2.1. Dynamic model of the robot with input saturation

Generally, a collaborative robot dynamic model is often expressed by a second-order nonlinear equation. The general dynamic formulation in the robot actuator joint space can be illustrated by [Reference Roy, Roy and Kar32]:

(1) \begin{equation} \mathbf{M(q)}\ddot{\mathbf q}+\mathbf{C(q,q)}\dot{\mathbf q}+\mathbf{g}(\mathbf q)+\mathbf{J}^{T}\mathbf{F}_{e}=\Delta \mathbf{\tau } \end{equation}

where $\Delta \mathbf{\tau }= sat(\mathbf{\tau })-\mathbf{\tau }_d$ ; $\mathbf q,\dot{\mathbf q},\ddot{\mathbf q}\in \mathbb{R}^n$ are generalized actuate joint position, velocity, and acceleration, respectively; $\mathbf{M(q)}\in \mathbb{R}^{n\times n}$ and $\mathbf{C(q,{q})}\in \mathbb{R}^{n\times n}$ are the symmetric positive definite inertia matrix and the Coriolis-centrifugal matrix of the robot, and $\mathbf{g}(\mathbf q)\in \mathbb{R}^{n\times 1}$ denotes the gravity vector; $\mathbf{F_e}\in \mathbb{R}^{6\times 1}$ represents the external force caused by the contact with the environment; $\mathbf{J}$ is the Jacobian matrix of the robot; the nonlinear saturation torque $sat(\mathbf{\tau })$ is given by

(2) \begin{equation} sat(\mathbf{\tau }^{i}) = \begin{cases} \mathbf{\tau }_{max}^{i}, &\text{if $\mathbf{\tau } ^{i}\gt \mathbf{\tau } _{max}^{i}$}, \\ \mathbf{\tau }^{i}, &\text{if $\mathbf{\tau } _{min}^{i}\lt \mathbf{\tau } ^{i}\lt \mathbf{\tau } _{max}^{i}$},\\ \mathbf{\tau }_{min}^{i}, &\text{if $\mathbf{\tau }^{i}\lt \mathbf{\tau } _{min}^{i}$}.\\ \end{cases} \end{equation}

where $i=1,2\dots n$ and $\mathbf{\tau }$ represents the vector of the input torque without saturation. $\mathbf{\tau } _{max}^{i}$ and $\mathbf{\tau } _{min}^{i}$ are the upper and lower bounds of permissible torque input limited by actuator joint constraints, respectively. $\mathbf{\tau }_d\in \mathbb{R}^{n\times 1}$ denotes the unknown disturbance in input torque.

Property 1. The matrix $\mathbf{M}$ is a symmetric positive inertia matrix constrained by the following:

(3) \begin{equation} \begin{aligned} m_{1}\left \| \textbf{x}\right \|^{2}\leq \textbf{x}^{T}\mathbf{M}\textbf{x}\leq m_{2}\left \|\textbf{x} \right \|^2 \end{aligned} \end{equation}

in which $m_{1}$ and $m_{2}$ are the lower and upper bound limits, and $\textbf{x}$ can be any n-dimensional vector given as $\textbf{x}\in \mathbb{R}^{n \times 1}$ .

Property 2. Matrix $\mathbf{C(q,{q})}$ , vector $\mathbf{g(q)}$ and vector $\mathbf{\tau }_{d}$ in (1) are bounded as:

(4) \begin{equation} \begin{aligned} \left \|\mathbf{C}(\mathbf q,\dot{\mathbf q}) \right \|&\leq \mathbf{C}_m \left \|\dot{\mathbf{q}} \right \|, \\ \left \|\mathbf{g}(\mathbf q) \right \|& \leq \mathbf{g}_m, \\ \left \|\mathbf{\tau }_{d} \right \|&\leq \mathbf{\tau }_{m}. \end{aligned} \end{equation}

in which $C_{m}$ , $\mathbf{g}_{m}$ and $\mathbf{\tau }_{m}$ are positive constants.

Property 3. $\dot{\mathbf{M}}-2\mathbf{C}$ is a skew symmetric matrix satisfying the following equality:

(5) \begin{equation} \textbf{x}^{T}[\dot{\mathbf{M}}-2\mathbf{C}]\textbf{x}=0 \end{equation}

where $\textbf{x}$ is any n-dimensional vector given as $\textbf{x}\in \mathbb{R}^{n \times 1}$ , and $\textbf{M}$ , $\textbf{C}$ are the metrics in (1).

2.2. Dynamic of image-based visual servoing

Image-based visual servoing (IBVS) is an external sensor-based controller. It takes cameras as the primary sensors and estimates the pose of the robot directly from the visual information [Reference Zhu, Briot and Chriette33]. The general kinematic relationship between the twist of the robot and the velocity of the image features can be described by [Reference Mariottini, Oriolo and Prattichizzo34]:

(6) \begin{equation} \dot{\mathbf s}=\mathbf{L}_s\dot{\mathbf{p}}_c \end{equation}

where $\mathbf{s}\in \mathbb{R}^m$ is the vector of $m$ image features and $\mathbf{L}_s$ is the well-known interaction matrix related to $\mathbf s$ [Reference Chaumette and Hutchinson35]. $\dot{\mathbf{p}}_c$ denotes the spatial relative camera-object velocity of the robot. It can be transformed to the actuator velocities $\dot{\mathbf q}$ with the Jacobian matrix $\mathbf{J}$ . Therefore, the relationship between $\dot{\mathbf s}$ and $\dot{\mathbf q}$ can be illustrated as:

(7) \begin{equation} \dot{\mathbf s}=\mathbf{L}_s^c\mathbf{T}_e\mathbf{J}\dot{\mathbf q} \end{equation}

where $^c\mathbf{T}_e$ represents the transform matrix from the kinematic screw to the camera frame. $\mathbf{J}_s$ is defined as $\mathbf{J}_s=\mathbf{L}_s^c\mathbf{T}_e\mathbf{J}$ . Then, Eq.7 can be rearranged in the following equation:

(8) \begin{equation} \dot{\mathbf q}=\mathbf{J}_s^{+}\dot{\mathbf s} \end{equation}

in which $^{+}$ here represents the pseudo-inverse of the matrix.

The application of visual feedback in the dynamic model of robots was proposed by [Reference Zhang and Ostrowski36] in 1999 for the first time. In order to generate the actuator torque, the image feature accelerations have been applied to the IBVS dynamic model [Reference Fusco, Kermorgant and Martinet37, Reference Ozawa and Chaumette38]. By differentiating (8), the IBVS interaction model can be expressed as:

(9) \begin{equation} \ddot{\mathbf q}=\mathbf{J}_{\mathbf s}^{+}\ddot{\mathbf s}+\dot{\mathbf{J}_{\mathbf s}^{+}}\dot{\mathbf s} \end{equation}

Inserting (6) and (9) into (1), the IBVS dynamic model can be written as:

(10) \begin{equation} sat(\mathbf{\tau })=\mathbf{M}\mathbf{J}^+_s\ddot{\mathbf s}+\mathbf{M}\dot{\mathbf{J}_{s}^{+}}\dot{\mathbf s}+\mathbf{C}\mathbf{J}_{s}^{+}\dot{\mathbf s}+\mathbf{g}+\mathbf{J}^{T}\mathbf{F}_e+\mathbf{\tau }_d \end{equation}

Multiplying both sides of (10) with the term ${\mathbf{J}_s ^+ }^T$ , the general dynamic Image-based visual servoing model can be rearranged as:

(11) \begin{equation} \overline{sat(\mathbf{\tau })}=\overline{\mathbf{M}}\ddot{\mathbf s}+\overline{\mathbf{C}}\dot{\mathbf s}+\overline{\mathbf{g}}+\overline{\mathbf{J}^{T}\mathbf{F}}_e+\overline{{\mathbf{\tau }}_d} \end{equation}

where $\overline{sat(\mathbf{\tau })}={\mathbf{J}^+_s}^Tsat(\mathbf{\tau }),\overline{\mathbf{M}}={\mathbf{J}^+_s}^T \mathbf{M}\mathbf{J}^+_s, \overline{\mathbf{C}}={\mathbf{J}^+_s}^T(\mathbf{M}\dot{\mathbf{J}_{s}^{+}}+\mathbf{C}\mathbf{J}_{s}^{+}),\overline{\mathbf{g}}={\mathbf{J}^+_s}^T \mathbf{g},\overline{\mathbf{J}^{T}\mathbf{F}}_e={\mathbf{J}^+_s}^T\mathbf{J}^{T}\mathbf{F}_e$ and $\overline{{\mathbf{\tau }}_{d}}={\mathbf{J}^+_s}^T\mathbf{\tau }_d$ . Similarly, for the dynamic image-based visual servoing model of collaborative robots, some properties have already been demonstrated in ref. [Reference Wang, Liu, Chen and Zhang39], and are presented as follows:

Property 4. The matrix $\overline{\mathbf{M}}$ is a positive symmetric matrix and bounded as:

(12) \begin{equation} m_3\|\mathbf s\|^{2}\leq \mathbf s^T\overline{\mathbf{M}}\mathbf s\leq m_4\|\mathbf s\|^{2},\forall \mathbf s \in \mathbf{R}^{m\times 1}. \end{equation}

in which $m_{3}$ and $m_{4}$ are the lower limit constant and upper limit constant [Reference Wang, Liu, Chen and Zhang39].

Property 5. The disturbance torque $\overline{\mathbf{\tau }_{d}} \in \mathbb{R}^{m\times 1}$ in the feature space is bounded as:

(13) \begin{equation} \left \|\overline{\mathbf{\tau }_{d}} \right \|\leq \mathbf{\tau }_{sm} \end{equation}

It has been mentioned in Section 1 that for a collaborative robot, its task trajectory is predefined. Therefore, we can promise that the robot does not encounter Type 1 singularity and Type 2 singularity, and the visual servoing controller does not encounter controller singularity during the task [Reference Merlet40, Reference Briot and Martinet41], which means that the matrices $\mathbf{J}$ and $\mathbf{L}_{s}$ are all full rank.

4.1. Auxiliary dynamic model for input saturation compensation

Firstly, we construct the auxiliary dynamic model, which is used to establish the sliding function in the following section. Considering the impacts of the input saturation, the auxiliary dynamic vector, in this case, can be defined as:

(23) \begin{equation} \triangle \mathbf{r}=\overline{\mathbf{M}}^{-1}\overline{\triangle \mathbf{\tau }}\in \mathbb{R}^{m\times 1} \end{equation}

where $\overline{\triangle \mathbf{\tau }}=\overline{sat(\mathbf{\tau })}-\overline{\mathbf{\tau }}$ .

Then, the following auxiliary dynamic is proposed as follows:

(24) \begin{equation} \left \{ \begin{aligned} \dot{\xi _{i,1}}&=-h_{i,1}\xi _{i,1}+\xi _{i,2}, \\ \dot{\xi _{i,2}}&=-h_{i,2}\xi _{i,2}+\triangle{r_{i}}. \end{aligned} \right . \end{equation}

where $i=1,2,\cdot \cdot \cdot m$ . $\xi _{i,1},\xi _{i,2}$ are the state variables of the auxiliary dynamic model.

Thus, we rewrite (24) in the matrix form:

(25) \begin{equation} \dot{\xi _i}=\mathbf{H}_i\xi _i+\mathbf{E}_i\triangle r_i \end{equation}

where $\xi _i=[\xi _{i,1},\xi _{i,2}]$ , $\mathbf{H}_i=diag\{-h_{i,1},-h_{i,2}\}$ , and $\mathbf{E}_i=[0,1]^T$ . $h_{i,1},h_{i,2}$ are positive and selected to make $\mathbf{H}_i$ become a Hurwitz matrix. By defining $\boldsymbol{\xi }=[{\xi }_1^T\cdot \cdot \cdot{\xi }_n^T]^T\in \mathbb{R}^{2m\times 1}$ , (25) can be rewritten as:

(26) \begin{equation} \dot{\boldsymbol{\xi }}=\mathbf{H}\boldsymbol{\xi }+\mathbf{E}\triangle \mathbf{r} \end{equation}

where $\mathbf{H}=diag\{\mathbf{H}_1\cdot \cdot \cdot,\mathbf{H}_m\}\in \mathbb{R}^{2m\times 2m}$ , $\mathbf{E}=[\mathbf{E}_1^T\cdot \cdot \cdot,\mathbf{E}_m^T]\in \mathbb{R}^{2m\times m}$

Defining $\xi^{\prime}_1=[\xi _{1,1}\cdot \cdot \cdot \xi _{1,m}]^T$ and $\xi^{\prime}_2=[\xi _{2,1}\cdot \cdot \cdot \xi _{2,m}]^T$ , we transform (23) into the following form:

(27) \begin{equation} \left \{ \begin{aligned} \dot{\xi^{\prime}_1}&=-\hat{\mathbf{H}}_1\xi^{\prime}_1+\xi^{\prime}_2,\\ \dot{\xi^{\prime}_2}&=-\hat{\mathbf{H}}_2\xi^{\prime}_2+\triangle \mathbf{r}. \end{aligned} \right . \end{equation}

with $\hat{\mathbf{H}}_1=diag\{h_{1,1}\cdot \cdot \cdot h_{n,1}\}\in \mathbb{R}^{m\times m}$ , $\hat{\mathbf{H}}_2=diag\{h_{1,2}\cdot \cdot \cdot h_{n,2}\}\in \mathbb{R}^{m\times m}$

Inserting (23) into (26), we obtain:

(28) \begin{equation} \ddot{\xi^{\prime}_1}=-\hat{\mathbf{H}}_1\dot{\xi^{\prime}_1}-\hat{\mathbf{H}}_2\xi^{\prime}_2+\overline{\mathbf{M}}^{-1}\overline{\triangle \mathbf{\tau }} \end{equation}

In the following section, (28) is applied to the design of the adaptive sliding mode controller.

4.2. Leakage-type adaptive sliding mode controller

Considering the auxiliary dynamic compensation proposed above, the error vector of this sliding model controller can be defined as:

(29) \begin{equation} e_{c}=e_{sf}-\xi^{\prime}_1 \end{equation}

where $e_{sf}=\mathbf{s}-\mathbf{s}_{r}$ . $\mathbf s$ is the real-time vision feedback and $\mathbf{s}_r$ is the reference feature trajectory in Section 3.

With (29), the linear sliding function can be defined as:

(30) \begin{equation} \mathbf{r}=\dot{e}_c+\Lambda e_c \end{equation}

where $\Lambda =diag\{\lambda _1\cdot \cdot \cdot \lambda _m\}$ , $\lambda _i\gt 0$ .

Differentiating (30) and combining it with (11) and (28), we have:

(31) \begin{equation} \begin{aligned} \dot{\mathbf{r}}&= \ddot{e}_c+\Lambda \dot{e}_c\\ &= \ddot{\mathbf{s}}-\ddot{\mathbf s}_{r}-\ddot{\xi^{\prime}_1}+\Lambda \dot{e}_c\\ &= \ddot{\mathbf s}-\ddot{\mathbf s}_{r}+\hat{\mathbf{H}}_1\dot{\xi^{\prime}_1}+\hat{\mathbf{H}}_2\xi^{\prime}_2-\overline{\mathbf{M}}^{-1}\overline{\triangle \mathbf{\tau }}+\Lambda \dot{e}_c\\ &= \overline{\mathbf{M}}^{-1}(\overline{sat(\mathbf{\tau })}-\overline{\mathbf{C}}\dot{s}-\overline{\mathbf{g}}-\overline{\mathbf{J}^{T}\mathbf{F}_e}-\overline{\mathbf{\tau }_d})- \ddot{\mathbf s}_r+ \hat{\mathbf{H}}_1\dot{\xi^{\prime}_1}+ \hat{\mathbf{H}}_2\xi^{\prime}_2-\overline{\mathbf{M}}^{-1}\overline{\triangle \mathbf{\tau }}+\Lambda \dot{e}_c\\ &= \overline{\mathbf{M}}^{-1}(\overline{\mathbf{\tau }}-\overline{\mathbf{{C}}}\dot{\mathbf s}-\overline{\mathbf{g}}-\overline{\mathbf{J}^{T}\mathbf{F}_e}-\overline{\mathbf{\tau }_d})-\ddot{\mathbf s}_r+\hat{\mathbf{H}}_1\dot{\xi^{\prime}_1}+\hat{\mathbf{H}}_2\xi^{\prime}_2+\Lambda \dot{e}_c. \end{aligned} \end{equation}

This system is stable when $\dot{\mathbf r}=0$ and the external disturbance is ignored ( $\overline{\mathbf{\tau }_d}=0$ ). Therefore, the following equivalent input torque can be obtained:

(32) \begin{equation} \overline{\mathbf{\tau }_0}=\overline{\mathbf{C}}\dot{\mathbf{s}}+\overline{\mathbf{g}}+\overline{\mathbf{J}^{T}\mathbf{F}_e}+\overline{\mathbf{M}}(\ddot{\mathbf{s}}_r-\hat{\mathbf{H}}_1\dot{\xi^{\prime}_1}-\hat{\mathbf{H}}_2\xi^{\prime}_2-\Lambda \dot{e}_c) \end{equation}

Moreover, to solve the problem of the external disturbance of the system, we propose the following control law:

(33) \begin{equation} \overline{\mathbf{\tau }_1}=\overline{\mathbf{M}}({-}\mathbf{K} sgn(\mathbf{r})-\mathbf{\Phi } \mathbf{r}) \end{equation}

where $\boldsymbol{\Phi }=diag\{\phi _1\cdot \cdot \cdot,\phi _m\}$ , $\phi _i\gt 0$ . $\mathbf{K}=diag\{k_1\cdot \cdot \cdot,k_m\}$ is the adaptive parameter. By denoting $\mathbf{r}=[r_1\cdot \cdot \cdot,r_m]^T$ , the update law of $k_{i}$ [Reference Roy, Baldi and Fridman42] can be expressed as:

(34) \begin{equation} \dot{k}_i=|{r_i}|-\mu _i k_i \end{equation}

with $\mu _i\gt 0$ ( $i=1\cdots n$ ) and the initial values of adaptive parameters $k_i(0)\geq 0$ .

Finally, the overall output of this adaptive sliding mode controller can be constructed as:

(35) \begin{equation} \overline{\mathbf{\tau }}=\overline{\mathbf{\tau }_0}+\overline{\mathbf{\tau }_1} \end{equation}

4.3. Stability analysis

In this section, the stability analysis and achievable performance of the control scheme in Fig 2 are presented.

Property 6. For any given vector $\mathbf{x}=[x_1\cdot \cdot \cdot,x_m]^T\in \mathbb{R}^m$ and $\mathbf{y}=[y_1\cdot \cdot \cdot,y_m]^T\in \mathbb{R}^m$ , the following inequality holds:

(36) \begin{equation} \mathbf{x}^Ty\leq \|\mathbf{x}\|\|\mathbf{y}\| \end{equation}

Property 7. For any given vector $\mathbf{x}=[x_1\cdot \cdot \cdot,x_m]^T\in \mathbb{R}^m$ , the following inequality holds:

(37) \begin{equation} \sum _{i=1}^m|x_i|\geq \sqrt{\sum _{i=1}^m \mathbf{x}^2_i}=\|\mathbf{x}\|\ \end{equation}

Property 8. For any given vector $\mathbf{x}\in \mathbb{R}^m$ and a symmetric positive definite matrix $\mathbf{M}\in \mathbb{R}^{m\times m}$ , the following inequality holds:

(38) \begin{equation} \lambda _{min}(\mathbf{R})\mathbf{x}^T \mathbf{x}\leq \mathbf{x}^T\mathbf{R}\mathbf{x}\leq \lambda _{max}(\mathbf{R})\mathbf{x}^T \mathbf{x} \end{equation}

where $\lambda _{min}(\mathbf{R})$ and $\lambda _{max}(\mathbf{R})$ are the minimal and maximal eigenvalues of matrix $\mathbf{R}$ .

Lemma 4.1. Define a positive Lyapunov function of state $\mathbf r$ as $\mathbf{V}(\mathbf{r})$ satisfying the following inequality:

(39) \begin{equation} \dot{\mathbf{V}}(\mathbf{r})+\mathbf{\Theta }\mathbf{V}(\mathbf{r})\leq \zeta \end{equation}

where $\mathbf{\Theta }\gt 0$ and $\zeta \gt 0$ . For any initial condition of the Lyapunov function, $\mathbf{V}(r)$ can converge to a finite region as:

(40) \begin{equation} \mathbf{V}(\mathbf r)\leq \frac{\zeta }{\mathbf{\Theta }(1-\epsilon )} \end{equation}

and the convergence time of $\mathbf{V}(\mathbf r)$ is limited by the following inequality as:

(41) \begin{equation} t\leq \frac{1}{\mathbf{\Theta }\epsilon }\ln \frac{\mathbf{\Theta }(1-\epsilon )\mathbf{V}(0)}{\zeta } \end{equation}

where $\epsilon \in (0,1)$ and $\mathbf{V}(0)$ is the initial statement of Lyapunov function. The proof can be found from [Reference Shao, Zheng, Wang, Wang and Liang31].

Lemma 4.2. Given a differential function as:

(42) \begin{equation} \mathbf{h}(e,t)=(\frac{d}{dt}+\nu )^{w-1} e \end{equation}

where $\nu \gt 0$ and $e=[e,\dot{e}\cdots,e^{m-1}]^T\in \mathbb{R}^m$ . When $\mathbf{h}(e,t)$ is constrained as $|\mathbf{h}(e,t)|\leq \ \boldsymbol{\Theta }$ , the following inequality holds:

(43) \begin{equation} |e^{(i)}|\leq (2\nu )^{i}\frac{\mathbf{\Theta }}{\nu ^{m-1}} \end{equation}

where $i=0,1 \cdots m-1$ . The proof can be found from [Reference Slotine and Li44].

Then, we design the Lyapunov function as follows:

(44) \begin{equation} \mathbf{V}=\frac{1}{2}\mathbf r^T\mathbf r+\frac{1}{2}(k-\bar{k})^T(k-\bar{k}) \end{equation}

where $k=[k_1\cdot \cdot \cdot k_n]^T$ . Differentiating (44) and combining it with (30) to (35), the derivative of $\mathbf{V}$ is given by:

(45) \begin{equation} \begin{aligned} \dot{\mathbf{V}}&=\mathbf r^T\dot{\mathbf r}+(k-\bar{k})^T\dot{k}\\[3pt] &=\mathbf r^T(\overline{\mathbf{M}}^{-1}(\overline{\mathbf{\tau }}-\overline{\mathbf{C}}\dot{\mathbf s}-\overline{\mathbf{g}}-\overline{\mathbf{J}^{T}\mathbf{F}_e}- \overline{\mathbf{\tau }_d})-\ddot{s}_r+\hat{\mathbf{H}}_1\dot{\xi^{\prime}_1}+\hat{\mathbf{H}}_2\xi^{\prime}_2+\Lambda \dot{e}_c)+(k-\bar{k})^T\dot{k}\\[3pt] &=\mathbf r^T(\overline{\mathbf{M}}^{-1}(\overline{\mathbf{\tau }_1}-\overline{\mathbf{\tau }_d}))+(k-\bar{k})^T\dot{k}\\[3pt] & =\mathbf r^T({-}\mathbf{K} sgn(\mathbf r)-\mathbf{\Phi } \mathbf r-\overline{\mathbf{M}}^{-1}\overline{\mathbf{\tau }_d})+(k-\bar{k})^T((\mathbf r)_{||}-\mathbf{\mu } k). \end{aligned} \end{equation}

where $(\mathbf r)_{||}=[|r_1|\cdot \cdot \cdot |r_m|]^T$ and $\mathbf{\mu }=diag\{\mu _1 \cdots \mu _m\}$ from (34). In addition, considering Properties 4 to 7, (45) can be rewritten as:

(46) \begin{equation} \begin{aligned} \dot{\mathbf{V}}\leq &-\sum ^m_{i=1} k_i|{\mathbf{r}}_i|-{\mathbf{r}}^T\mathbf{\Phi }{\mathbf{r}}+\sum ^m_{i=1}m_{4}{\mathbf{\tau }_{sm,i}}|{\mathbf{r}}_i|+(k-\bar{k})^T({\mathbf{r}}_{||}-\mathbf{\mu } k)\\[3pt] \leq &-\sum ^m_{i=1} k_i|{\mathbf{r}}_i|-{\mathbf{r}}^T\mathbf{\Phi }{\mathbf{r}}+(k-\bar{k})^T({\mathbf{r}}_{||}-\mathbf{\mu } k)\\[3pt] &= -{\mathbf{r}}^T\mathbf{\Phi }{\mathbf{r}}-(k-\bar{k})^T\mathbf{\mu } k. \end{aligned} \end{equation}

Then, by adopting Property 8, (46) becomes:

(47) \begin{equation} \begin{aligned} \dot{\mathbf{V}}\leq &-{\mathbf{r}}^T\mathbf{\Phi }{\mathbf{r}}-\frac{1}{2}(k-\bar{k})^T\mu (k-\bar{k})+\frac{1}{2}\bar{k}^T\mathbf{\mu } \bar{k}\\ \leq &-\frac{\lambda _{min}(\mathbf{\Phi )}}{\lambda _{max}(\mathbf{\overline{M}})}{\mathbf{r}}^T\mathbf{\overline{M}}{\mathbf{r}}-\frac{1}{2}\lambda _{min}(\mathbf{\mu })(k-\bar{k})^T(k-\bar{k})+\frac{1}{2}\bar{k}^T\mathbf{\mu } \bar{k}\\ \leq &-min\{\frac{2\lambda _{min}(\mathbf{\Phi })}{\lambda _{max}(\mathbf{\overline{M}})},\lambda _{min}(\mathbf{\mu })\}\mathbf{V}+\frac{1}{2}\bar{k}^T\mathbf{\mu } \bar{k}\\ &= -\mathbf{\Theta }\mathbf{V}+{\zeta }. \end{aligned} \end{equation}

where $\Theta =min\{\frac{2\lambda _{min}(\mathbf{\Phi })}{\lambda _{max}(\mathbf{\overline{M}})},\lambda _{min}(\mathbf{\mu })\}$ and ${\zeta }=\frac{1}{2}\bar{k}^T\mathbf{\mu } \bar{k}$ .

With Assumption 1, the boundary of the sliding variable $\mathbf{r}$ can be obtained:

(48) \begin{equation} \frac{1}{2}\lambda _{min}(\mathbf{\overline{M}})\|\mathbf{{r}}\|^2\leq \frac{1}{2}\mathbf{{r}}^T\mathbf{\overline{M}}\mathbf{{ r}}\leq \frac{\boldsymbol{\zeta }}{\mathbf{\Theta }(1-\epsilon )} \end{equation}

where $\epsilon \in (0,1)$ .

Simplify (48) and take Lemma 4.1 into consideration, the boundary of $\mathbf{{r}}$ can be described as:

(49) \begin{equation} |\mathbf{{r}}_i|\leq \|\mathbf{{r}}\|\leq \sqrt{\frac{2\zeta }{\lambda _{min}(\overline{\mathbf{M}})\mathbf{\Theta }(1-\epsilon )}} \end{equation}

then the convergence time is bounded as follows:

(50) \begin{equation} t\leq \frac{1}{\boldsymbol{\Theta }\epsilon }\ln \frac{\boldsymbol{\Theta }(1-\epsilon )\mathbf{V}(0)}{\zeta } \end{equation}

From (30) and according to Lemma 4.2 for $w=2$ and $i=1$ , the error $e_{c}$ in (29) is bounded as:

(51) \begin{equation} \left |e_c\right |\leq \frac{\hat{r}_i}{\nu _i} \end{equation}

where $\nu _i$ is a positive constant and $\hat{r}$ is the upper limit of $\left | r_{i}\right |$ .

Therefore, with the proposed controller (35), the linear sliding function $\mathbf{{r}}$ and $e_c$ asymptotically converge to a limited region in a finite time.

5.1. Collaborative robot and interaction environment model

The simulations are performed in MATLAB with the robot system toolbox. The physical CAD model of the 6-DOF desktop manipulator STEP SD7/700 is applied, and all the robot transformations are perfectly known using the robotic system toolbox. The initial joint angles of STEP SD7/700 are set to be $\mathbf{q}(0) = [0,0,0,0,-1.4708,0.5236]^T$ rad.

The structure of this collaborative robot is shown in Fig. 3. The origin of the global coordinates is the centre of the robot base. The centre of the interactive plane is $P_i(0.3,0,0.4)$ m. The 4-point image features are chosen as the image feature in this simulation, given as $A_1(0.55,0.05,0.3)$ m, $A_2(0.55,-0.05,0.3)$ m, $A_3(0.45,-0.05,0.3)$ m, and $A_4(0.45,0.05,0.3)$ m, in the global frame, respectively.

In the simulation, We employ the eye-in-hand configuration and fix the camera to the end-effector of the manipulator (see Fig. 3). The camera’s resolution is $1024 \times 1024$ pixels, and focal lengths along the x and y axis are 10 mm. The ratios between the size of a pixel and unit length are 100 pixels/mm along the two axes of the pixel plane. The grinding tip is used to perform the task on the interactive plane, and the force sensor is used to get the feedback of the interaction force.

Considering the real polishing environment, the model of the interaction force between the robot and the interactive plane in this simulation is an MSD model given as 16 and we set $\mathbf{K}_e = 10000$ N/m, $\mathbf{D}_e = 0.57$ Ns/m, and $\mathbf{B}_e = 0.001$ Ns $^2$ /m.

5.2. Definition of the position trajectory and noise from feedback

To perform the polishing task, we define a trajectory to reach the workspace and avoid the singularity location. It consists of two phases: the approaching phase and the interaction phase.

For the approaching phase, the robot’s initial position is $(0.398,0,0.6339)$ m, and the desired position is $(0.3,0.05,0.4005)$ m. We defined the time-varying position trajectory in the Cartesian space as:

(52) \begin{equation} \mathbf{p}_d = \begin{bmatrix} 0.3\\ 0.05\\ 0.4005+\frac{(2-t)^3}{80} \end{bmatrix} \end{equation}

where $t \in [0,2]$ s. After the approaching phase, the robot manipulator remains stable at $(0.3,0.05,0.4005)$ m for $1$ sec. Then, during the interaction phase, the desired force and position trajectory are given simultaneously. For the polish force command, we set it to be 10 N normal to the interaction plane all the time, and the predefined time-varying trajectory in the Cartesian space is given as:

(53) \begin{equation} \mathbf{p}_d = \begin{bmatrix} 0.3+0.05sin(\frac{t-3}{2})\\ 0.05cos(\frac{t-3}{2})\\ 0.4\end{bmatrix} \end{equation}

where $t \in [3,20]$ s.

In order to prove the robustness and adaptive ability of the proposed controller, some noise and uncertainties are added to the system. First, we added uncertainty of $5\%$ on the diagonal element of the inertia matrix given as:

(54) \begin{equation} \hat{\mathbf{M}} = \mathbf{M}+0.05\mathbf{M}_d \end{equation}

where $\mathbf{M}_d = diag\{ M_{1,1},M_{2,2},M_{3,3},M_{4,4},M_{5,5},M_{6,6}\}$ and $M_{i,i}, i = 1\cdots,6$ are the diagonal elements of inertia matrix of $\mathbf{M}$ . In addition, by using the Matlab function wgn, white Gaussian noise with a maximum of $0.1$ pixel is added to the image features, a noise of 0.1 N is added to the force feedback, and the power of wgn function is set as $-32$ . Considering the possible ageing of the robot motor, a $1\%$ random noise is given as the unknown disturbance $\mathbf{\tau }_d$ in 1.

5.3. Simulation design

For the force control law (22), we set $k_{j,i}=0.08$ , $k_{p,i}=0.02$ and $i=1,\cdots 6$ . Three sets of controllers are proposed in this simulation:

• Controller 1: The leakage-type adaptive sliding mode controller (LTASMC) we developed in (35). We set $h_{i,1}=13$ , $h_{i,2}=50$ , $k_i(0)=0$ , $\phi _i=10$ , $\mu _i=200$ , $\lambda _i=5$ , ( $i = 1,\cdots 8$ ).

• Controller 2: The adaptive sliding mode controller proposed in ref. [Reference Gao, An, Proctor and Bradley45]. This controller replaces the adaptive control law in (33) with follows:

  (55) \begin{equation} k_i = a_i|r_i| \end{equation}
  where $a_i = 1$ , $i=1,\cdots 8$ , and other elements are the same as Controller 1.

• Controller 3: The parallel vision/force controller (PVFC) [Reference Zhu, Huang, Qiu, Zheng and Gong8]. The input torque is given as:

  (56) \begin{equation} \mathbf{\tau }=\mathbf{\tau }_0+\mathbf{\tau }_1+\mathbf{\tau }_2 \end{equation}
  and
  (57) \begin{equation} \begin{aligned} \mathbf{\tau }_0 &=\mathbf{J}_s^T(\overline{\mathbf{C}}\dot{\mathbf{s}}+\overline{\mathbf{g}}+\overline{\mathbf{J}^{T}\mathbf{F}_e}),\\ \mathbf{\tau }_1 &= \hat{\mathbf{M}}\mathbf{J}_s^+(\ddot{s}_d-K_{dv}\dot{e}_s-K_{pv}*e_s),\\ \mathbf{\tau }_2 &= -\hat{\mathbf{M}}\mathbf{J}^+(K_{pf} e_f+K_{if}\int ^t_0 e_f d\tau ). \end{aligned} \end{equation}
  where we set $K_{dv} = 20$ , $K_{pv} = 50$ , $K_{pf} = 0.01$ , and $K_{if} = 0.03$ .

Three cases are designed to illustrate the robustness and adaptability of these three controllers:

• Case 1: The robot system works without any uncertainty, disturbance, or input saturation.

• Case 2: The robot system works with input saturation given as $\mathbf{\tau }_{max}^i = 8$ Nm and $\mathbf{\tau }_{min}^i = -8$ Nm where $i=1,\cdots 6$ . In this case, we only focus on the approaching phase. Two controllers (the proposed controller with and without the auxiliary dynamic model (ADM)) are used to demonstrate the performance of the proposed controller to handle the actuator saturation. Uncertainties and disturbances are not added to the robot system.

• Case 3: Uncertainties of the inertia parameters and disturbance of the image features, force feedback, and actuator torque are added to the robot model and the input saturation is given as $\mathbf{\tau }_{max}^i = 12$ Nm and $\mathbf{\tau }_{min}^i = -12$ Nm where $i=1,\cdots 6$ .

5.4. Analysis of the results

The results of three simulations are given as follows:

Case 1: The desired and actual image trajectory, position errors, and input torque are presented in Fig. 4. The force command and actual force feedback are plotted in Fig. 5(a). In Fig. 4(a), 4(d), and Fig. 4(g), the red dashed line represents the desired image features. The four solid-coloured lines represent the actual image features in the pixel plane, and $P_1, P_2, P_3, P_4$ are the initial image features of the robot. From Fig. 4(a), 4(d), and Fig. 4(g), we can find that three controllers can track the predefined image features trajectory in Case 1, but the converging speed of three controllers are different. As is shown in Fig. 4(b), Fig. 4), and F ig.4(h), the position errors of the proposed controller and the conventional adaptive sliding mode controller converge to a small neighbourhood of zero quickly within $0.6$ sec. However, for the parallel vision/force controller, it takes $0.8$ sec to converge to the desired trajectory. When the force command is applied and the desired trajectory is changed, the convergence time of the parallel vision/force controller is $0.5$ sec and that of the proposed controller. The conventional adaptive sliding mode controller is about $1$ sec. The maximal force errors of the proposed controller, conventional adaptive sliding mode controller, and parallel vision/force controller are $0.01$ N, $0.017$ N, and $0.025$ N, respectively, and the position errors of the three controllers are within the order of $10^{-4}$ m. From Fig. 4(c), Fig. 4(f), and 4(i), we see that the chattering of the conventional adaptive sliding mode controller in input torque is serious. However, with the help of the leak-type adaptive law, the chattering of the proposed controller is significantly reduced. From Fig. 4 and Fig. 5(a), it is clear that the control accuracy of the proposed controller and conventional adaptive sliding mode controller is better than that of the parallel vision/force controller and compared to the conventional adaptive sliding mode controller, the chatter of the input torque under the proposed controller is eliminated.

Figure 4. The simulation results of case 1 ((a), (d), and (g) show the desired trajectories and actual trajectories of three controllers in the pixel plane, (b), (e), and (h) show the trajectory tracking errors of three controllers, (c), (f) and (i) show the input torque of three controllers.).

Figure 5. The desired and actual force control performance of three controllers in case 1 and case 3.

Case 2: The simulation results of Case 2 are given in Fig. 6 and Fig. 7. As is shown in Fig. 7, the input torque of the robot under two controllers is bounded due to the input saturation, and the converging speed of the proposed controller of Case 2 is slower than that of Case 1. From Fig. 6, the robot under the proposed controller with the auxiliary dynamic model converges faster than without the auxiliary dynamic model in the X and Y-axes, while the convergence rate of the proposed controller with the auxiliary dynamic model in the Z-axis is similar to that without the auxiliary dynamic model. Numerical results demonstrate that the proposed controller with the auxiliary dynamic model converges faster than without the auxiliary dynamic model and our proposed controller is effective in handling input saturation.

Figure 6. The trajectory tracking errors of LTASMC (with and without auxiliary dynamic model) in case 2.

Figure 7. The input torque results of LTASMC (with and without auxiliary dynamic model) in case 2.

Figure 8. The trajectory tracking errors of three controllers ((a), (b), and (c)) and input torque of three controllers ((d), (e), and (f)) in case 3.

Case 3: From 8 and Fig. 5(b), it is clear that with the help of the vision-admittance-based reference trajectory generator, the proposed controller, and the conventional adaptive sliding mode controller can simultaneously track the predefined image feature trajectory and force command with system uncertainty and input saturation. From Fig. 8, the convergence times of the proposed controller, conventional adaptive sliding mode controller, and parallel vision/force controller are 0.7 sec, 0.8 sec, and 0.9 sec respectively. Fig. 8(d), Fig. 8(e), and Fig. 8(f) show that the proposed controller has the smallest chatter, while conventional adaptive sliding mode controller and parallel vision/force controller have similar chatter. As is shown in Fig. 5b, the force converging speed of Case 3 is similar to that of Case 2. However, when the system converges, the maximal force error of the proposed controller is the smallest given as $0.1$ N, and that of conventional adaptive sliding mode controller and parallel vision/force controller are $0.2$ N and $0.5$ N, respectively. In Case 3, the robot system under the proposed controller is more robust to systematic uncertainties and feedback noise compared with the other two controllers. Furthermore, the proposed controller performs better than the others in handling input chatter and control accuracy.

Generally, from the simulation results, we conclude this section by pointing out that in Case 1, by tracking the image feature trajectory obtained from the proposed vision-admittance-based model, we can achieve accurate predefined trajectory tracking and compliant contact force control at the same time. In Case 2, the auxiliary dynamic model we added to the LT adaptive sliding mode controller effectively solves the input saturation. In Case 3, the robustness and adaptability of the proposed controller are validated.