2.1. Coding and control techniques of ref. [Reference Parsa and Farhadi16]

The proposed coding technique in ref. [Reference Parsa and Farhadi16] is based on the linearization method [Reference Slotine and Li20]. By implementing this method, we presented an encoder, decoder, and a sufficient condition on the length of transmitted packets, $R_t$ , at each linearization zone that guaranteed almost sure asymptotic state tracking of the family of the equivalent linear dynamic systems, which were resulted from the linearizing the nonlinear dynamic system (1). Note that as in each linearized zone we deal with a new linear dynamic system, the length of the transmitted packet is different in different linearized zones. Hence, we denote the length of transmitted packet in the linearized zone $j$ by $R_{[j]}$ . Having that, the proposed coding technique works as follows (for the simplicity of the presentation consider the scalar case).

At the time instant $t = 0$ , we notice that $X_0 \in [\!-L_0, L_0]$ , where the upper bound $L_0$ is known for both encoder and decoder; and we fix the rate to be $\bar{R}_{[0]}$ . Then, using the coding technique that will be described very shortly, the reconstruction of $X_0$ denoted by $\hat{\bar{X}}_{0}$ is obtained at the decoder. Due to the existence of the feedback acknowledgment, the encoder also reconstructs $\hat{\bar{X}}_{0}$ . Then, at this time instant ( $t=0$ ), the encoder and decoder linearize the nonlinear dynamic system at the working point ( $\hat{\bar{X}}_{0},U_0)$ , $U_0=0$ , which results in a state space system matrix $A_{[0]}$ and $B_{[0]}$ for the equivalent linear model. Then, the encoder and decoder partition the interval $[\!-L_0, L_0]$ into $2^{R_{[0]}}$ equal sized, nonoverlapping subintervals and the center of each subinterval is chosen as the index of that interval ( $\gamma _0, \gamma _1,. . ., \gamma _{2^{R_{[0]}}-1}$ ). Subsequently, the index of the subinterval that includes $X_0$ (e.g., $\gamma _{j0}$ where $j\ 0 \in \{0,1,\ldots,2^{R_{[0]}}-1\}$ ) is encoded into $R_{[0]}$ bits and transmitted to the decoder through the packet erasure channel. If the decoder receives this $R_{[0]}$ bits successfully, it identifies the index of the subinterval where $X_0$ lives in; and the value of this index is chosen as $\hat{X}_0$ . Therefore, the decoding error for this case is bounded above by $|X_0 - \hat{X}_0| \leq V_0 =\frac{L_0}{2^{R_{[0]}}}$ . But if erasure occurs, then $\hat{X}_0 = 0$ ; and therefore, $|X_0 - \hat{X}_0| \leq V_0 = L_0$ . Hence, the decoding error can be represented as follows:

(3) \begin{align} |E_0|\doteq |X_0-\hat{X}_0| \leq V_0=M_0L_0;\ \nonumber \\[3pt]M_0 =\left \{\begin{array}{cc} \dfrac{1}{2^{R_{[0]}}},\ \ \Pr \left(M_0=\dfrac{1}{2^{R_{[0]}}}\right)= 1-\alpha \\ \\[-8pt]1, \ \ \ \Pr (M_0=1)=\alpha \ \ \ \ \ \end{array} \right. \end{align}

Note that $\hat{\bar{X}}_{0}$ is constructed similarly by implementing $\bar{R}_{[0]}$ bits. The procedure for the reconstruction of $\hat{\bar{X}}_{0}$ may be repeated several times until we have a successful transmission for the reconstruction of $\hat{\bar{X}}_{0}$ . At the time instant $t = 1$ , the encoder encodes $X_1-A_{[0]}\hat{X}_0-B_{[0]}U_0$ . To encode this signal, the interval $[\!-L_1, L_1]$ is calculated as follows: $|X_1-A_{[0]}\hat{X}_0-B_{[0]}U_0|=|A_{[0]}X_0-A_{[0]}\hat{X}_0|=|A_{[0]}||X_0-\hat{X}_0|\leq |A_{[0]}|V_0=L_1.$ Then, the encoder and decoder partition the interval $[\!-L_1, L_1]$ into $2^{R_{[0]}}$ equal sized, nonoverlapping subintervals, and the center of each subinterval is chosen as the index of that interval. When the encoder observes the signal $X_1-A_{[0]}\hat{X}_0-B_{[0]}U_0$ , the index of the subinterval that includes $X_1-A_{[0]}\hat{X}_0-B_{[0]}U_0$ (e.g., $\gamma _{j1}$ where $j1 \in \{0,1,\ldots,2^{R_{[0]}}-1\}$ ) is encoded into $R_{[0]}$ bits and transmitted to the decoder through the packet erasure channel. Subsequently, the decoder constructs $\hat{X}_1$ as follows: $ \hat{X}_1 =\gamma _{j1}+A_{[0]}\hat{X}_0+B_{[0]}U_0$ , if $M_1=\frac{1}{2^{R_{[0]}}}$ with the probability of $\Pr (M_1=\frac{1}{2^{R_{[0]}}})= 1-\alpha$ ; and $\hat{X}_1 =A_{[0]}\hat{X}_0+B_{[0]}U_0$ , if $M_1=1$ with the probability of $\Pr (M_1=1)=\alpha$ . Therefore,

(4) \begin{align} |E_1|\doteq |X_1-\hat{X}_1| \leq V_1=M_1L_1;\ \nonumber \\[3pt]M_1 =\left \{\begin{array}{l} \dfrac{1}{2^{R_{[0]}}},\ \ \Pr \left(M_1=\dfrac{1}{2^{R_{[0]}}}\right)= 1-\alpha \\ \\[-8pt] 1, \ \ \ \Pr (M_1=1)=\alpha \ \ \ \ \ \end{array} \right. \end{align}

For the next step ( $t =2$ ), if $|E_1|\leq |E_0|$ , this procedure continues with the equivalent state space matrices $A_{[0]}$ and $B_{[0]}$ and the packet length $R_{[0]}$ ; but if $|E_1| \gt |E_0|$ , then the encoder linearizes the nonlinear dynamic system at the new working point $(\hat{X}_1,U_1)$ that results in the state space system matrices $A_{[1]}$ and $B_{[1]}$ of the equivalent linear model (i.e., the system (5) with $j=1$ ). The encoder by sending $R_{[1]}\neq R_{[0]}$ bits through the packet erasure channel informs the decoder that a new linearization has been applied. Therefore, the decoder performs the same linearization; and the rest of the procedure continues with the new matrices $A_{[1]}$ and $B_{[1]}$ .

(5) \begin{align} \left \{\begin{array}{cc}X_{t+1}= A_{[0]}X_t+B_{[0]}U_t; \ t\in [0,t_1), \\[3pt]X_{t+1}= A_{[1]}X_t+B_{[1]}U_t; \ t\in [t_1,t_2) \ \ \ \ \ \ \ \ \\[3pt] \ \ \ \ \ \vdots \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\[3pt] X_{t+1}= A_{[j]}X_t+B_{[j]}U_t; \ t\in [t_j,t_{j+1}), \ \ j\in N_{+} \ \ \ \\[3pt] Y_t = X_t \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{array} \right. \end{align}

By following a similar procedure, as described above, the sequence $\hat{X}_0,\hat{X}_1,\hat{X}_0,\ldots$ are constructed at the decoder. The vector case ( $X_t\in \mathbb{R}^n$ ) was treated similarly in ref. [Reference Parsa and Farhadi16]. It has been proved in ref. [Reference Parsa and Farhadi16] that if ${\triangle t}_j\doteq t_{j+1}-t_j$ ( $j \in N_+$ )s are sufficiently large and the packet lengths $R_t=R_{[j]}$ ( $t\in [t_j,t_{j+1})$ ) satisfy the following inequalities:

(6) \begin{align} (1 - \alpha )R_{[j]} \gt \sum _{i=1}^{n}\max \{0, \log _{2}|\lambda _i(A_{[j]})|\}; \ & \forall j\in{\mathbb{N}_+}, \end{align}

then using the above coding technique we have almost sure asymptotic state tracking. Note that the first condition (i.e., ${\triangle t}_j\doteq t_{j+1}-t_j$ for $j \in{\mathbb{N}}_+$ is sufficiently large) is satisfied if the sampling period is small compared with the linearization period.

Now, for the nonlinear system (1) suppose that for each linearized equivalent system, there exists a matrix $K_{[j]}$ such that the matrix $A_{[j]}+B_{[j]}K_{[j]}$ is a stable matrix (e.g., $K_{[j]}=-B_{[j]}^{+}A_{[j]}$ , where $B_{[j]}^{+}$ is the Pseudo inverse). Then, using the proposed coding technique and the controller $U_t=K_{[j]}\hat{\tilde{X}}_t + W_{[j],t}$ where $W_{[j],t}\doteq -B_{[j]}^{+}(A_{[j]}{\mathcal R}_t-{\mathcal R}_{t+1})$ and $\hat{\tilde{X}}_t \doteq \hat{X}_t-{\mathcal R}_t$ for each linearized system, we have almost sure asymptotic reference tracking [Reference Parsa and Farhadi16].

2.2. Coding and control techniques of ref. [Reference Parsa and Farhadi17]

The coding and control techniques of ref. [Reference Parsa and Farhadi17] work differently. The coding and control techniques of ref. [Reference Parsa and Farhadi17] are not based on the linearization method and throughout we transmit with a fixed rate $R$ rather than a variable rate $R_{[j]}$ . Reference [Reference Parsa and Farhadi17] assumes the nonlinear dynamic system $F(X_t,U_t)$ is the control affine, that is, $F(X_t,U_t)=F(X_t)+BU_t$ , where $F(.)$ is a piece wise nondecreasing or nonincreasing function (see Fig. 3). The coding technique of ref. [Reference Parsa and Farhadi17] is similar to the coding technique of ref. [Reference Parsa and Farhadi16] for each linearized zone; except it considers the following dynamic $X_{t+1}=F(X_t)+BU_t$ and instead of encoding $\gamma _j$ it encodes $\eta _j=F^{-1}(\gamma _j)$ . It has been proved in ref. [Reference Parsa and Farhadi17] that under some mild conditions on the transmission rate $R$ , the proposed coding technique results in the almost sure asymptotic tracking of the state trajectory. It has been also proved that the control vector $U_t=-B^{+}(F(\hat{X}_t)-{\mathcal R}_{t+1})$ results in the almost sure asymptotic reference tracking.

Figure 3.

Equal sized, nonoverlapping subintervals for encoding with rate $R=2$ using the coding technique of ref. [Reference Parsa and Farhadi17].

Figure 4.

The robotic manipulator considered as the case study of the paper.

2.3. Applications of the proposed techniques on two-link robotic manipulators

Now, we implement these coding and control techniques to the case study of the paper, which is a two-link robotic manipulator shown in Fig. 4. This type of manipulators are very common in manufacturing industries. Its dynamic is described by the following Eq. [Reference Slotine and Li20] (for the simplicity of the presentation, we drop the continuous time dependency index).

(7) \begin{align} H(q)\ddot{q}+C(q,\dot{q})\dot{q}+g(q)=\tau, \end{align}

where

\begin{equation*}H(q)=\left(\begin{array}{c@{\quad}c}H_{11} & H_{12}\\ \\[-8pt] H_{21} & H_{22} \end {array}\right), \ \ddot {q}=\left(\begin{array}{c} \ddot {q}_1 \\ \\[-8pt] \ddot {q}_2 \end{array}\right),\end{equation*}

\begin{equation*} \ C(q,\dot {q})= \left(\begin{array}{c@{\quad}c}-h\dot {q}_2 & -h\dot {q}_1-h\dot {q}_2 \\ \\[-8pt] h\dot {q}_1 & 0 \end {array}\right), \ \dot {q}=\left(\begin{array}{c} \dot {q}_1 \\ \\[-8pt] \dot {q}_2 \end{array}\right), \end{equation*}

\begin{equation*} g(q)=\left(\begin{array}{c} g_1 \\ \\[-8pt] g_2 \end {array}\right), \ \ \tau =\left(\begin{array}{c} \tau _1 \cr \tau _2 \end{array}\right), \end{equation*}

where

\begin{align*} H_{11} &= m_1 l_{C1}^2+I_1+m_2[l_1^2+l_{C2}^2+2l_1 l_{C2} \cos (q_2)]+I_2 \\[4pt]H_{22} &= m_2 l_{C2}^2+I_2 \\[4pt] H_{21}&=H_{12}=m_2 l_1 l_{C2} \cos (q_2)+I_2+m_2 l_{C2}^2 \\[4pt] h &= m_2 l_1 l_{C2} \sin (q_2) \\[4pt] g_1 &= m_1 l_{C1} g \cos (q_1)+m_2 g [l_{C2} \cos (q_1+q_2)+l_1\cos (q_1)]\\[4pt] g_2 &= m_2 l_{C2} g \cos (q_1+q_2), \end{align*}

and $m_1$ is the mass of the fist arm, $m_2$ is the mass of the second arm, and $g$ is the earth gravity acceleration coefficient (i.e., $g=$ 9.81).

In order to implement the coding and control techniques of refs. [Reference Parsa and Farhadi16] and [Reference Parsa and Farhadi17] to the above manipulator, it is required that we rewrite the above dynamic in the state space form. To do that, we define the following states:

\begin{align*} x^{(1)}=q_1, \ x^{(2)}=q_2, \ x^{(3)}=\dot{q}_1, \ x^{(4)}=\dot{q}_2. \end{align*}

Subsequently, we have the following state space representation for the above manipulator:

\begin{align*} \dot{x}^{(1)} &= x^{(3)} \\[4pt]\dot{x}^{(2)} &= x^{(4)} \\[4pt] \dot{x}^{(3)} &= \frac{num_3}{H_{12} H_{21}-H_{11} H_{22}} \\[4pt] num_3 &= \tau _2 H_{12}-\tau _1 H_{22}-g_2 H_{12}-h H_{12}{x^{(3)}}^2-h H_{22}{x^{(4)}}^2\\[4pt]&\quad +g_1 H_{22}-2h H_{22} x^{(3)} x^{(4)} \\[4pt] \dot{x}^{(4)} &= \frac{num_4}{H_{22} H_{11}-H_{21} H_{12}}\\[4pt] num_4 &= \tau _2 H_{11}-\tau _1 H_{21}-g_2 H_{11}-h H_{11}{x^{(3)}}^2-h H_{21}{x^{(4)}}^2 \\[4pt]&\quad +g_1 H_{21}-2h H_{21} x^{(3)} x^{(4)}. \end{align*}

The measurement data from this robotic manipulator is the vector $Y$ and the control commands is the vector $U$ described as follows:

\begin{equation*}Y=\left(\begin{array}{c}x^{(1)} \\ \\[-8pt] x^{(2)} \\ \\[-8pt] x^{(3)} \cr x^{(4)} \end {array}\right), \ U=\left(\begin{array}{c} \tau _1 \\ \\[-8pt] \tau _2 \end{array}\right).\end{equation*}

Because the communication network is a digital channel, we need to sample the measurement data with a fixed period of $T$ seconds and apply the control commands via the so called Zero Order Hold (ZOH) [Reference Ogata21] with the update period of $T$ seconds. Hence, in the block diagram of Fig. 1 we should deal with the following equivalent discrete time dynamic system for the above manipulator for the implementation of the coding and control techniques of refs. [Reference Parsa and Farhadi16] and [Reference Parsa and Farhadi17]. The following equivalent discrete time model is obtained by implementing the Euler’s approximation method [Reference Ogata21]:

\begin{align*} x^{(1)}_{t+1} &= T x_t^{(3)} +x^{(1)}_t \dot{=} \,\,f^{(1)}\\[4pt]x^{(2)}_{t+1} &= T x_t^{(4)}+ x^{(2)}_t \dot{=} \,\,f^{(2)} \\[4pt] x^{(3)}_{t+1} &= T\left(\frac{num_3}{H_{12} H_{21}-H_{11} H_{22}}\right)+x^{(3)}_t \dot{=} \,\,f^{(3)} \\[4pt] x^{(4)}_{t+1} &= T\left(\frac{num_4}{H_{22} H_{11}-H_{21} H_{12}}\right)+x^{(4)}_t \dot{=} \,\,f^{(4)} \end{align*}

Hence, the matrices $A_{[j]}$ and $B_{[j]}$ for the $j$ th linearized system to be used in the coding and control techniques of ref. [Reference Parsa and Farhadi16] are given by the following matrices.

\begin{align*} A_{[j]}&= \left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c}\dfrac{\partial f^{(1)}}{\partial x^{(1)}} & \dfrac{\partial f^{(1)}}{\partial x^{(2)}} & \dfrac{\partial f^{(1)}}{\partial x^{(3)}}& \dfrac{\partial f^{(1)}}{\partial x^{(4)}} \\ \\[-4pt] \dfrac{\partial f^{(2)}}{\partial x^{(1)}} & \dfrac{\partial f^{(2)}}{\partial x^{(2)}} & \dfrac{\partial f^{(2)}}{\partial x^{(3)}}& \dfrac{\partial f^{(2)}}{\partial x^{(4)}} \\ \\[-4pt] \dfrac{\partial f^{(3)}}{\partial x^{(1)}} & \dfrac{\partial f^{(3)}}{\partial x^{(2)}} & \dfrac{\partial f^{(3)}}{\partial x^{(3)}}& \dfrac{\partial f^{(3)}}{\partial x^{(4)}} \\ \\[-4pt] \dfrac{\partial f^{(4)}}{\partial x^{(1)}} & \dfrac{\partial f^{(4)}}{\partial x^{(2)}} & \dfrac{\partial f^{(4)}}{\partial x^{(3)}}& \dfrac{\partial f^{(4)}}{\partial x^{(4)}}\end{array}\right)|_{(\hat{X}_{t_j},U_{t_j})} \\ \\[-4pt]&=\left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & T & 0 \\ \\[-8pt] 0 & 1 & 0 & T\\ \\[-8pt] a_{31} & a_{32} & a_{33} & a_{34}\\ \\[-8pt] a_{41} & a_{42} & a_{43} & a_{44} \end{array}\right)|_{(\hat{X}_{t_j},U_{t_j})}, \end{align*}

\begin{align*} a_{31} &= \frac{1}{H_{11}H_{22}-H_{12}H_{21}}\left(\frac{\partial g_2}{\partial q_1}H_{12}-\frac{\partial g_1}{\partial q_1}H_{22}\right) \\[4pt]a_{32} &= \frac{\partial f_3}{\partial q_2} \\[4pt]a_{33} &= \frac{T.2H_{12}hx^{(3)}+T.2H_{22}hx^{(4)}}{H_{11}H_{22}-H_{12}H_{21}}+1 \\[4pt]a_{34} &= \frac{T.2H_{22}h(x^{(3)}+x^{(4)})}{H_{11}H_{22}-H_{12}H_{21}} \\[4pt]a_{41} &= \frac{-1}{H_{11}H_{22}-H_{12}H_{21}}\left(\frac{\partial g_2}{\partial q_1}H_{11}-\frac{\partial g_1}{\partial q_1}H_{21}\right)\\[4pt]a_{42}&= \frac{\partial f_4}{\partial q_2} \\[4pt] a_{43} &= \frac{-T.2H_{11}hx^{(3)}-T.2H_{21}hx^{(4)}}{H_{11}H_{22}-H_{12}H_{21}} \\[4pt] a_{44}&= 1-\frac{T.2H_{21}h(x^{(3)}+x^{(4)})}{H_{11}H_{22}-H_{12}H_{21}}. \end{align*}

\begin{align*} B_{[j]} &= \left(\begin{array}{c@{\quad}c} \dfrac{\partial f^{(1)}}{\partial \tau _1} & \dfrac{\partial f^{(1)}}{\partial \tau _2} \\ \\[-6pt] \dfrac{\partial f^{(2)}}{\partial \tau _1} & \dfrac{\partial f^{(2)}}{\partial \tau _2} \\ \\[-6pt] \dfrac{\partial f^{(3)}}{\partial \tau _1} & \dfrac{\partial f^{(3)}}{\partial \tau _2} \\ \\[-6pt] \dfrac{\partial f^{(4)}}{\partial \tau _1} & \dfrac{\partial f^{(4)}}{\partial \tau _2} \end{array}\right)|_{(\hat{X}_{t_j},U_{t_j})} \\ \\[-6pt]&= \left(\begin{array}{cc}0 & 0\\ \\[-8pt] 0 & 0 \\ \\[-6pt] \dfrac{T H_{22}}{H_{11}H_{22}-H_{12}H_{21}} & \dfrac{-T H_{12}}{H_{11}H_{22}-H_{12}H_{21}} \\ \\[-6pt] \dfrac{-TH_{21}}{H_{11}H_{22}-H_{12}H_{21}} & \dfrac{T H_{11}}{H_{11}H_{22}-H_{12}H_{21}} \end{array}\right)|_{(\hat{X}_{t_j},U_{t_j})}. \end{align*}

Following that a linear controller for each linearized zone in the form of $U_{t}=-B^{+}_{[j]}A_{[j]}\hat{\tilde{X}}_t+W_{[j],t}$ , $W_{[j],t}=-B^{+}_{[j]}(A_{[j]}{\mathcal R}_t-{\mathcal R}_{t+1})$ , $\hat{\tilde{X}}_t=\hat{X}_t-{\mathcal R}_t$ is obtained for the teleoperation of the two-link manipulator.

Now, let $X_t=\begin{pmatrix}x^{(1)}_t \cr x^{(2)}_t \cr x^{(3)}_t \cr x^{(4)}_t \end{pmatrix}$ , $\hat{X}_t=\begin{pmatrix}\hat{x}^{(1)}_t \cr \hat{x}^{(2)}_t \cr \hat{x}^{(3)}_t \cr \hat{x}^{(4)}_t \end{pmatrix}$ (which is the decoder output), $d_1=\bar{H}_{12} \bar{H}_{21}-\bar{H}_{11} \bar{H}_{22}$ and $d_2=\bar{H}_{22} \bar{H}_{11}-\bar{H}_{21} \bar{H}_{12}$ , where

\begin{align*} \bar{H}_{11} &= m_1 l_{C1}^2+I_1+m_2[l_1^2+l_{C2}^2+2l_1 l_{C2} \cos (x^{(2)}_t)]+I_2 \\[3pt] \bar{H}_{22} &= m_2 l_{C2}^2+I_2 \\[3pt] \bar{H}_{21}&=\bar{H}_{12}=m_2 l_1 l_{C2} \cos (x^{(2)}_t)+I_2+m_2 l_{C2}^2 \\[3pt] \bar{h} &= m_2 l_1 l_{C2} \sin (x^{(2)}_t) \\[3pt] \bar{g}_1 &= m_1 l_{C1} g \cos (x^{(1)}_t)+m_2 g [l_{C2} \cos (x^{(1)}_t+x^{(2)}_t)\\[3pt] &\quad +l_1\cos (x^{(1)}_t)]\\[3pt] \bar{g}_2 &= m_2 l_{C2} g \cos (x^{(1)}_t+x^{(2)}_t). \end{align*}

Let also $F(X_t)=\begin{pmatrix} f^{(1)} \cr f^{(2)} \cr \bar{f}^{(3)} \cr \bar{f}^{(4)} \end{pmatrix}$ , where

\begin{align*} \bar{f}^{(3)}&=\frac{num_5}{d_1}, \\[3pt]num_5&=-\bar{g}_2 \bar{H}_{12}-\bar{h} \bar{H}_{12}{x^{(3)}_t}^2-\bar{h} \bar{H}_{22}{x^{(4)}_t}^2+\bar{g}_1 \bar{H}_{22}\\[3pt]&\quad -2\bar{h} \bar{H}_{22} x^{(3)}_t x^{(4)}_t \\[3pt] \bar{f}^{(4)}&=\frac{num_6}{d2}, \\[3pt] num_6&=-\bar{g}_2 \bar{H}_{11}-\bar{h} \bar{H}_{11}{x^{(3)}_t}^2-\bar{h} \bar{H}_{21}{x^{(4)}_t}^2+\bar{g}_1 \bar{H}_{21}\\[3pt]&\quad -2\bar{h} \bar{H}_{21} x^{(3)}_t x^{(4)}_t. \end{align*}

Also, let

\begin{equation*}\bar {B}=\left(\begin{array}{c@{\quad}c} 0 & 0 \\ \\[-8pt] 0 & 0 \\ \\[-5pt] -\dfrac {\bar {H}_{22}}{d_1} & \dfrac {\bar {H}_{12}}{d_1} \\ \\[-5pt] -\dfrac {\bar {H}_{21}}{d_2} & \dfrac {\bar {H}_{11}}{d_2}\end {array}\right).\end{equation*}

Then, the nonlinear controller equipped with the coding technique of ref. [Reference Parsa and Farhadi17] that results in the teleoperation of the two-link manipulator is given by

\begin{equation*}U_t=-\bar {B}^{+}(F(\hat {X}_t)-{\mathcal R}_{t+1}).\end{equation*}

3.1. The performance under the coding and control techniques of ref. [Reference Parsa and Farhadi16]

Under the above conditions, for the case of $\alpha =0$ and $T=0.01$ , we implement the aforementioned coding and control techniques of [Reference Parsa and Farhadi16] to track the reference signals for $q_1$ , which is $\frac{\pi }{8}$ , and $q_2$ as shown in Fig. 5 by dashed trajectory. For the time interval of 0 to 15 s, the nominal dynamic is simulated and for the interval of 15 to 23.75 s the actual dynamic is simulated, then for the time interval of 23.75 to 32.5 the nominal dynamic is simulated and for the next time interval of 32.5 to 41.25 the actual dynamic is simulated and so on and so forth. That is, the manipulator frequently picks up a predefined object and moves it and then returns to pick up and move another similar object. When it carries this object, its dynamic is described by the actual dynamic and when it returns to pick up another similar object, its dynamic is the nominal dynamic. This is a typical scenario for the robotic manipulator of the case study as it very often moves a predefined load and after dropping this load, it returns to its initial position to pick up, move, and drop another similar load. Figure 5 also illustrates the trajectory for $q_2$ taken by the manipulator (solid line). As it is clear from this figure, we have an excellent quality of the reference tracing. We have similar excellent quality of reference tracking for $q_1$ . But, this excellent quality comes with a price. Figures 6 and 7 illustrate the required torques for this reference tracking. Due to the operational constrains, the robotic manipulator of the case study obviously is not able to execute these required torques; and what we get are the results reported in Figs. 8 and 9.

Figure 5.

The reference signal and the trajectory taken by the manipulator for $q_2$ when the reference trajectory is nonsmooth, $\alpha =0$ , $T=0.01$ , and the coding and control techniques of ref. [Reference Parsa and Farhadi16] are used.

Figure 6.

The required torque for joint 1 when reference trajectories are nonsmooth, $\alpha =0$ , $T=0.01$ , and the coding and control techniques of ref. [Reference Parsa and Farhadi16] are used.

Figure 7.

The required torque for joint 2 when reference trajectories are nonsmooth, $\alpha =0$ , $T=0.01$ , and the coding and control techniques of ref. [Reference Parsa and Farhadi16] are used.

Figure 8.

The reference signal and the trajectory taken by the manipulator for $q_1$ subject to the the operational constraints when reference trajectories are nonsmooth, $\alpha =0$ , $T=0.01$ , and the coding and control techniques of [Reference Parsa and Farhadi16] are used.

Figure 9.

The reference signal and the trajectory taken by the manipulator for $q_2$ subject to the operational constraints when reference trajectories are nonsmooth, $\alpha =0$ , $T=0.01$ , and the coding and control techniques of ref. [Reference Parsa and Farhadi16] are used.

Now, in order to satisfy the operational constrains, we implement the smooth version of the reference signals as they are shown in Figs. 10 and 11. Figures 12 and 13 illustrate the required torques for having this excellent quality of the reference tracking. As it is clear from these figures, by smoothing the reference signals, the upper bound constraints on torques are satisfied; but the rate constraints are not satisfied. Nevertheless, when these torques are applied by the manipulator to its joints by considering the rate and the upper bound constraints, we get a very similar result for the reference tracking as shown in Figs. 10 and 11 with the applied torques shown by Figs. 14 and 15. This indicates that our modified method, which is based on smoothing the reference signal results in a satisfactory performance. Note that whenever the value of the required torque passes the upper bound constraint, the upper bound is applied by the manipulator. Similarly, when the rate of change of torque passes its bound, the rate of change of torque is limited to the plus or the minus of the maximum allowable rate of change of torque (e.g., $\pm$ 0.03 for $T=0.01$ ).

Figure 10.

The smooth version of the reference signal and the trajectory taken by the manipulator for $q_1$ when $\alpha =0$ , $T=0.01$ , and the coding and control techniques of ref. [Reference Parsa and Farhadi16] are used.

Figure 11.

The smooth version of the reference signal and the trajectory taken by the manipulator for $q_2$ when $\alpha =0$ , $T=0.01$ , and the coding and control techniques of ref. [Reference Parsa and Farhadi16] are used.

Figure 12.

The required torque for joint 1 when reference trajectories are smooth, $\alpha =0$ , $T=0.01$ , and the coding and control techniques of ref. [Reference Parsa and Farhadi16] are used.

Figure 13.

The required torque for joint 2 when reference trajectories are smooth, $\alpha =0$ , $T=0.01$ , and the coding and control techniques of ref. [Reference Parsa and Farhadi16] are used.

Figure 14.

The applied torque on joint 1 by considering the operational constraints when reference trajectories are smooth, $\alpha =0$ , $T=0.01$ , and the coding and control techniques of ref. [Reference Parsa and Farhadi16] are used.

Figure 15.

The applied torque on joint 2 by considering the operational constraints when reference trajectories are smooth, $\alpha =0$ , $T=0.01$ , and the coding and control techniques of ref. [Reference Parsa and Farhadi16] are used.

Now, we evaluate the telepresence and teleoperation performances in the presence of severe random packet dropout with the erasure probability of $\alpha \in [0,1)$ . That is, when sensor data is not received at the destination with the probability of $\alpha .100$ $\%$ . For the fair comparison, we define the following Root Sum Square Error (RSSE) criterion [Reference Parsa and Farhadi17]:

\begin{equation*}RSEE=\sqrt {\sum _{t=0}^{t=50/T}((x^{(1)}_t-r^{(1)}_t)^2+(x^{(2)}_t-r^{(2)}_t)^2)}.\end{equation*}

For the case of $T=0.01$ seconds and $\alpha =0$ (Fig. 10 to Fig. 15), $RSSE=0.0568$ (without considering the operational constraints) and $RSSE^*=1.5921$ (by considering the operational constraints). For the other cases, the $RSSE$ s are reported in Table II. Table II indicates that even for the extremely severe communication imperfections, the quality of the reference tracking is as excellent as the case of $\alpha =0$ . Nevertheless, this comes with a price for the severe cases. While for very small $\alpha$ , the length of transmitted packets is 4 or 5 bits, for $\alpha =0.99$ , this number is 20 or 21 bits.

Table II.

$RSSE$ s for $T=0.01$ when the coding and control techniques of [Reference Parsa and Farhadi16] are used. $RSSE$ : Without considering the operational constraints. $RSSE^*$ : By considering the operational constraints.

It is desirable to increase the sampling period to increase the life time of the processors and also be able to use cheaper processors. Nevertheless, it has been shown in ref. [Reference Parsa and Farhadi16] for autonomous vehicles that by increasing the sampling period the quality of the reference tracking is getting worse. This is a direct consequence of the stability requirement of switching systems as discussed in ref. [Reference Parsa and Farhadi16]. For the case of $\alpha =0.5$ , the effects of different sampling periods in $RSSE$ are reported in Table III. From Table III, it is clear that by increasing the sampling period, the quality of the reference tracking is getting worse, as it is expected.

Table III.

RSSE for $\alpha =0.5$ when the coding and control techniques of [Reference Parsa and Farhadi16] are used.

3.2. The performance of the coding and control techniques of ref. [Reference Parsa and Farhadi17]

Computer simulation illustrates that the coding and control techniques of [Reference Parsa and Farhadi17] for the reference trajectories, as shown in Figs. 10 and 11, are not able to satisfy the bound constraints on torques. Therefore, throughout this section, we focus on the reference trajectories as shown in Fig. 16 for $q_2$ and $\dfrac{\pi }{2}$ for $q_1$ . Figures 17 and 18 illustrate the applied torques on manipulator joints for tracking these reference trajectories. Note that the coding and control techniques of ref. [Reference Parsa and Farhadi17] cannot satisfy the bound constraints on torques for the reference trajectories as shown in Figs. 10 and 11; and the coding and control techniques of ref. [Reference Parsa and Farhadi16] cannot also satisfy the bound constraints on torques for the reference trajectories as shown above.

Tables IV and V are the counterparts of Tables II and III for the above reference trajectories when different $\alpha$ s and sample periods are implemented.

Table IV.

$RSSE$ s for $T=0.01$ when the coding and control techniques of [Reference Parsa and Farhadi17] are used. $RSSE$ : Without considering the operational constraints. $RSSE^*$ : By considering the operational constraints.

Figure 16.

The smooth version of the reference signal and the trajectory taken by the manipulator for $q_2$ when $\alpha =0$ , $T=0.01$ , and the coding and control techniques of ref. [Reference Parsa and Farhadi17] are used.

Figure 17.

The applied torque on joint 1 by considering the operational constraints when reference trajectories are smooth, $\alpha =0$ , $T=0.01$ , and the coding and control techniques of ref. [Reference Parsa and Farhadi17] are used.

Figure 18.

The applied torque on joint 2 by considering the operational constraints when reference trajectories are smooth, $\alpha =0$ , $T=0.01$ , and the coding and control techniques of ref. [Reference Parsa and Farhadi17] are used.

Table V.

RSSE for $\alpha =0.5$ when the coding and control techniques of [Reference Parsa and Farhadi17] are used.