of Theorem 2. In the case that $F_{r} \neq 0$ and $F_{l} \neq 0$ , substituting equations (15) and (16) into equation (10) leads to

\begin{matrix} H_{r} (q_{r}) {\overset{\cdot}{s}}_{r} + S_{r} (q_{r}, {\overset{\cdot}{q}}_{r}) s_{r} \\ = - Ω_{r} N_{r}^{*} (χ_{r}) {\bar{τ}}_{r} + Φ_{r} - {\bar{τ}}_{r} + Y_{d, r} Δ a_{d, r} - Ξ_{s, r} s_{r} \\ - Ξ_{r, r} {\hat{J}}_{r}^{T} r_{r} + Ξ_{e, r} {\hat{J}}_{r}^{T} {\overset{\cdot}{e}}_{r} - \tanh (s_{r} / ν) {\hat{Φ}}_{r} + F_{r} \end{matrix}

(36)

\begin{matrix} H_{l} (q_{l}) {\overset{\cdot}{s}}_{l} + S_{l} (q_{l}, {\overset{\cdot}{q}}_{l}) s_{l} \\ = - Ω_{l} N_{l}^{*} (χ_{l}) {\bar{τ}}_{l} + Φ_{l} - {\bar{τ}}_{l} + Y_{d, l} Δ a_{d, l} - Ξ_{s, l} s_{l} \\ - Ξ_{r, l} {\hat{J}}_{l}^{T} r_{l} + Ξ_{e, l} {\hat{J}}_{l}^{T} {\overset{\cdot}{e}}_{l} - \tanh (s_{l} / ν) {\hat{Φ}}_{l} - F_{l} \end{matrix}

(37)

Define the Lyapunov-like function candidate as $V_{2} (t) = V_{1} (t) + Θ_{m} - \int_{0}^{t} s_{r}^{T} F_{r} d δ + Θ_{l} + \int_{0}^{t} s_{l}^{T} F_{l} d δ$ , where $Θ_{m} - \int_{0}^{t} s_{r}^{T} F_{r} d δ + Θ_{l} + \int_{0}^{t} s_{l}^{T} F_{l} d δ \geq 0$ is guaranteed. Following the proof of Theorem 2, we can also obtain that $\lim_{t \to \infty} e_{i} \to 0$ , $\lim_{t \to \infty} {\overset{\cdot}{e}}_{i} \to 0$ and $\lim_{t \to \infty} {\overset{\cdot}{x}}_{i} \to 0$ for $i = {r, l}$ .▪

Simulation

In this section, the numerical simulations are presented to verify the performance of proposed telecontrol scheme by a pair of 2-DOF manipulators with revolute joints.

Models of robot kinematics, dynamics, actuator, and quantizer

The physical parameters of the teleoperator are listed in Table 1, where $m_{i, 1}$ , $m_{i, 2}$ are the masses for the links and $l_{i, 1}$ , $l_{i, 2}$ are the lengths of links.

Table 1.

Physical parameters of remote and local robot.

	$m_{i, 1}$ (kg)	$m_{i, 2}$ (kg)	$l_{i, 1}$ (m)	$l_{i, 2}$ (m)
Remote robot	2.9	0.6	0.4	0.3
Local robot	3.0	0.8	0.35	0.3

Let the joint angles be $q_{i} = [q_{i, 1}, q_{i, 2}]^{T}$ , the angle velocities be ${\overset{\cdot}{q}}_{i} = [{\overset{\cdot}{q}}_{i, 1}, {\overset{\cdot}{q}}_{i, 2}]^{T}$ , and the robotic kinematics parameters in Property 1 be $a_{k, i} = [a_{k, i}^{1}, a_{k, i}^{2}]^{T} = [l_{i, 1}, l_{i, 2}]^{T}$ . Then, the robot Jacobian $J_{i} (q_{i})$ and $Y_{k, i}$ can be derived as follows⁴⁰

J_{i} = [\begin{matrix} - a_{k, i}^{1} \sin (q_{i, 1}) - a_{k, i}^{2} \sin (q_{i, 1} + q_{i, 2}) & a_{k, i}^{2} \sin (q_{i, 1} + q_{i, 2}) \\ a_{k, i}^{1} \cos (q_{i, 1}) + a_{k, i}^{2} \cos (q_{i, 1} + q_{i, 2}) & a_{d, i}^{2} c os (q_{i, 1} + q_{i, 2}) \end{matrix}]

(38)

Y_{k, i} = [\begin{matrix} - \sin (q_{i, 1}) {\overset{\cdot}{q}}_{k, 1} & \sin (q_{i, 1} + q_{i, 2}) ({\overset{\cdot}{q}}_{i, 1} + {\overset{\cdot}{q}}_{i, 2}) \\ \cos (q_{i, 1}) {\overset{\cdot}{q}}_{k, 1} & \cos (q_{i, 1} + q_{i, 2}) ({\overset{\cdot}{q}}_{i, 1} + {\overset{\cdot}{q}}_{i, 2}) \end{matrix}]

(39)

Let the robotic dynamics parameters in Property 4 be $a_{d, i} = [a_{d, i}^{1}, a_{d, i}^{2}, a_{d, i}^{3}, a_{d, i}^{4}, a_{d, i}^{5}]^{T}$ with $a_{d, i}^{1} = l_{i, 1}^{2} m_{i, 2} + l_{i, 1}^{2} (m_{i, 1} + m_{i, 2})$ , $a_{d, i}^{2} = l_{i, 1} l_{i, 2} m_{i, 2}$ , $a_{d, i}^{3} = l_{i, 2}^{2} m_{i, 2}$ , $a_{d, i}^{4} = l_{i, 2} m_{i, 2}$ , $a_{d, i}^{5} = l_{i, 1} (m_{i, 1} + m_{i, 2})$ . Then, the robot dynamics and the dynamic regression matrix $Y_{d, i}$ can be represented as follows¹¹

H_{i} (q_{i}) = [\begin{matrix} a_{d, i}^{1} + 2 a_{d, i}^{2} \cos (q_{i, 2}) & a_{d, i}^{3} + a_{d, i}^{2} \cos (q_{i, 2}) \\ a_{d, i}^{3} + a_{d, i}^{2} \cos (q_{i, 2}) & a_{d, i}^{3} \end{matrix}]

(40)

S_{i} (q_{i}, {\overset{\cdot}{q}}_{i}) = [\begin{matrix} - 2 a_{d, i}^{2} \sin (q_{i, 2}) {\overset{\cdot}{q}}_{i, 2} & - a_{d, i}^{2} \sin (q_{i, 2}) {\overset{\cdot}{q}}_{i, 2} \\ a_{d, i}^{2} \sin (q_{i, 2}) {\overset{\cdot}{q}}_{i, 1} & 0 \end{matrix}]

(41)

G_{i} (q_{i}) = [\begin{matrix} a_{d, i}^{4} g (\cos (q_{i, 1}) + \cos (q_{i, 2})) + a_{d, i}^{5} \cos (q_{i, 1}) \\ a_{d, i}^{4} g (\cos (q_{i, 1}) + \cos (q_{i, 2})) \end{matrix}]

(42)

\begin{array}{l} Y_{d, i} (q_{i}, {\dot{q}}_{i}, {\overset{\cdot\cdot}{q}}_{i}) \\ = [\begin{matrix} {\overset{\cdot\cdot}{q}}_{r, i, 1} & Y_{i, 21} & {\overset{\cdot\cdot}{q}}_{r, i, 2} & Y_{d, 14} & g \cos (q_{i, 1}) \\ 0 & Y_{i, 21} & {\overset{\cdot\cdot}{q}}_{i, 1} + {\overset{\cdot\cdot}{q}}_{i, 2} & Y_{d, 24} & 0 \end{matrix}] \end{array}

(43)

where $g = 9.8$ is the acceleration of gravity, $Y_{i, 21} = 2 \cos (q_{i, 2}) {\overset{\cdot\cdot}{q}}_{i, 1} + \cos (q_{i, 2}) {\overset{\cdot\cdot}{q}}_{i, 2} - \sin (q_{i, 2}) {\overset{\cdot}{q}}_{i, 2}^{2} - 2 \sin (q_{i, 2}) {\overset{\cdot}{q}}_{i, 1} {\overset{\cdot}{q}}_{i, 2}$ , $Y_{i, 21} = \cos (q_{i, 2}) {\overset{\cdot\cdot}{q}}_{i, 1} + \sin (q_{i, 2}) {\overset{\cdot}{q}}_{i, 2}^{2}$ , and $Y_{d, 14} = Y_{d, 24} = g \cos (q_{i, 1}) + \cos (q_{i, 2})$ . In the consideration of this article, actuators in equation (2) are suffered from unknown constraint simulated as follows

k_{i, j} = {\begin{matrix} 1 + 0.1 \sin (Q_{i, j} (τ_{i, j})) & if Q_{i, j} (τ_{i, j}) \geq 2.5 \\ 0 & if - 2.5 \leq Q_{i, j} (τ_{i, j}) \leq 2.5 \\ 1 - 0.05 \cos (Q_{i, j} (τ_{i, j})) & if Q_{i, j} (τ_{i, j}) \leq - 2.5 \end{matrix}

(44)

Finally, the quantizer parameters in equation (6) are shown in Table 2.

Table 2.

Model parameters of hysteresis quantizer.

$u_{min}$	$ϵ$	$φ$
0.5	0.8	1/9

Control performance of the proposed bilateral telecontrol scheme with quantized input

For the implement of adaptive controllers and parameter updating laws, the following control gains are adopted: $Λ = 5$ , $Ξ_{s, i} = 2 I_{2}$ , $Ξ_{r, i} = 5 I_{2}$ , $Ξ_{e, i} = 8 I_{2}$ , $Π_{d, i} = 10 I_{5}$ , $Π_{k, i} = 4 I_{2}$ , $Π_{w, i} = 1 I_{2}$ , $ε = [4, 4]^{T}$ , and $Π_{T, i} = 2 I_{2}$ , where $I_{K}$ denotes the $k \times k$ identity matrix. The Nussbaum-type function $N (χ) = χ^{2} \cos (χ)$ is utilized. In this simulation, the initial position and velocity are $q_{r} (0) = [0.3, 0.4]^{T}$ , $q_{l} (0) = [0.7, 0.8]^{T}$ , ${\overset{\cdot}{q}}_{r} (0) = [0, 0]^{T}$ , ${\overset{\cdot}{q}}_{l} (0) = [0, 0]^{T}$ , ${\hat{a}}_{d, i} (0) = 0$ , ${\hat{a}}_{k, i} (0) = 0$ , $χ_{i} = 0$ , and ${\hat{Φ}}_{i} = 0$ . The time delays are set as $T_{r} = 0.3 s$ and $T_{l} = 0.4 s$ . The simulation results of X/Y positions, position errors, velocity signals, external force $F_{r}$ , and the quantized control signals are presented in Figure 2. It can be observed that the proposed bilateral telecontrol scheme can all guarantee the stability of the robotic teleoperator system, in the sense that the signals in closed-loop system are bounded. Moreover, the control objective is ensured in the sense that position errors between the remote and local robots converge to zero. Note that the positions of the remote and local robots converge to values but not zero. From Figure 2(d), the velocity signals are shown to converge to zero. Thus, we can conclude that the position tracking and velocity tracking for the teleoperation system with input modulated by quantizer are implemented successfully.

Figure 2.

Performance results of adaptive telecontrol scheme: (a) position on X-axle, (b) position on Y-axle, (c) position error, (d) velocity signals, (e) external forces, and (f) control signal.

Conclusion

In this article, we analyzed the control problem of bilateral teleoperator system with hysteresis-type quantized input. By utilizing the sector-bound property, a decomposition-based method is applied to decompose the quantized signal into a linear part and a nonlinear disturbance-like part such that the control design is feasible. Moreover, an adaptive telecontroller integrated with Nussbaum-gain function is proposed to achieve position and velocity tracking of teleoperator system even the nonlinear couplings between actuator mechanism and quantized operations inherently exist in the robot dynamics. The stability of the closed-loop system is then analyzed by the Lyapunov theory in the sense of parameter boundedness. Finally, simulation results are presented to illustrate the effectiveness and performance of the adaptive telecontrol approach.

The problem of adaptive bilateral telecontrol problem was investigated in this article. However, it is well known that fuzzy and neural control are also widely used in robot system. Therefore, one of the possible future works would be adaptive fuzzy and neural network control^41–46 for nonredundant teleoperators.

Footnotes

Handling Editor: Chenguang Yang

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 in part by the National Natural Science Foundation of China under grant 61573108, in part by National Natural Science Foundation of China (U1501251), in part by the Natural Science Foundation of Guangdong Province 2016A030313715, in part by the Natural Science Foundation of Guangdong Province through the Science Fund for Distinguished Young Scholars under grant S20120011437, and in part by the Ministry of Education of New Century Excellent Talent under grant NCET-12-0637.

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