3.1. Control design

In this section, an UDE-based tracking controller is designed for uncertain robot manipulators with input saturation and output constraints. This section is divided into three steps. In step 1, we combine NSDF and backstepping method to derive the virtual control law $\alpha _1$ and auxiliary variable $\eta _1$ . In step 2, the control law $\tau$ and auxiliary variable $\eta _2$ are derived by combining UDE and backstepping method, and the stability of robot manipulator is proved by Lyapunov method. In step 3, we construct an auxiliary system using $\eta _1$ and $\eta _2$ , and the stability of the auxiliary system is proved by Lyapunov method.

Step1: To solve the output constraints, two NSDF vectors $\zeta =[\zeta _1,\ldots,\zeta _m]^T$ and $\zeta _d=[\zeta _{d1},\ldots,\zeta _{dm}]^T$ are defined as follows:

(7) \begin{equation} \zeta _i=K\frac{y_i(t)}{(F_{i1}(t)+y_i(t))(F_{i2}(t)-y_i(t))},i=1,\ldots,m \end{equation}

(8) \begin{equation} \zeta _{di}=K\frac{y_{di}(t)}{(F_{i1}(t)+y_{di}(t))(F_{i2}(t)-y_{di}(t))},i=1,\ldots,m \end{equation}

where $K$ is a positive constant.

Define $e_1=\zeta -\zeta _d$ and $v_1=e_1-\eta _1$ and consider the Lyapunov function $V_1=\frac{1}{2}v_1^Tv_1$ . The time derivative of $V_1$ is:

(9) \begin{equation} \dot{V_1}=v_1^T(\dot{\zeta }-\dot{\zeta _d}-\dot{\eta _1})\\ \end{equation}

Taking the time derivative of each term in $\zeta$ and $\zeta _d$ based on (7) and (8), we get

(10) \begin{equation} \dot{\zeta }_i=\mu _{1i}\dot{y}+\mu _{2i},i=1,\ldots,m \end{equation}

(11) \begin{equation} \dot{\zeta }_{di}=\mu _{d1i}\dot{y}_d+\mu _{d2i},i=1,\ldots,m \end{equation}

where

(12) \begin{equation} \mu _{1i}=K\frac{F_{i1}(t)F_{i2}(t)+y_i(t)^2}{(F_{i1}(t)+y_i(t))^2(F_{i2}(t)-y_i(t))^2} \end{equation}

(13) \begin{equation} \mu _{2i}=K\frac{(\dot{F}_{i1}(t)F_{i2}(t)+F_{i1}(t)\dot{F}_{i2}(t)+(\dot{F}_{i2}(t)-\dot{F}_{i1}(t))y_i(t))y_i(t)}{(F_{i1}(t)+y_i(t))^2(F_{i2}(t)-y_i(t))^2} \end{equation}

(14) \begin{equation} \mu _{d1i}=K\frac{F_{i1}(t)F_{i2}(t)+y_{di}(t)^2}{(F_{i1}(t)+y_{di}(t))^2(F_{i2}(t)-y_{di}(t))^2} \end{equation}

(15) \begin{equation} \mu _{d2i}=K\frac{(\dot{F}_{i1}(t)F_{i2}(t)+F_{i1}(t)\dot{F}_{i2}(t)+(\dot{F}_{i2}(t)-\dot{F}_{i1}(t))y_{di}(t))y_{di}(t)}{(F_{i1}(t)+y_{di}(t))^2(F_{i2}(t)-y_{di}(t))^2} \end{equation}

Then we can obtain the time derivative of $\zeta$ and $\zeta _d$ as follows:

(16) \begin{equation} \dot{\zeta }=\mu _1\dot{y}(t)+\mu _2 \end{equation}

(17) \begin{equation} \dot{\zeta _d}=\mu _{d1}\dot{y}_d(t)+\mu _{d2} \end{equation}

where $\mu _1=diag[\mu _{11},\ldots,\mu _{1m}] \in R^{m \times m}$ , $\mu _{2}=[\mu _{21},\ldots,\mu _{2m}]^T\in R^m$ , $\mu _{d1}=diag[\mu _{d11},\ldots,\mu _{d1m}]\in R^{m \times m}$ , $\mu _{d2}=[\mu _{d21},\ldots,\mu _{d2m}]^T\in R^m$ .

Substituting (6), (16) and (17) into (9), we have

(18) \begin{align} \dot{V}_1 \nonumber & =v_1^T(\mu _1\dot{y}+\mu _2-\mu _{d1}\dot{y}_d-\mu _{d2}-\dot{\eta }_1)\\[5pt] \nonumber & =v_1^T(\mu _1\dot{x}_1+\mu _2-\mu _{d1}\dot{y}_d-\mu _{d2}-\dot{\eta }_1)\\[5pt] & =v_1^T(\mu _1x_2+\mu _2-\mu _{d1}\dot{y}_d-\mu _{d2}-\dot{\eta }_1) \end{align}

Defining $e_2=x_2-\alpha _1$ and $v_2=e_2-\eta _2$ , (18) can be further transformed as

(19) \begin{equation} \dot{V_1}=v_1^T(\mu _1(v_2+\eta _2+\alpha _1)+\mu _2-\mu _{d1}\dot{y}_d-\mu _{d2}-\dot{\eta }_1) \end{equation}

The virtual control law $\alpha _1$ and auxiliary variable $\eta _1$ are chosen as

(20) \begin{equation} \alpha _1=\mu _1^{-1}(-c_1e_1+\mu _{d1}\dot{y}_d-\mu _2+\mu _{d2}) \end{equation}

(21) \begin{equation} \dot{\eta }_1=-c_1\eta _1+\mu _1\eta _2 \end{equation}

where $c_1$ is a design constant.

Substituting (20) and (21) into (19) results in

(22) \begin{equation} \dot{V_1}=-c_1v_1^Tv_1+v_2^T\mu _1^Tv_1 \end{equation}

Step2: Considering the Lyapunov function $V_2=V_1+\frac{1}{2}v_2^Tv_2$ and taking its time derivative based on (6) and (22), we have

(23) \begin{align} \dot{V_2} \nonumber & =\dot{V_1}+v_2^T(\dot{e}_2-\dot{\eta }_2)\\[5pt] \nonumber & =-c_1v_1^Tv_1+v_2^T\mu _1^Tv_1+v_2^T(\dot{x}_2-\dot{\alpha _1}-\dot{\eta }_2)\\[5pt] & =-c_1v_1^Tv_1+v_2^T(\mu _1^Tv_1+M_x^{-1}(J^{+T}U(\tau )+D_x-C_xx_2-G_x)-\dot{\alpha _1}-\dot{\eta }_2) \end{align}

where the model uncertainties and external disturbance $D_x$ is unknown. According to (6), $D_x$ can be denoted as

(24) \begin{equation} D_x=M_x\dot{x}_2+C_xx_2+G_x-J^{+T}U(\tau ) \end{equation}

which shows that the unkown $D_x$ can be obtained from the known system dynamics and control signal. However, in order not to have the inputs cancel out, (24) can not be subtituted to (23) directly. Assuming that the frequency range of a signal is limited, the signal can be estimated using a filter with appropriate bandwidth information, then the procedure of UDE-based control design given in ref. [Reference Zhong and Rees8] is adopted to handle the uncertainty so that a control law is derived. Assume that $G_f(s)=diag[G_{f1}(s),\ldots,G_{fm}(s)] \in R^{m \times m}$ is a strictly proper stable filter with the unity gain and zero phase shift over the spectrum of $D_x$ and zero gain elsewhere. Then, $D_x$ can be estimated by

(25) \begin{align} \hat{D}_x \nonumber & =\mathcal{L}^{-1}(G_f(s))*D_x\\[5pt] & =\mathcal{L}^{-1}(G_f(s))*(M_x\dot{x}_2+C_xx_2+G_x-J^{+T}U(\tau )) \end{align}

where $\hat{D}_x$ is the estimate of $D_x$ . The estimation error $\tilde{D}_x$ is defined as follows:

(26) \begin{equation} \tilde{D}_x=D_x-\hat{D}_x \end{equation}

According to (25) and (26), the laplace transform of $\tilde{D}_x$ is represented as

(27) \begin{align} \tilde{D}_x(s) \nonumber & =D_x(s)-\hat{D}_x(s)\\[5pt] & =D_x(s)(I-G_f(s)) \end{align}

where $\tilde{D}_x(s)$ , $D_x(s)$ and $\hat{D}_x(s)$ denote the Laplace transform of $\tilde{D}_x$ , $D_x$ and $\hat{D}_x$ , respectively. Taking the inverse Laplace transform of (27), there is

(28) \begin{equation} \tilde{D}_x=D_x*(I-G_f(s)) \end{equation}

Since $D_x$ is bounded and the UDE filter $G_f(s)$ is designed close to unity over the spectrum of $D_x$ , the estimation error $\tilde{D}_x$ is bounded according to (28). Hence, it is reasonably assumed that

(29) \begin{equation} \left \|\tilde{D}_x\right \| \le \delta _1 \end{equation}

where $\delta _1$ is a positive scalar. Then we can further obtain that $\hat{D}_x$ is bounded based on (26).

According to ref. [Reference Zhong and Rees8], $G_f(s)=diag[G_{f1}(s),\ldots,G_{fm}(s)]$ is chosen as a first-order low-pass filter.

(30) \begin{equation} G_f(s)=diag\left[\frac{1}{T_1s+1},\ldots,\frac{1}{T_ms+1}\right] \end{equation}

where $1/{T_i},i=1,\ldots,m$ denote the bandwidth of $1/({T_is+1})$ . Although this will cause some error in the estimation, the steady-state estimation error is always zero because $G_f(0)=I$ . The amplitude gain of the filter is

(31) \begin{equation} \lvert G_{fi}(jw)\rvert =\frac{1}{\sqrt{T_i^2w^2+1}} \in (0,1),i=1,\ldots,m \end{equation}

where $w$ is the frequency of the input signal to the filter.

According to Remark 3, (23) and (25) can be rewritten as

(32) \begin{equation} \dot{V_2}=-c_1v_1^Tv_1+v_2^T(\mu _1^Tv_1+M_x^{-1}(J^{+T}\tau +J^{+T}\Delta u+D_x-C_xx_2-G_x)-\dot{\alpha _1}-\dot{\eta }_2) \end{equation}

(33) \begin{align} \hat{D}_x \nonumber & =\mathcal{L}^{-1}(G_f(s))*(M_x\dot{x}_2+C_xx_2+G_x-J^{+T}\tau -J^{+T}\Delta u)\\[5pt] & =\mathcal{L}^{-1}(G_f(s))*(M_x\dot{x}_2+C_xx_2+G_x-J^{+T}\tau )-\mathcal{L}^{-1}(G_f(s))*(J^{+T}\Delta u) \end{align}

The control law $\tau$ is selected as

(34) \begin{equation} J^{+T}\tau =C_xx_2+G_x+M_x(\dot{\alpha _1}-\mu _1^Te_1-c_2e_2)-\frac{\left(v_2^TM_x^{-1}\right)^T}{2r_1^2}-\mathcal{L}^{-1}(G_f(s))*(M_x\dot{x}_2+C_xx_2+G_x-J^{+T}\tau ) \end{equation}

where $c_2$ and $r_1$ are design constants.

Solving (34), we can further obtain

(35) \begin{align} \tau \nonumber & =J^{T}\left(\mathcal{L}^{-1}((I-G_f(s))^{-1})*\left(C_xx_2+G_x+M_x(\dot{\alpha _1}-\mu _1^Te_1-c_2e_2)-\frac{\left(v_2^TM_x^{-1}\right)^T}{2r_1^2}\right)\right)\\[5pt] & \quad -J^{T}(\mathcal{L}^{-1}((I-G_f(s))^{-1}G_f(s))*(M_x\dot{x}_2+C_xx_2+G_x)) \end{align}

The auxiliary variable $\eta _2$ is selected as

(36) \begin{equation} \dot{\eta _2}=-c_2\eta _2-\mu _1^T\eta _1+M_x^{-1}J^{+T}\Delta u+M_x^{-1}\mathcal{L}^{-1}(G_f(s))*(-J^{+T}\Delta u) \end{equation}

Combining (32), (33), (35) and (36), we have

(37) \begin{equation} \dot{V_2}=-c_1v_1^Tv_1-c_2v_2^Tv_2+v_2^TM_x^{-1}\tilde{D}_x-\frac{v_2^TM_x^{-1}\left(v_2^TM_x^{-1}\right)^T}{2r_1^2} \end{equation}

According to Young’s inequality, there is

(38) \begin{align} v_2^TM_x^{-1}\tilde{D}_x \nonumber & \le \frac{v_2^TM_x^{-1}\left(v_2^TM_x^{-1}\right)^T}{2r_1^2} + \frac{r_1^2\tilde{D}_x^T\tilde{D}_x}{2}\\[5pt] & \le \frac{v_2^TM_x^{-1}\left(v_2^TM_x^{-1}\right)^T}{2r_1^2} + \frac{r_1^2\delta _1^2}{2} \end{align}

Combined with (38), (37) becomes

(39) \begin{equation} \dot{V}_2 \le -c_1v_1^Tv_1-c_2v_2^Tv_2+\frac{r_1^2\delta _1^2}{2} \end{equation}

Defining $\rho _1=2min\left \{c_1,c_2\right \}$ and $\lambda _1=\frac{r_1^2\delta _1^2}{2}$ , we can further get

(40) \begin{equation} \dot{V}_2 \le -\rho _1 V_2+\lambda _1 \end{equation}

Solving (40), we can obtain that

(41) \begin{equation} 0\le V_2(t)\le \frac{\lambda _1}{\rho _1}+\left(V_2(0)-\frac{\lambda _1}{\rho _1}\right)e^{-\rho _1t} \end{equation}

where $V_2(0)$ is the initial value of $V_2(t)$ . Based on (41), we can further get

(42) \begin{equation} lim_{t\rightarrow \infty }V_2(t) \le \frac{\lambda _1}{\rho _1} \end{equation}

Step3: To solve the input saturation, an auxiliary system has been set up:

(43) \begin{equation} \begin{array}{l} \left \{ \begin{array}{l} \dot{\eta }_1=-c_1\eta _1+\mu _1\eta _2\\[5pt] \dot{\eta }_2=-c_2\eta _2-\mu _1^T\eta _1+M_x^{-1}J^{+T}\Delta u+M_x^{-1}\mathcal{L}^{-1}(G_f(s))*(-J^{+T}\Delta u) \end{array} \right. \end{array}\\[5pt] \end{equation}

In order to prove the stability of system (43), a Lyapunov function $V_a=\frac{1}{2}(\eta _1^T\eta _1+\eta _2^T\eta _2)$ is considered. Taking the time derivative of $V_a$ , we have

(44) \begin{align} \dot{V}_a \nonumber & =\eta _1^T(-c_1\eta _1+\mu _1\eta _2)+\eta _2^T(-c_2\eta _2-\mu _1^T\eta _1+M_x^{-1}+M_x^{-1}\mathcal{L}^{-1}(G_f(s))*(-J^{+T}\Delta u))\\[5pt] & =-c_1\eta _1^T\eta _1-c_2\eta _2^T\eta _2+\eta _2^TM_x^{-1}J^{+T}\Delta u+\eta _2^TM_x^{-1}\mathcal{L}^{-1}(G_f(s))*(-J^{+T}\Delta u) \end{align}

Using Young’s inequality, we have

(45) \begin{equation} \eta _2^TM_x^{-1}J^{+T}\Delta u \le \frac{\eta _2^T\eta _2}{2}+\frac{(M_x^{-1}J^{+T}\Delta u)^T(M_x^{-1}J^{+T}\Delta u)}{2} \end{equation}

(46) \begin{equation} \eta _2^TM_x^{-1}\mathcal{L}^{-1}(G_f(s))*(-J^{+T}\Delta u) \le \frac{\eta _2^T\eta _2}{2}+\frac{(M_x^{-1}\mathcal{L}^{-1}(G_f(s))*(-J^{+T}\Delta u))^T(M_x^{-1}\mathcal{L}^{-1}(G_f(s))*(-J^{+T}\Delta u))}{2} \end{equation}

Combined with (45) and (46), (44) becomes

(47) \begin{align} \dot{V}_a \nonumber & \le -c_1\eta _1^T\eta _1-(c_2-1)\eta _2^T\eta _2+\frac{(M_x^{-1}J^{+T}\Delta u)^T(M_x^{-1}J^{+T}\Delta u)}{2}\\[5pt] & +\frac{(M_x^{-1}\mathcal{L}^{-1}(G_f(s))*(-J^{+T}\Delta u))^T(M_x^{-1}\mathcal{L}^{-1}(G_f(s))*(-J^{+T}\Delta u))}{2} \end{align}

According to cauchy inequality, for any matrix $P \in R^{ m\times n}$ and vector $Q \in R^n$ , the following inequality holds:

(48) \begin{equation} (PQ)^T(PQ) \le \left \|P\right \|_F^2\left \|Q\right \|^2 \end{equation}

According to (48), we have

(49) \begin{align} (M_x^{-1}J^{+T}\Delta u)^T(M_x^{-1}J^{+T}\Delta u) \nonumber & \le \left \|M_x^{-1}\right \|_F^2\left \|J^{+T}\Delta u\right \|^2\\[5pt] & \le \delta _2^2\delta _3^2 \end{align}

(50) \begin{align} (M_x^{-1}\mathcal{L}^{-1}(G_f(s))*(-J^{+T}\Delta u))^T(M_x^{-1}\mathcal{L}^{-1}(G_f(s))*(-J^{+T}\Delta u)) \nonumber & \le \left \|M_x^{-1}\right \|_F^2\left \|\mathcal{L}^{-1}(G_f(s))*(-J^{+T}\Delta u)\right \|^2\\[5pt] & \le \delta _3^2\delta _4^2 \end{align}

where $\delta _2$ is the upper bound of $\left \|J^{+T}\Delta u\right \|$ , $\delta _3$ is the upper bound of $\left \|M_x^{-1}\right \|_F$ and $\delta _4$ is the upper bound of $\left \|\mathcal{L}^{-1}(G_f(s))*(-J^{+T}\Delta u)\right \|$ . $\mathcal{L}^{-1}(G_f(s))*(-J^{+T}\Delta u)$ is bounded because $J^{+T}\Delta u$ is bounded and the amplitude gain of filter (31) is bounded.

Substituting (49) and (50) into (47) yileds

(51) \begin{align} \dot{V_a} \nonumber & \le -c_1\eta _1^T\eta _1-(c_2-1)\eta _2^T\eta _2\\[5pt] \nonumber & +\frac{\left \|M_x^{-1}\right \|_F^2\left \|J^{+T}\Delta u\right \|^2}{2}\\[5pt] \nonumber & +\frac{\left \|M_x^{-1}\right \|_F^2\left \|\mathcal{L}^{-1}(G_f(s))*(J^{+T}\Delta u)\right \|^2}{2}\\[5pt] & \le -c_1\eta _1^T\eta _1-(c_2-1)\eta _2^T\eta _2+\frac{(\delta _2^2+\delta _4^2)\delta _3^2}{2} \end{align}

Defining $\rho _2=2min\left \{c_1,(c_2-1)\right \}$ and $\lambda _2=\dfrac{(\delta _2^2+\delta _4^2)\delta _3^2}{2}$ , we can further get

(52) \begin{equation} \dot{V_a} \le -\rho _2V_a+\lambda _2 \end{equation}

Solving the inequality (52), we have

(53) \begin{equation} 0 \le V_a(t)\le \frac{\lambda _2}{\rho _2}+\left(V_a(0)-\frac{\lambda _2}{\rho _2}\right)e^{-\rho _2t} \end{equation}

where $V_a(0)$ is the initial value of $V_a(t)$ . Based on (53), we can further get

(54) \begin{equation} lim_{t\rightarrow \infty }V_a(t) \le \frac{\lambda _2}{\rho _2} \end{equation}

3.2. Stability analysis

Proof of Theorem 1: The proof is divided into two parts. One part proves that all signals in the closed-loop system are bounded and the output constraints are not violated. The other part proves that the tracking error converges to a small neighborhood around zero.

Part 1: The formula (41) and (53) can be scaled to

(55) \begin{equation} 0\le V_2(t)\le \frac{\lambda _1}{\rho _1}+V_2(0)e^{-\rho _1t} \end{equation}

(56) \begin{equation} 0 \le V_a(t)\le \frac{\lambda _2}{\rho _2}+V_a(0)e^{-\rho _2t} \end{equation}

which mean that for $t \ge max\left \{0,\frac{1}{\rho _1}\ln \left(\frac{V_2(0)\rho _1}{\lambda _1}\right)\right \}$ , the signals $v_1$ and $v_2$ will remain within the compact sets $\Omega _{v_1}$ and $\Omega _{v_2}$ , respectively, defined by:

(57) \begin{equation} \Omega _{v_1}\;:\!=\;\left \{v_1 \in R^m | \left \|v_1\right \| \le 2 \sqrt{\frac{\lambda _1}{\rho _1}} \right \} \end{equation}

(58) \begin{equation} \Omega _{v_2}\;:\!=\;\left \{v_2 \in R^m | \left \|v_2\right \| \le 2 \sqrt{\frac{\lambda _1}{\rho _1}} \right \} \end{equation}

similarly, for $t \ge max\left \{0,\frac{1}{\rho _2}\ln \left(\frac{V_a(0)\rho _2}{\lambda _2}\right)\right \}$ , the signals $\eta _1$ and $\eta _2$ will remain within the compact sets $\Omega _{\eta _1}$ and $\Omega _{\eta _2}$ , respectively, defined by:

(59) \begin{equation} \Omega _{\eta _1}\;:\!=\;\left \{\eta _1 \in R^m | \left \|\eta _1\right \| \le 2 \sqrt{\frac{\lambda _2}{\rho _2}} \right \} \end{equation}

(60) \begin{equation} \Omega _{\eta _2}\;:\!=\;\left \{\eta _2 \in R^m | \left \|\eta _2\right \| \le 2 \sqrt{\frac{\lambda _2}{\rho _2}} \right \} \end{equation}

Because $v_1$ , $v_2$ , $\eta _1$ and $\eta _2$ are bounded, $e_1=v_1+\eta _1$ and $e_2=v_2+\eta _2$ are bounded. According to Assumption 3 and (8), $\zeta _d$ is bounded. Then we can further derive that $\zeta = e_1+\zeta _d$ is bounded. Based on (7) and the boundness of $\zeta$ , the system output $y$ is bounded and the output constraints are not violated. Due to Assumption 2 and the boundness of $y_i$ and $y_{di}$ , it is obvious that $\mu _1$ , $\mu _2$ , $\mu _{d1}$ and $\mu _{d2}$ are bounded from (12) to (15). Furthermore, the boundness of $\alpha _1$ is guaranteed from (20). Then we can derive that $x_2=e_2+\alpha _1$ is bounded. Finally, $\tau$ is bounded based on (35). All signals in the closed-loop system are bounded.

Part 2: Next we discuss the tracking error $e_3=y-y_d=[e_{31},\ldots,e_{3m}]^T$ . According to (7), (8) and $e_1=\zeta -\zeta _d$ , the i th element of $e_1$ can be described as

(61) \begin{equation} e_{1i}=K\frac{y_i(t)}{(F_{i1}(t)+y_i(t))(F_{i2}(t)-y_i(t))}-K\frac{y_{di}(t)}{(F_{i1}(t)+y_{di}(t))(F_{i2}(t)-y_{di}(t))},i=1,..,m \end{equation}

Let $\epsilon _1=(F_{i1}(t)+y_i(t))(F_{i2}(t)-y_i(t))(F_{i1}(t)+y_{di}(t))(F_{i2}(t)-y_{di}(t))$ and $\epsilon _2=F_{i1}(t)F_{i2}(t)+y_i(t)y_{di}(t)$ , (61) can be expressed as

(62) \begin{equation} \epsilon _1e_{1i}=K\epsilon _2e_{3i} \end{equation}

where $\epsilon _1$ and $\epsilon _2$ are positive and bounded based on Assumptions 2 and 3. Defining $\epsilon =\epsilon _1/(K\epsilon _2)$ , (62) can be rewritten as

(63) \begin{equation} \epsilon e_{1i}=e_{3i} \end{equation}

Taking the absolute of (63), we can further get

(64) \begin{align} \lvert e_{3i}\rvert \nonumber & \le \bar{\epsilon } \lvert e_{1i} \rvert \\[5pt] \nonumber & \le \bar{\epsilon } \left \|e_1\right \|\\[5pt] & \le \bar{\epsilon }(\left \|v_1\right \|+\left \|\eta _1\right \|) \end{align}

where $\bar{\epsilon }$ is the upper bound of $\epsilon$ .

According to (57), (59) and (64), for $t \ge max\left \{0,\dfrac{1}{\rho _1}\ln \left(\dfrac{V_2(0)\rho _1}{\lambda _1}\right),\dfrac{1}{\rho _2}\ln \left(\dfrac{V_a(0)\rho _2}{\lambda _2}\right)\right \}$ , the tracking error $e_{3i}$ will remain within the compact set $\Omega _{e_{3i}}$ :

(65) \begin{equation} \Omega _{e_{3i}}\;:\!=\;\left \{e_{3i} \in R | |e_{3i}| \le 4\bar{\epsilon } \sqrt{max\left \{\frac{\lambda _1}{\rho _1},\frac{\lambda _2}{\rho _2}\right \} } \right \} \end{equation}

This completes the proof of Theorem 1.

3.3. Selection of control parameters

In order to obtain good tracking performance, the selection criteria of corresponding control parameters are given as follows:

(1) The backstepping design parameters $c_1$ , $c_2$ can affect the convergent rate of tracking error. Large values of $c_1$ and $c_2$ bring faster error convergence, but the control amplitude will increase if too large $c_1$ and $c_2$ are chosen. The design parameter $r_1$ should be a small positive constant to guarantee good tracking performance. In our simulation, $c_1$ , $c_2$ and $r_1$ are chosen as 40, 100 and 1, respectively.

(2) The design parameter $K$ is introduced in NSDF. Increasing $K$ value can reduce the steady-state error, but will bring large control amplitude. In our simulation, $K$ is chosen as 0.01.

(3) The design parameters $T_i, i=1,..,m$ in (30) should be selected small to ensure small estimation error. In our simulation, $T_1$ and $T_2$ are chosen as 0.014.

4.1. Simulation platform

Let the position of manipulator’s joint $q=[q_1,q_2]^T$ . Let $m_i$ and $l_i$ be the mass and length of link $i$ , $l_{ci}$ be the distance from joint $i-1$ to the center of mass of inertia of link $i$ , and $I_i$ be the moment of inertia of link $i$ about axis coming out of the page passing through the center of mass of link $i, i= 1,2$ . The dynamic model of the two-link robot manipulator [Reference He, Chen and Yin14] can be described as formula (1), where

\begin{equation*} M_0(q) =\left [\begin{array}{c@{\quad}c}{M_{11}}&{M_{12}}\\[5pt]{M_{21}}&{M_{22}} \end{array}\right ] \end{equation*}

\begin{equation*} C_0(q,\dot{q}) =\left [\begin{array}{c@{\quad}c}{C_{11}}&{C_{12}}\\[5pt]{C_{21}}&{C_{22}} \end{array}\right ] \end{equation*}

\begin{equation*} G_0(q) =\left [\begin{array}{c}{G_{11}}\\[5pt]{G_{21}}\end{array}\right] \end{equation*}

and

\begin{equation*} M_{11} =m_1l_{c1}^2+m_2(l_1^2+l_{c2}^2+2l_1l_{c2}cosq_2)+I_1+I_2 \end{equation*}

\begin{equation*} M_{12} =m_2(l_{c2}^2+l_1l_{c2}cosq_2)+I_2 \end{equation*}

\begin{equation*} M_{21} =m_2(l_{c2}^2+l_1l_{c2}cosq_2)+I_2 \end{equation*}

\begin{equation*} M_{22} =m_2l_{c2}^2+I_2 \end{equation*}

\begin{equation*} C_{11} =-m_2l_1l_{c2}\dot{q}_2sinq_2 \end{equation*}

\begin{equation*} C_{12} =-m_2l_1l_{c2}(\dot{q}_1+\dot{q}_2)sinq_2 \end{equation*}

\begin{equation*} C_{21} =m_2l_1l_{c2}\dot{q}_1sinq_2 \end{equation*}

\begin{equation*} C_{22} =0 \end{equation*}

\begin{equation*} G_{11} =(m_1l_{c2}+m_2l_1)gcosq_1+m_2l_{c2}gcos(q_1+q_2) \end{equation*}

\begin{equation*} G_{21} =m_2l_{c2}gcos(q_1+q_2) \end{equation*}

The model uncertainties are given as $\Delta M_0(q)=-0.1M_0(q)$ , $\Delta C_0(q,\dot{q})=-0.2C_0(q,\dot{q})$ , $\Delta G_0(q)=-0.1G_0(q)$ . The external disturbance is chosen as $\tau _d=[sin(t),sin(t)]^T$ . $UD=[UD_1,UD_2]^T$ denotes model uncertainties and external disturbance. The jacobian matrix is given as follows:

\begin{align*} J(q) & =\left [\begin{array}{c@{\quad}c}{-l_1sin(q_1)-l_2sin(q_1+q_2)}&{-l_2sin(q_1+q_2)}\\[5pt]{l_1cos(q_1)+l_2cos(q_1+q_2)}&{l_2cos(q_1+q_2)} \end{array}\right ] \end{align*}

The robot manipulator parameters are listed in Table I.

Table I.

Parameters of the robot.

Table II.

Controller parameters.

Figure 1.

Tracking performance.

Figure 2.

Tracking error.

Figure 3.

Control input.

Figure 4.

Model uncertainties and external disturbances estimation curve.

Then system (1) can be converted to (5) based on (4). The initial position and velocity of the robot manipulator are given as $x_1(0)=[0.235,0.205]^T$ and $x_2(0)=[0,0]^T$ . The initial state value of auxiliary system are set to $\eta _1=[0,0]^T$ and $\eta _2=[0,0]^T$ . $y=[y_1,y_2]^T$ denotes the system output. The desired trajectory is given as $y_d=[0.2sin({\pi }t),0.2cos({\pi }t)]^T$ . $e_3=[e_{31},e_{32}]^T=y-y_d$ denotes the trajectory tracking error. The output constraint functions are selected as $F_{11}=0.21-0.03sin({\pi }t)$ , $F_{12}=0.21+0.03cos({\pi }t)$ , $F_{21}=0.21-0.03cos({\pi }t+{\pi }/2)$ and $F_{22}=0.21+0.03cos({\pi }t+{\pi }/2)$ . The input saturation values are chosen as $U_m=[U_{m1},U_{m2}]^T=[15,5]^T$ .

To further evaluate the performance of the proposed controller, we select two comparative controllers from the literature. The adaptive NN tracking controller [Reference Wu, Huang, Wang and Wang41] is designed as

\begin{equation*} \tau _x =-\overline{M}_x(K_2z_2+L^{-1}\hat{w}^T\phi \left(Z)+\hat{D}-\dot{\alpha }+\vartheta _1+P_2\vartheta _2+z_2\left \|\vartheta _2\right \|^2\left \|\overline{M}_x^{-1}\right \|^4\lambda _1\hat{\Delta }_{\tau }+\varphi \right) \end{equation*}

and the inverse dynamics control algorithm based on the chattering-free continuous sliding-mode controller (IDCSMC) [Reference Asar, Elawady and Sarhan42] is designed as

\begin{equation*} \tau =M_0J^{-1}(\ddot{y}_d-\dot{J}\dot{q}+K_ss(t))+C_0\dot{q}+G_0-\hat{d}(t) \end{equation*}

where the sliding surface vector $s$ is defined as

\begin{equation*} s =K_d(\dot{y}_d(t)-\dot{y}(t))+K_p(y_d(t)-y(t)) \end{equation*}

4.2. Simulation results

In this section, we present the simulation results of the proposed controller, the adaptive NN controller and IDCSMC. The controller parameters are listed in Table II. The simulation time is 20 s.

Figure 1 shows the curves of the system outputs $y_1$ and $y_2$ under the control of proposed controller, adaptive NN controller and IDCSMC, respectively. Similarly, the curves of tracking errors $e_{31}$ and $e_{32}$ are shown in Fig. 2. Figure 3 shows the curves of control inputs $\tau _1$ and $\tau _2$ . The curves of UDE estimation are given in Fig. 4. According to the simulation results, we can get the following conclusions. (1) All the three controllers can ensure the system output tracks on the given trajectory. (2) The proposed control achieves the smallest steady-state tracking error. (3) The proposed controller and adaptive NN controller can satisfy input saturation and output constraints, but IDCSMC can’t. (4) UDE can estimate the model uncertainties and external disturbances precisely.

4.3. Performance comparisons

For performance analysis, we use root mean square (RMS) and maximum (MAX) values of the sampled tracking error $e(i)$ for comparisons [Reference etal43, Reference etal44], which are defined as

\begin{equation*} RMS(e) =\sqrt{\sum _{i=1}^{N}\frac{e^2(i)}{N}} \end{equation*}

\begin{equation*} MAX(e) =maximum(|e|) \end{equation*}

where $N$ is the number of sampled tracking error. $N$ is chosen as 5001 and the sample time is 0.01 s. Sampling starts at $t=0s$ . Table III shows the performance comparisons of three controllers. The proposed controller achieves the smallest RMS values for both two coordinates. In particular, the performance improvements in terms of RMS error for the proposed controller compared with the adaptive NN controller and IDCSMC are 3 $\%$ , 27 $\%$ for $e_{31}$ and 88 $\%$ , 87 $\%$ for $e_{32}$ , respectively. Also, when t $\gt$ 2s, the performance improvements in terms of MAX error for the proposed controller compared with the adaptive NN controller and IDCSMC are 93 $\%$ , 95 $\%$ for $e_{31}$ and 87 $\%$ , 89 $\%$ for $e_{32}$ , respectively. The results show that the proposed control method has better robustness and tracking performance compared with the other two controllers.

Table III.

Performance comparisons of controllers.