# FPGA Applied to Latency Reduction for the Tactile Internet
## 5 Simulated Tactile Internet Model
Finalizing the data stream associated with the forward channel, it can be said that the \(\mathbf{\hat{a}}(n)\) signal is formed by an estimate of the spatial position generated by the OP, \(\mathbf{\hat{a}}(n)\), i.e.,  
\[\mathbf{\hat{a}}(n)=\left[\hat{x}^{OP}(n),\hat{y}^{OP}(n),\hat{z}^{OP}(n) \right]. \tag{54}\]  
The interaction of the PHANToM Omni, SD, with ENV can vary from free movement to physical contact. When some kind of physical contact occurs, the SD detects the touch and sends this information back to the HSD. As per the model detailed in Figure 4 the ENV sends back to SD the information associated with the contact force in the spatial plane, expressed here as,  
\[\mathbf{o}(n)=\left[F_{x}^{ENV}(n),F_{y}^{ENV}(n),F_{z}^{ENV}(n)\right]. \tag{55}\]  
The value associated with the contact force information can be measured directly through SD-coupled force sensors or indirectly estimated through other types of sensors that may be SD-coupled or inserted into the environment [52]. In the case of the model presented in Figure 4, the SD sends to HSD the objects surface's spatial positions through sensors spread in the ENV. The signal expressed as  
\[\mathbf{s}^{OBJ}(n)=\left[x^{OBJ}(n),y^{OBJ}(n),z^{OBJ}(n)\right] \tag{56}\]represents the spatial position of the closest object from the SD tool. Thus, based on the information already described, every \(n\)-th time \(t_{s}\) the SD sends to the HSD a signal characterized by the array \(\mathbf{g}(n)\) expressed as  
\[\mathbf{g}(n)=\Big{[}\boldsymbol{\theta}^{SD}(n),\mathbf{s}^{OBJ}(n)\Big{]}. \tag{57}\]  
In the HSD, when the signal \(\mathbf{g}(n)\) is received, the Split module separates the \(\boldsymbol{\theta}^{SD}(n)\) signal and sends it to the FCS and the FK-HSD hardware module. In addition, the signal \(\mathbf{s}^{OBJ}(n)\) is sent to the FB-HSD hardware module, as detailed in Figure 4. The FK-HSD hardware module performs the forward kinematics calculation similarly to FK-HMD and thus the current spatial position of the SD tool in the environment, ENV, can be obtained. Every \(n\)-th instant \(t_{s}\) FK-HSD generates a signal expressed as  
\[\mathbf{l}(n)=[x^{ENV}(n),y^{ENV}(n),z^{ENV}(n)] \tag{58}\]  
where \(x^{ENV}(n)\), \(y^{ENV}(n)\) and \(z^{ENV}(n)\) are the spatial position of the tool in the ENV module from \(\boldsymbol{\theta}^{SD}(n)\). The FBF-HSD hardware module implements the calculations associated with the generation of the feedback force from the contact between the tool and the object. Based on the work presented in [52] the contact force, represented by the \(\mathbf{h}(n)\) signal, can be expressed as  
\[\mathbf{h}(n)=\Big{[}F_{x}^{HSD}(n),F_{y}^{HSD}(n),F_{z}^{HSD}(n)\Big{]}, \tag{59}\]  
where  
\[F_{x}^{HSD}(n)=h_{x}(n)\Big{(}x^{OBJ}(n)-x^{ENV}(n)\Big{)}, \tag{60}\]  
\[\mathbf{u}^{HSD}(n)=\left[\tau_{1}^{HSD}(n),\tau_{2}^{HSD}(n),\tau_{3}^{HSD}(n)\right] \tag{53}\]0  
and  
\[\mathbf{u}^{HSD}(n)=\left[\tau_{1}^{HSD}(n),\tau_{2}^{HSD}(n),\tau_{3}^{HSD}(n)\right] \tag{53}\]1  
In these equations, the constants \(h_{x}(n)\), \(h_{y}(n)\) and \(h_{z}(n)\) represent the elasticity coefficients associated with the object. It is important to note that in this model the \(\mathbf{h}(n)\) signal is a synthesized version of the real force value here characterized by the \(\mathbf{o}(n)\) array. After the feedback force calculation process, as illustrated in Figure 4, the \(\mathbf{h}(n)\) signal is transmitted to the HMD via the backwards channel (BC) which, similarly to FC, adds latency and noise.