## 2 Materials and Methods
### Control Architecture  
The physical representation of the proposed system is shown in Figure 2a. As can be observed, a PC is used to process the grabbed images from the camera mounted on-board of the MAV though the video transmitter (Eachine TX801 5.8 GHz) and the command signals to the MAV through a bi-directional USB radio communication (Crazyradio PA 2.4 GHz). After computing in the PC the camera posture estimation and identifying the ArUco marker, the relative MAV posture is then given as new measurement data to the linear Kalman filter algorithm. As shown in Figure 2, the estimated pose of the MAV is then used as a reference signal to the compute the error signal. In order to reduce the error, the PID controllers the compute the commanded control signal in order to align the MAV pose towards the target location. As a first approach, we only considered the yaw rotation (\(\Psi\)-angle) and the \(x\) and \(z\) translations. The on-board controller consists of two PID controllers in a cascade. A rule of thumb for cascaded PID controllers is that in order to have a stable and robust control scheme, the inner loop should operate at a higher frequency than the outer loop [13]. If the outer loop runs at a lower frequency than the inner loop, synchronization problems may occur, which entails in that the steady state value of the outer loop is reached before the inner loop has computed its output response, causing instability [13]. In this case, the on-board attitude and attitude rate controllers operate at 250 and 500 Hz, respectively, and the off-board controllers operate at about 10 Hz. The set of off-board PID controllers has three different command variables: the relative yaw angle and translation in x- and z direction. In order to determine the best suitable experimental tuning method for the PID controllers, three different methods  
Figure 2: (**a**) Overview of the proposed off-board vision-based control system; (**b**) schematic diagram for the vision-based control of the MAV. were tested: Ziegler-Nichols, Lambda, and Approximate M-constrained Integral Gain Optimisation (AMIGO). The Ziegler-Nichols method was published in 1942 [14] and was developed by performing a large number of simulations on pneumatic analogue machines [15]. The parameters are determined from a step response analysis according to the Table 1. The AMIGO method was introduced in 2004. It has similarities to the Ziegler-Nichols method, but is instead based on simulations performed with computers [16]. The parameters are determined from a step response experiment according to Table 2. The lambda method differs from the methods above by a parameter that the user can adjust themselves. The method is widely used in the paper industry and was developed in the 1960s. The original method only dealt with the PI controller, but it is possible to derive formulas for PID controllers as well. The lambda method does not provide parameters for integrating processes [15]. For the PID controller in parallel, the parameters are taken from a step response according to Table 3.  
As is shown in Tables 1-3, the times calculated are T\({}_{63\%}\), T\({}_{100\%}\), and \(L\). \(L\) is the dead time of the system and this is calculated by determining the point of intersection of the previously calculated maximum derivative and the time axis. T\({}_{100\%}\) is obtained by calculating the intersection point of the maximum derivative with the line Ys, then obtained a time that is T\({}_{100\%}+L\). To get T\({}_{100\%}\), \(L\) is then subtracted. T\({}_{63\%}\) is obtained by examining when the first measurement data point exceeds the value of 0.63 Ys. The time for that measurement data point is then time T\({}_{63\%}+L\), the same as for T\({}_{100\%}\), the dead time \(L\) must be subtracted away. In the lambda method, the factor T\({}_{\rm cl}\) = T\({}_{63\%}\) will be used to obtain an aggressive regulation similar to the results of Ziegler-Nichols and AMIGO.