# FPGA Applied to Latency Reduction for the Tactile Internet
## 5 Simulated Tactile Internet Model
After the transmission through the forward channel, here called FC, the signal received by the HSD can be expressed as  
\[\mathbf{v}(n)=\left[x^{HSD}(n),y^{HSD}(n),z^{HSD}(n)\right]. \tag{45}\]  
Based on Equation (5) the spatial coordinate signal received by HSD can be expressed as  
\[x^{HSD}(n)=x^{HMD}\Big{(}n-d_{x}^{f}(n)\Big{)}+r_{x}^{f}(n), \tag{46}\]  
\[y^{HSD}(n)=y^{HMD}\Big{(}n-d_{y}^{f}(n)\Big{)}+r_{y}^{f}(n), \tag{47}\]  
and  
\[z^{HSD}(n)=z^{HMD}\Big{(}n-d_{z}^{f}(n)\Big{)}+r_{z}^{f}(n) \tag{48}\]  
where \(d_{x}^{f}(n)\), \(d_{y}^{f}(n)\), \(d_{z}^{f}(n)\), \(r_{x}^{f}(n)\), \(r_{y}^{f}(n)\) and \(r_{z}^{f}(n)\) are the delays and noises associated with CF. As, in this case, the Slave PHANToM Omni, SD, copies the movement of the master PHANToM Omni, MD, it is necessary for the HSD to perform a feedback control system on the three joints of the PHANToM Omni slave, here expressed as  
\[\boldsymbol{\theta}^{SD}(n)=\left[\theta_{1}^{SD}(n),\theta_{2}^{SD}(n),\theta _{3}^{SD}(n)\right] \tag{49}\]  
that is, \(\theta_{1}^{SD}(n)\), \(\theta_{2}^{SD}(n)\), \(\theta_{3}^{SD}(n)\) are control variables associated with DS. The control system illustrated in Figure 4 as FCS shall minimize the error, \(\mathbf{e}^{FCS}(n)\), between \(\boldsymbol{\theta}^{SD}(n)\) and the reference signal \(\boldsymbol{\theta}^{HSD}(n)\) characterized as  
\[\boldsymbol{\theta}^{HSD}(n)=\left[\theta_{1}^{HSD}(n),\theta_{2}^{HSD}(n), \theta_{3}^{HSD}(n)\right] \tag{50}\]where  
Figure 4:0  
and  
\[\left[ \begin{array}{c}e_{1}^{ FCS}(n)\\ e_{2}^{ FCS}(n)\\ e_{3}^{ FCS}(n)\end{array} \right]=\left[ \begin{array}{c}\theta_{1}^{HSD}(n)\\ \theta_{2}^{HSD}(n)\\ \theta_{3}^{ HSD}(n)\end{array} \right]-\left[ \begin{array}{c}\theta_{1}^{ SD}(n)\\ \theta_{2}^{SD}(n)\\ \theta_{3}^{SD}(n)\end{array} \right]. \tag{52}\]  
The \(\boldsymbol{\theta}^{SD}(n)\) signal is obtained from the SD via sensors (encoders) at the SD joints and the \(\boldsymbol{\theta}^{HSD}(n)\) signal is obtained from the IK-HSD hardware module shown in Figure 4. This hardware module implements all inverse kinematics equations presented in Section 4.2, i.e., Equations (18)-(25). There are several techniques and approaches that can be used in the FCS module ranging from more traditional techniques such as a proportional-integral-derivative controller [44] to more innovative artificial intelligence based techniques [45; 46]. The CPD-HSD and JPD-HSD modules, illustrated in Figure 4, represent the algorithms of prediction and detection in cartesian space and joints, respectively. These modules are responsible for minimizing the latency and noise added by the FC associated with the tactile internet system (Equations (46)-(48)). Depending on the prediction and detection technique used, the HSD may use only one of the modules, namely the CPD-HSD or JPD-HSD. There is still no consensus about whether the Cartesian space or joints is the best for minimizing latency and noise inserted by the channel. There are several works in the literature that present proposals using only one of the spaces and proposals that try to use the information from both simultaneously. Similarly to the FCS module, approaches ranging from the more traditional techniques up to more innovative techniques based on artificial intelligence have been used in the CPD-HSD and JPD-HSD modules [47; 48; 49; 50; 51]. Thus, it can be said that \(\boldsymbol{\theta}^{HSD}(n)\) is an estimate of the \(\mathbf{b}(n)\) signal generated by the MD. At each \(n\)-th time, the FCS acts on the SD through the \(\mathbf{u}(n)\) signal, detailed in Figures 1 and 4, which in the case of the PHANToM Omni can be expressed as  
\[\mathbf{u}^{HSD}(n)=\left[\tau_{1}^{HSD}(n),\tau_{2}^{HSD}(n),\tau_{3}^{HSD}(n)\right] \tag{53}\]  
where \(\tau_{i}^{HSD}(n)\) is the \(i\)-th torque applied every \(i\)-th joint. The FCS will act as a tracking mechanism, making the SD follow the path traveled by the MD.