2.1. Dynamics modeling of remote system

The dynamics of the teleoperation system for master and slave in the task space can be modeled as follows:

(1) \begin{equation} M_{m}\!\left(x_{m}\right)\ddot{x}_{m} + C_{m}\!\left(x_{m},\dot{x}_{m}\right)\dot{x}_{m} + G_{m}\!\left(x_{m}\right) + D_{m} =\, f_{m} \end{equation}

(2) \begin{equation} M_{s}\!\left(x_{s}\right)\ddot{x}_{s} + C_{s}\!\left(x_{s},\dot{x}_{s}\right)\dot{x}_{s} + G_{s}\!\left(x_{s}\right) + D_{s} =\, f_{s} \end{equation}

where $x_{m}, \dot{x}_{m}, \ddot{x}_{m}$ and $x_{s}, \dot{x}_{s}, \ddot{x}_{s}$ are the position, velocity, and acceleration signals of end-effort for master and slave manipulators, $M_{m}(x_{m})$ and $M_{s}(x_{s})$ are the inertia matrices, $C_{m}(x_{m},\dot{x}_{m})$ and $C_{s}(x_{s},\dot{x}_{s})$ account centrifugal/Coriolis terms, $G_{m}(x_{m})$ and $G_{s}(x_{s})$ are the gravitational matrices, $D_{m}$ and $D_{s}$ are the modeling errors and external disturbance, and $f_{m}$ and $f_{s}$ are the control input of master and slave devices.

2.2. RBFNN

RBFNN is used to approximation manipulator uncertainties to handle the uncertainty issue in the dynamic model. The following introduces the definition of the RBF neural network utilized in this article:

(3) \begin{equation} F\mathit{(}\vartheta \mathit{)} = W^{T}S\!\left(\vartheta \right) + \varepsilon \end{equation}

where $\vartheta$ denotes the input of the neural network, $W=[\omega _{1},\omega _{2},\ldots \omega _{n}]^{T}$ is the ideal weight parameter, $n$ is the number of RBFNN nodes, and $\varepsilon (\vartheta )$ is the approximation errors. $S(\vartheta )=[s_{1}(\vartheta ),s_{2}(\vartheta ),\ldots s_{n}(\vartheta )]^{T}$ is the Gaussian basis function in the form as

(4) \begin{equation} s_{i}\!\left(\vartheta \right)=\exp \!\left({-}\frac{\left(\vartheta -\mathrm{c}_{i}\right)^{2}}{b_{i}^{2}}\right) \end{equation}

where $\mathrm{c}_{i}, \mathrm{b}_{i}$ are the center and width of the neuron. The ideal weight vector $W^{*}$ is an artificial quantity for the purposed method, which aims to minimize the value of $W$ .

(5) \begin{equation} W^*=\text{argmin}\!\left\{|\varepsilon |\right\} \end{equation}

2.3. General DMPs model

DMPs is used to trajectory learning. DMPs consists of two main components: (1) spring-damper-type equation which draws our system to the target and (2) a forcing term that gets the desire behaviors.

The DMPs model is first introduced as

(6) \begin{equation} \left\{\begin{array}{l} \tau \dot{v}=\alpha _{z}\!\left(\beta _{z}\!\left(g-x\right)-v\right)+f(s)\\ \tau \dot{x}=v\\ \tau \dot{s}={-}\alpha _{s}s,s_{0}=1 \end{array}\right. \end{equation}

where $\alpha _{z},$ $\beta _{z}$ are the positive parameters, $x$ is the position variable, $g$ is the goal point, $v$ is the velocity, $\dot{v}$ is the acceleration, $\tau \gt 0$ is the time constant, $s$ is a phase variable that avoid explicit time dependence, the initial value of $s$ is set as 1, and $\alpha _{s}$ is the factor to modify the converging time

The forcing term $f(s)$ is defined as

(7) \begin{equation} f(s)=\frac{\sum _{i=1}^{N} \psi _{i} w_{i}}{\sum _{i=1}^{N} \psi _{i}} s \!\left(g-y_{0}\right) \end{equation}

The forcing term has the components that consist of N Gaussian basis functions, enable the encoding of demonstrated trajectory, where $y_{0}$ is the starting position state, $w_{i}$ is the column of the weight vector, $\psi _{i}$ is Gaussian radial basis function, where $\psi _{i}(s)=\exp\! ({-}h_{i}(\mathrm{s}-\mathrm{c}_{i})^{2}), \mathrm{c}_{i}$ is the center of Gaussian kernels and $h_{i}$ is the variance. The vector $w_{i}$ can be trained with supervised learning algorithms such as locally weighted regression.

The calculating process is proposed to minimize the error function as following:

(8) \begin{equation} \min \!\left\{\,f^{t}(s)-\,f(s)\right\} \end{equation}

where $f(s)$ is an item calculated by the trajectory in demonstration, and $f^{t}(s)$ represents the target value as following:

(9) \begin{equation} f^{t}(s)=\tau \dot{v}-\alpha _{z}\!\left(\beta _{z}(g-x)-v\right) \end{equation}

2.4. General IBLF

Consider the strict feedback nonlinear system described as

(10) \begin{equation} \begin{cases} \dot{x}_{1} =\,f_{1}+g_{1}x_{2}\\ \dot{x}_{2} =\,f_{2}+g_{2}u\\ y =x_{1} \end{cases} \end{equation}

where $f_{1},f_{2},g_{1},g_{2}$ are smooth functions, $x_{1},x_{2}$ are the states, and $u$ and $y$ are the input and output.

Introduce the IBLF candidate in the following:

(11) \begin{equation} V_{1}=\int _{0}^{z_{1}}\frac{\rho k_{c}^{2}}{k_{c}^{2}-\left(\rho +x_{r}\right)^{2}}d\rho \end{equation}

where $k_{c}$ is the constant, $x_{r}$ is the variable, $\rho$ is a member of integrating, and the error variable $z_{1}= x_{1}-x_{r}, z_{2}= x_{2}-\alpha, \alpha$ is a continuously differentiable function.

The time derivative of $V_{1}$ is given by

(12) \begin{equation} \dot{V}_{1}=\frac{z_{1}k_{c}^{2}}{k_{c}^{2}-x_{1}^{2}}\dot{z}_{1}+\frac{\partial V_{1}}{\partial x_{r}}\dot{x}_{r} \end{equation}

where

(13) \begin{equation} \frac{\partial V_{1}}{\partial x_{r}}=z_{1}\!\left(\frac{k_{c}^{2}}{k_{c}^{2}-x_{1}^{2}}-\lambda \right) \end{equation}

(14) \begin{equation} \lambda =\frac{k_{c}}{2z_{1}}\ln \frac{\left(k_{c}+z_{1}+x_{r}\right)\left(k_{c}-x_{r}\right)}{\left(k_{c}-z_{1}-x_{r}\right)\!\left(k_{c}+x_{r}\right)} \end{equation}

The virtual control variable $\alpha$ can be designed as follows:

(15) \begin{equation} \alpha =\frac{1}{g_{1}}\!\left(-k_{1}z_{1}-f_{1}+\frac{g_{1}\!\left(k_{c}^{2}-x_{1}^{2}\right)\dot{x}_{r}\lambda }{k_{c}^{2}}\right) \end{equation}

where $k_{1}$ is positive constant. Substituting (15) into (12), we can get

(16) \begin{equation} \dot{V}_{1}=-\frac{k_{1}z_{1}^{2}k_{c}^{2}}{k_{c}^{2}-x_{1}^{2}}+\frac{g_{1}z_{1}z_{2}k_{c}^{2}}{k_{c}^{2}-x_{1}^{2}} \end{equation}

Then, a Lyapunov function candidate is chosen as follows:

(17) \begin{equation} V_{2}=V_{1}+\frac{1}{2}z_{2}^{2} \end{equation}

The time derivative of $V_{2}$ is given by

(18) \begin{equation} \dot{V}_{2}=-\frac{k_{1}z_{1}^{2}k_{c}^{2}}{k_{c}^{2}-x_{1}^{2}}+\frac{g_{1}z_{1}z_{2}k_{c}^{2}}{k_{c}^{2}-x_{1}^{2}}+z_{2}\!\left(f_{2}+g_{2}u-\dot{\alpha }\right) \end{equation}

The control law is designed as

(19) \begin{equation} u=\frac{1}{g_{2}}\!\left(-f_{2}+\dot{\alpha }-k_{2}z_{2}-\frac{g_{1}z_{1}k_{c}^{2}}{k_{c}^{2}-x_{1}^{2}}\right) \end{equation}

3.1. DMPs based on IBLF

The expression of DMPs as (6) can be revised as nonlinear system as follows:

(20) \begin{equation} \left\{\begin{array}{l} \dot{x}_{1}=g_{1}x_{2}\\[2pt] \dot{x}_{2}=\,f_{2}+u+\Delta u\\[2pt] \tau \dot{s}=-\alpha _{s}s,s(0)=1\\[2pt] \dot{\tau }=-\gamma _{0}\!\left(\tau -\tau _{0}\right)+\gamma _{1}\tau v^{T}\sigma (v),\tau _{0}=1\\[2pt] y=x_{1} \end{array}\right. \end{equation}

where $x_{1},x_{2},g_{1}$ represent the $x,v,1/\tau$ in Eq. (5), $f_{2}=(\alpha _{z}(\beta _{z}(g-x)-v)+\gamma \sigma (v))/\tau, u$ is the forcing function $f(s), \Delta u$ is a term added by IBLF, $\gamma, \gamma _{0}, \gamma _{1}$ are the positive constants, to allow the velocity being close to the limit while still not exceeding it, the $\sigma _{i}$ is designed as

(21) \begin{equation} \sigma _{i}\!\left(v_{i}\right)=\frac{v_{i}}{\left(A_{i}-v_{i}\right)\!\left(B_{i}+v_{i}\right)} \end{equation}

where $A_{i}$ , $B_{i}$ are the positive constants.

Similar to the general form of IBLF, we define $z_{1}=x_{1}-x_{r}, z_{2}=x_{2}-\alpha, x_{r}$ is the desired state in control system, while there is no such state in DMPs, so here we introduce the motion generated by DMPs method without any limits as the desired state:

(22) \begin{equation} \left\{\begin{array}{c} \dot{x}_{r}=g_{1}v_{c}\\[3pt] \dot{v}_{c}=\dfrac{\alpha _{z}\!\left(\beta _{z}\!\left(g-x_{r}\right)-v_{c}\right)+f(s)}{\tau } \end{array}\right. \end{equation}

Theorem 1: The output constraint is never violated, and all closed loop signals are confined if the following conditions are met for the DMPs function represented by (20).

(23) \begin{equation} \left\{\begin{array}{l} \alpha =\dfrac{1}{g_{1}}\!\left(-k_{1}z_{1}+\dfrac{g_{1}\!\left(k_{c}^{2}-x^{2}\right)\dot{x}_{r}\lambda }{k_{c}^{2}}\right)\\[14pt] u+\Delta u=-f_{2}+\dot{\alpha }-k_{2}z_{2}-\dfrac{g_{1}z_{1}k_{c}^{2}}{k_{c}^{2}-x_{1}^{2}} \end{array}\right. \end{equation}

where $\lambda$ is introduced in (14)

To get the form of $\Delta u$ , we calculate $u$ without added term is

(24) \begin{equation} u=\dot{v}-f_{2} \end{equation}

So

(25) \begin{equation} \Delta u=-k_{2}z_{2}-\frac{g_{1}z_{1}k_{c}^{2}}{k_{c}^{2}-x_{1}^{2}}+\dot{\alpha }-\dot{v} \end{equation}

Proof:

We synthesize a Lyapunov function as

(26) \begin{equation} V^{c}=V_{1}+\frac{1}{2}z_{2}^{2} \end{equation}

where $V_{1}$ is the IBLF candidate in (11)

Then

(27) \begin{align} \dot{V}^{c}&=\dfrac{z_{1}k_{c}^{2}}{k_{c}^{2}-x^{2}}\dot{z}_{1}+\dfrac{\partial V_{1}}{\partial x_{r}}\dot{x}_{r}+\mathrm{z}_{2}^{\mathrm{T}}\dot{\mathrm{z}}_{2}\nonumber\\ &=\dfrac{z_{1}k_{c}^{2}}{k_{c}^{2}-x^{2}}\!\left(\dot{x}-\dot{x}_{r}\right)+\dfrac{\partial V_{1}}{\partial x_{r}}\dot{x}_{r}+\mathrm{z}_{2}^{\mathrm{T}}\!\left(\dot{v}-\dot{\alpha }\right)\nonumber\\ &=\dfrac{z_{1}k_{c}^{2}}{k_{c}^{2}-x^{2}}\!\left(g_{1}\!\left(z_{2}+\alpha \right)-\dot{x}_{r}\right)+\dfrac{\partial V_{1}}{\partial x_{r}}\dot{x}_{r}\nonumber\\&\quad + \mathrm{z}_{2}^{\mathrm{T}}\!\left(f_{2}+u+\Delta u-\dot{\alpha }\right) \end{align}

Taking the expressions of (16) into (20), we have

(28) \begin{align} \dot{V}^{c} & =\dfrac{g_{1}z_{2}z_{1}k_{c}^{2}}{k_{c}^{2}-x^{2}}-\dfrac{k_{1}z_{_{1}}^{2}k_{c}^{2}}{k_{c}^{2}-x^{2}}+z_{1}\lambda \dot{x}_{r}\nonumber\\&\quad -\dfrac{z_{1}k_{c}^{2}}{k_{c}^{2}-x^{2}}\dot{x}_{r}+z_{1}\!\left(\dfrac{k_{c}^{2}}{k_{c}^{2}-x_{1}^{2}}-\lambda \right)\dot{x}_{r}\nonumber\\ &\quad +\mathrm{z}_{2}^{\mathrm{T}}\!\left(f_{2}-f_{2}+\dot{\alpha }-k_{2}z_{2}-\dfrac{g_{1}z_{1}k_{c}^{2}}{k_{c}^{2}-x_{1}^{2}}-\dot{\alpha }\right)\nonumber\\ &=\dfrac{g_{1}z_{2}z_{1}k_{c}^{2}}{k_{c}^{2}-x^{2}}-\dfrac{k_{1}z_{_{1}}^{2}k_{c}^{2}}{k_{c}^{2}-x^{2}}+\mathrm{z}_{2}^{\mathrm{T}}\!\left(-k_{2}z_{2}-\dfrac{g_{1}z_{1}k_{c}^{2}}{k_{c}^{2}-x_{1}^{2}}\right)\nonumber\\ &=-\dfrac{k_{1}z_{_{1}}^{2}k_{c}^{2}}{k_{c}^{2}-x^{2}}-k_{2}z_{_{2}}^{2}\leq 0 \end{align}

where $k_{1},k_{2}$ are positive numbers. according to Lyapunov stability theorem, we know that above expression guarantees global stability and the global tracking convergence in the system.

3.2. Master controller design

The control command on the master robot is designed as an impedance controller, such that the position of the robot end effector can be moved by export, let us consider the modeling of the operating torque. In this paper, a damping-stiffness model is considered.

(29) \begin{equation} F_{e}=k_{m}\mathrm{e}_{m}+d_{m}\dot{\mathrm{e}}_{m} \end{equation}

where $k_{m}$ and $d_{m}$ denote designed damping and stiffness matrices, where $e_{m}=x_{md}-x_{m}, x_{md}$ is the desired state, in this remote corrective system, it is fixed value. For avoiding the slave robot moving off the edge of the surface during control, a variable stiffness is proposed

(30) \begin{equation} k_{m}=k_{\textit{min}}+\mathrm{e}^{-d\left(x_{m}\right)}\!\left(k_{\textit{max} }-k_{\textit{min} }\right) \end{equation}

where $\alpha$ is the proximity of distance to boundary, closer get to the boundary, the closer get to 1, where $d(x_{m})$ is the distance to the nearest edge, $k_{\textit{max} }, k_{\textit{min} }$ are positive numbers.

Then, the control torque can be designed as

(31) \begin{equation} f_{m}=\hat{\mathrm{G}}_{m}+\mathrm{u}_{m}+F_{e} \end{equation}

where $u_{m}$ is robust term, and ${\mathop{G}\limits^\frown}_{m}$ is the estimation of $G$ , which satisfies: $G-\hat{\mathrm{G}}_{m}=\varepsilon _{m}$ .

3.3. Slave controller design

Inspired by ref. [Reference Chen, Huang, Sun, Gu and Yao24], the desired position signal $x_{sd}$ can be derived via the slave trajectory creator. For simplicity, a filter as

(32) \begin{equation} V_{f}(s)=1/\!\left(1+\tau _{f}s\right)^{2} \end{equation}

It is used with the input of $x_{m}(t - T(t))$ to create the correction state $x_{sc}$ , then $x_{sd}=\Delta x_{sd}+x_{sdmp}$ , where $\Delta \mathrm{x}_{\mathrm{sd}}=x_{sc}-x_{sd}$ is the forward kinematic function of the Touch $x_{sdmp}$ is the path generate by DMPs.

Due to the disturb and errors in tracking, the trajectory generated by creator may not guarantee the $x$ in the constraint space all the time, we can obtain $x$ by a soft saturation function, then if the x cross the bound it will map the inside state. We derive the following via a saturation function to guarantee that the reference trajectory stays inside the limited region:

(33) \begin{equation} x_{\mathrm{sd}}=\left\{\begin{array}{l} x_{\mathrm{sd}},\left| x_{\mathrm{sd}}\right| \leq \eta k_{{c_{i}}}\\ \eta k_{{c_{i}}},x_{\mathrm{sd}}\lt -\eta k_{{c_{i}}}\\ \eta k_{{c_{i}}},x_{\mathrm{sd}}\gt \eta k_{{c_{i}}} \end{array}\right. \end{equation}

where $\eta$ is a constant very close to 1.

Because of the uncertainity of robot dynamic and model, reference trajectory cannot ensure the end-effector stay in the constrained space. IBLF method is introduced to ensure the constrains of the predefined task space met.

We can obtain the dynamics system of slave

(34) \begin{equation} \left\{\begin{array}{l} \dot{x}_{1}=x_{2}\\ \dot{x}_{2}=M_{s}^{-1}\!\left(f_{s}-D_{s}-G_{s}-C_{s}\dot{x}_{s}\right)\\ y=x_{1} \end{array}\right. \end{equation}

Then, we define $z_{1}=x_{1}-x_{sd}, z_{2}=x_{2}-\alpha$ , where $z_{1}, z_{2}$ are error variables, and $\alpha$ denotes the virtual control variable, we design it as

(35) \begin{equation} a = - k_{s1}z_{1} + \frac{\left(k_{c}^{2} - x_{1}^{2}\right)\dot{x}_{r}\lambda }{k_{c}^{2}} \end{equation}

where $k_{s1}$ is the positive number, $k_{c}$ is the limit in Cartesian space, and $\lambda$ is given in (14)

The control law is designed as

(36) \begin{equation} f_{s} = - \frac{z_{1}k_{c}^{2}}{k_{c}^{2} - x_{1}^{2}} - k_{s2}z_{2} + \hat{W}^{T}S(Z) + u_{s} \end{equation}

In the actual system, the dynamics parameters and are typically unknown. RBFNN is used to approximation manipulator uncertainties to handle the uncertainty issue in the dynamic model. The neural network’s input on the slave side can be chosen as $\mathrm{Z}=[\mathrm{x}_{\mathrm{s}},\dot{\mathrm{x}}_{\mathrm{s}},{\alpha},\dot{{\alpha} }]$ . RBF neural network is defined as

(37) \begin{equation} W^{T}S(Z) = M_{s}\dot{a} + C_{s}a + G_{s} + e_{s} \end{equation}

3.4. Stability analysis

Define the Lyapunov function of master and salve as

(38) \begin{equation} V_{m} = \frac{1}{2}\!\left(\dot{x}_{m}^{T}M_{m}\dot{x}_{m} + e_{m}^{T}k_{m}e_{m}\right) \end{equation}

(39) \begin{equation} V_{s} = V_{1} + \frac{1}{2}z_{2}^{T}M_{s}z_{2} + \frac{1}{2}tr\!\left(\tilde{W}_{s}^{T}G_{s}^{ - 1}\tilde{W}_{s}\right) \end{equation}

where $\tilde{W}=\hat{W}-W$

The Lyapunov candidate function of whole system as

(40) \begin{align} V&=V_{m}+V_{s}\nonumber\\ &=\dfrac{1}{2}\!\left(\dot{\mathrm{x}}_{m}^{T}\mathrm{M}_{m}\dot{\mathrm{x}}_{m}+e_{m}^{T}k_{m}\mathrm{e}_{m}\right)+\mathrm{V}_{1}+\dfrac{1}{2}\mathrm{z}_{2}^{\mathrm{T}}\mathrm{M}_{s}\mathrm{z}_{2}+\dfrac{1}{2}\mathrm{tr}\!\left(\tilde{\mathrm{W}}_{\mathrm{s}}^{\mathrm{T}}{\Gamma} _{\mathrm{s}}^{-1}\tilde{\mathrm{W}}_{\mathrm{s}}\right) \end{align}

Then, the derivative of $\mathrm{V}_{m}$ can be calculated as

(41) \begin{equation} \dot{\mathrm{V}}=\ddot{x}_{m}^{T}\mathrm{M}_{m}\dot{\mathrm{x}}_{m}+\dfrac{1}{2}\dot{x}_{m}^{T}\dot{M}_{m}\dot{\mathrm{x}}_{m}+\dot{e}_{m}^{T}k_{m}\mathrm{e}_{m}+\dot{\mathrm{V}}_{1} +\mathrm{z}_{2}^{\mathrm{T}}\mathrm{M}_{s}\dot{\mathrm{z}}_{2}+\dfrac{1}{2}\mathrm{z}_{2}^{\mathrm{T}}\dot{\mathrm{M}}_{s}\mathrm{z}_{2}+\mathrm{tr}\!\left(\tilde{\mathrm{W}}_{\mathrm{s}}^{\mathrm{T}}{\Gamma} _{\mathrm{s}}^{-1}\dot{\tilde{W}}_{\mathrm{s}}\right) \end{equation}

Combine (1), (2), (16), (41)

(42) \begin{align} \dot{\mathrm{V}} & =\dot{x}_{m}^{T}\!\left(f_{m}-\mathrm{C}_{m}\dot{\mathrm{x}}_{m}-\mathrm{G}_{m}-\mathrm{D}_{m}+\dfrac{1}{2}\dot{M}_{m}\dot{\mathrm{x}}_{m}\right)-\dot{e}_{m}^{T}k_{m}e_{m}\nonumber\\ &\quad -\dfrac{k_{1}z_{1}^{2}k_{c}^{2}}{k_{c}^{2}-x_{1}^{2}}+\dfrac{z_{1}z_{2}k_{c}^{2}}{k_{c}^{2}-x_{1}^{2}}+\mathrm{z}_{2}^{\mathrm{T}}\!\left(f_{s}-G_{s}-\mathrm{C}_{s}\dot{\mathrm{x}}_{\mathrm{s}}-D_{s}-\mathrm{M}_{s}\dot{\alpha }\right)\nonumber\\ &\quad +\dfrac{1}{2}\mathrm{z}_{2}^{\mathrm{T}}\dot{\mathrm{M}}_{s}\mathrm{z}_{2}+\mathrm{tr}\tilde{\mathrm{W}}_{\mathrm{s}}^{\mathrm{T}}{\Gamma} _{\mathrm{s}}^{-1}\dot{\tilde{W}} \end{align}

Substituting (31) (36) into (42), we have

(43) \begin{align} \dot{V} & =\dot{x}_{m}^{T}\!\left(\hat{\mathrm{G}}_{m}+\mathrm{u}_{m}+F_{e}-\mathrm{C}_{m}\dot{\mathrm{x}}_{m}-\mathrm{G}_{m}-\mathrm{D}_{m}+\dfrac{1}{2}\dot{M}_{m}\dot{\mathrm{x}}_{m}\right)-\dot{e}_{m}^{T}k_{m}e_{m}\nonumber\\ &\quad -\dfrac{k_{s1}z_{1}^{2}k_{c}^{2}}{k_{c}^{2}-x_{1}^{2}}+\dfrac{1}{2}\mathrm{z}_{2}^{\mathrm{T}}\!\left(\dot{\mathrm{M}}_{\mathrm{s}}-2\mathrm{C}_{\mathrm{s}}\right)\mathrm{z}_{2}\nonumber\\ &\quad +\mathrm{z}_{2}^{\mathrm{T}}\!\left(\dfrac{z_{1}k_{c}^{2}}{k_{c}^{2}-x_{1}^{2}}+f_{s}-D_{s}-G_{s}-\mathrm{C}_{\mathrm{s}}{\alpha} -\mathrm{M}_{\mathrm{s}}\!\left(\mathrm{q}\right)\dot{\alpha }\right)\nonumber\\ &\quad +\mathrm{tr}\tilde{\mathrm{W}}_{\mathrm{s}}^{\mathrm{T}}{\Gamma} _{\mathrm{s}}^{-1}\!\left(\dot{\hat{W}}_{s}-\dot{\mathrm{W}}_{\mathrm{s}}\right) \end{align}

then

(44) \begin{align} \dot{V}& = \dot{x}_{m}^{T}\mathit{(}\hat{G}_{m} + u_{m} + F_{e} - G_{m} - D_{m}\mathit{)} - \dot{e}_{m}^{T}k_{m}e_{m} - \frac{k_{s1}z_{1}^{2}k_{c}^{2}}{k_{c}^{2} - x_{1}^{2}}\nonumber\\ &\quad + z_{2}^{T}\!\left(\frac{z_{1}k_{c}^{2}}{k_{c}^{2} - x_{1}^{2}} + f_{s} - D_{s} - G_{s} - C_{s}a - M_{s}\!\left(q\right)\dot{a}\right) + tr\tilde{W}_{s}^{T}G_{s}^{ - 1}\dot{\hat{W}}_{s}\nonumber\\ & = \dot{x}_{m}^{T}\mathit{(}\hat{G}_{m} + u_{m} - G_{m} - D_{m}\mathit{)} + \dot{e}_{m}^{T}\mathit{(}F_{e} - k_{m}e_{m}\mathit{)} - \frac{k_{s1}z_{1}^{2}k_{c}^{2}}{k_{c}^{2} - x_{1}^{2}}\nonumber\\ &\quad - k_{s2}z_{2}^{2} + z_{2}^{T}\!\left(u_{s} + \hat{W}^{T}S (Z) - D_{s} - G_{s} - C_{s}a - M_{s}\!\left(q\right)\dot{a}\right) + tr\tilde{W}_{s}^{T}G_{s}^{ - 1}\dot{\hat{W}}_{s}\nonumber\\ & = - d_{m}\dot{e}_{m}^{2} - \frac{k_{s1}z_{1}^{2}k_{c}^{2}}{k_{c}^{2} - x_{1}^{2}} - k_{s2}z_{2}^{2} + \dot{x}_{m}^{T}\!\left(u_{m} - e_{m} - D_{m}\right)\nonumber\\ &\quad + z_{2}^{T}\!\left(u_{s} - D_{s} - e_{s}\right) + tr\tilde{W}_{s}^{T}\!\left(G_{s}^{ - 1}\dot{\hat{W}}_{s} + z_{2}^{T}S\!\left(Z\right)\right) \end{align}

Furthermore, the robust term in a controller, which is used to deal with estimating error, external disturbance, and modeling error, can be created as

(45) \begin{equation} \mathrm{u}_{m}=-\!\left(d_{m}+{\unicode[Times]{x025B}} _{mb}\right)sgn\!\left(\dot{x}_{m}\right) \end{equation}

where $\| \mathrm{D}_{m}\| \leq d_{\mathrm{m}}, \| {\unicode[Times]{x025B}} _{m}\| \leq {\unicode[Times]{x025B}} _{mb}, d_{\mathrm{m}}$ and ${\unicode[Times]{x025B}} _{mb}$ are positive constants.

(46) \begin{equation} \mathrm{u}_{\mathrm{s}}=-\!\left(d_{\mathrm{s}}+{\unicode[Times]{x025B}} _{sb}\right)\mathrm{sgn}\!\left(z_{2}\right) \end{equation}

where $\| D_{s}\| \leq d_{s}, \| {\unicode[Times]{x025B}} _{s}\| \leq {\unicode[Times]{x025B}} _{sb}$ $d_{s}$ and ${\unicode[Times]{x025B}} _{sb}$ are positive constants.

Thus, the adaptive law, which is used to estimate the RBF neural network parameters online and real time, can be designed as

(47) \begin{equation} \dot{\hat{W}}_{s}=-{\Gamma} _{s}\mathrm{S}\!\left(\mathrm{Z}\right)\mathrm{z}_{2}^{\mathrm{T}} \end{equation}

then

(48) \begin{align} \dot{\mathrm{V}} & \leq -\dot{e}_{m}^{T}D\dot{\mathrm{e}}_{m}-\left| \dot{\mathrm{x}}_{m}^{T}\right| \!\left(d_{m}+{\unicode[Times]{x025B}} _{mb}+\varepsilon _{m}+D_{m}\right)\nonumber\\ &\quad -\frac{k_{1}z_{1}^{2}k_{c}^{2}}{k_{c}^{2}-x_{1}^{2}}-K_{2}\mathrm{z}_{2}^{2}+\left| \mathrm{z}_{2}^{\mathrm{T}}\right| \!\left(d_{s}+{\unicode[Times]{x025B}} _{sb}+\varepsilon _{s}+D_{s}\right)\nonumber\\ &\leq 0 \end{align}

According to the Lyapunov stability theorem, the Lyapunov function is uniformly positive defined, its derivative is negative defined. The aforementioned formula ensures global stability and the convergence of global tracking in the system employing the suggested controller.

4.1. Constrained control effect

In this part, we apply the designed controllers to verify work of the proposed algorithm. The experiments are performed on the two touch robots. First, a demonstration trajectory is given by operator, then the slave robot moves along the trajectory and the master robot is operated to modify the trajectory of the slave robot.

For the master robot, the parameters are chosen as $k_{min}=10, k_{\max }=100, d_{m}+\varepsilon _{mb}=0.1, D=1,$ ${\Gamma} _{m}=1, x_{md}=[50,0,60]$ is near the mid of its workspace. For the slave robot, control parameters are chosen as $k_{s1}=10, k_{c}=60$ and $k_{s2}=15, d_{s}+\varepsilon _{sb}=0.1, {\Gamma} _{s}=1$ . The slave robot receives the correction information from the master after filter and then applies the soft saturation function to generate the desired trajectory of the end effector in task space. Parameter of soft saturation function is designed as $\eta =0.98$ , the weight parameters of the RBF NN are initialized as 0, and the centers of the functions are distributed in the interval [0, 1]. The human operates the master device toward boundaries in the y-axes orderly.

The experiment results are shown in Fig. 3, our suggested controller guarantees that the end-effector tracks the reference trajectory in real time while operating inside the restricted area. Operator can feel the resistance when moving away from the hold position which prevent accidental contact. When the operator forced the slave to cross the bound of $y=60$ , the soft saturation function and IBLF controller work together to ensure the end-effector stay below the limit.

Figure 3. Results of remote corrective control.

4.2. Trajectory learning

The second group of experiments aims to test the ILBF-based motion model. The ability of constrains of task space and the velocity are tested. A drawing task is designed for the test. The common parameters of DMPs are chosen as $\alpha _{s}=5, \alpha _{z}=10, \beta _{z}=100$ and others are set separately in each experiment.

To validate the correction performance of the IBLF-based DMPs, the parameters of IBLF are $k_{1}=1, k_{2}=10, k_{c}=8$ . The speed constraint item is added to the IBLF-based method, and the selected parameter is $A=100, B=-100, \gamma =10, \gamma _{0}=5, \gamma _{1}=10$ .

As shown in Fig. 4, during the operation, the track was corrected by teleoperation equipment, and an obstacle was added in the early stage of the experiment to make the movement within the edge. The corrected trajectory will be learned then, we hope that the learned trajectory will not cross the limit where we get the information from correction.

Figure 4. Motion generation of different methods.

Figure 5. Motion generation of different methods.

Then the corrected trajectory is learned by a classic DMPs method and a modified DMPs method proposed in this article. As shown in Fig. 5, The classical DMPs method is compared with the IBLF-based method. In the top of the figure, the generalization process is shown, both can learn the characteristics of the trajectory, and finally approach the target point. However, it shows that the red line across the limit around $x=-50$ . While the bule line always stays within the constraints. It can be clear expression in the mid of the figure. It is the relationship between time and location. We can see that the red line over the border at $t=0.4$ . Compared with the classical method, IBLF-based method can constrain the motion within the set range. However, due to the few selected neural network nodes, the local features cannot be perfectly expressed. The bottom of the picture shows the speed constraint capability. The velocity oscillation at $t=0.3$ is a confrontation to prevent crossing the obstacle and the desired trajectory. Comparing to the red line which is the velocity of the classic DMPs, the blue one can always stays within the constraints. It can be seen that velocity is able to stay within the limits comparing with the classic DMPs.