## 2 Methodology
### Haptic Rendering  
For the calculation of haptic forces, we used the same idea as for the forces felt in a manned aircraft. To deflect a control surface, an elevator in our case, the operator moves the control stick in either direction, forward/backward in our case. The force \(F\) exerted by the operator to move the control  
Figure 3: Response of the aerodynamic model of A UAV with negative and positive elevator inputs. (**a**) Effect on altitude. (**b**) Effect on pitch angle. surface, called the stick force, is proportional to the hinge moment \(H_{e}\) acting on the control surfaces. To move the control surface, the operator has to overcome the hinge moment by applying force. The hinge moment can be expressed as having the combined effects of the tail angle of attack \(\alpha_{t}\), elevator deflection \(\delta_{e}\), and tab angle \(\delta_{t}\) separately. The hinge moment coefficient \(C_{h_{e}}\) can be expressed as  
\[C_{h_{e}}=C_{h_{o}}+C_{h_{eq}}\alpha_{t}+C_{h_{eq}}\delta_{e}+C_{h_{eq}}\delta_ {t} \tag{3}\]  
where \(C_{h_{o}}\) is the residual moment, \(C_{h_{eq}}=\frac{dC_{h}}{d\alpha_{t}}\), \(C_{h_{eq}}=\frac{dC_{h}}{d\delta_{e}}\), and \(C_{h_{eq}}=\frac{dC_{h}}{d\delta_{t}}\). Precise hinge moment parameters are hard to find analytically, so wind tunnel tests are required to find these values for the optimal control system design. As mentioned earlier, the work done for displacing the control stick is equal to the work done in overcoming the hinge moment and moving the control surface to the desired deflection angle. This can be mathematically expressed as  
\[F=\frac{H_{e}\delta_{e}}{l_{s}\delta_{s}} \tag{4}\]  
where \(l_{s}\) and \(\delta_{s}\) are the length of control stick and stick deflection angle, respectively, and the \(\frac{\delta_{e}}{l_{s}\delta_{s}}\) part of Equation (4) is the gearing ratio \(G\) used to provide mechanical advantage in a control system. Now, we define the hinge moment as  
\[H_{e}=\frac{1}{2}C_{h_{e}}\rho V^{2}s_{e}c_{e} \tag{5}\]  
where \(\rho\) is air density, \(V\) is speed, \(s_{e}\) is the area of the hinge line and \(c_{e}\) is the chord length of the UAV whose values are state in Table 1.  
Using \(H_{e}\) in Equation (4), the relation for the control stick force becomes  
\[F=\frac{1}{2}\frac{\delta_{e}}{l_{s}\delta_{s}}C_{h_{e}}\rho V^{2}s_{e}c_{e}. \tag{6}\]  
In order to mimic the forces experienced by an operator of a manned aircraft in a flight simulation, forces are calculated and rendered using the relation expressed in Equation (6). It can be seen that the magnitude of force increases with the UAV size and square of speed. The control parameter in our case is \(\delta_{e}\) in Equation (2), which depends on the control stick deflection simulated by the haptic device. \(\delta_{e}\) is updated based on the input from the operator and flight parameters (forward and vertical velocity, pitch rate, angle and altitude) are calculated by solving the first order differential equations (FODEs) of the mathematical model. Pitch angle \(\theta\) is passed to the UAV model for the visualization of pitch rotation. The speed of the UAV is obtained from the forward and vertical velocities \(u\) and \(w\), respectively. Similarly, the altitude of the UAV is determined from Equation (2) using FODE. These flight parameters are then passed to the graphics and force rendering engines to update the visualization in real time and render forces through the haptic device to the operator. The graphical user interface shown in Figure 4 consists of the UAV model, an attitude indicator used to give immediate and direct indication to aircrafts pitch and bank in reference to an artificial horizon, and an altimeter to show the altitude of the UAV. The workflow of the developed system is shown in Figure 5. The operator continuously adjusts the UAV controls, using the control stick modeled using a 6 DOF haptic device, the motions of which are constrained to mimic those of the control stick of an aircraft.