## 5 Numerical Simulations  
As already mentioned in the introduction, the idea of this paper is to combine APF and SMC controllers and to replan online the trajectories in presence of obstacles, verifying the computational effort required for these algorithms. The simulation model is tested for a generic Chaser-Target combination involved in sequential flight phases, as in Figure 1. The Chaser is considered in stable initial conditions along an orbit of height \(h=500\) km, and the Target center of mass is located at the origin of the LVLH reference frame, and it is 16 km far from the Chaser. A cubic-shape Chaser (1.2 m) is considered with an initial mass of 600 kg, as in Table 1. The location of the four waypoints is summarized in Table 2.  
The proposed approach is compared with the combination of the APF and Linear Quadratic Regulator (LQR) controller for the first phase of the maneuver. An LQR controller is designed for the position dynamics, instead a constant attitude (i.e., the Body and LVLH frame aligned) is considered for this combination. External disturbances and sensor noise are included for both cases. The maximum value of the thruster errors (bias and random values) is equal to the 10% of the maximum thrust. The gain matrix of the LQR controller is set considering an ideal impulsive Hohmann maneuver. For this reason, the end point of the maneuver is about \(-8\) km, as in Figure 6. A PWM behavior is not evaluated for the LQR controller, so a continuous thrust is assigned. If an obstacle is included in the evaluation of the first maneuver, we can observe that, even if the safety radius is 1000 m, the position dynamics is deviated only when the Chaser is inside the safety zone, with a speed on the X axis that is about 6 m/s. A zero altitude is reached as desired. The ideal maneuver profile described in both Figure 1 and Table 2 is affected by three obstacles that the Chaser shall safely avoid, thanks to the artificial potential field algorithm definition. Obstacles are defined considering a center of application and a safety radius, as described in Section 3. As first approximation, all the obstacles are considered in a fixed known position. Their location is summarized in Table 3 and different safety radii are considered. \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Symbol** & **Unit** \\ \hline Initial mass & \(m_{c0}\) & 600 kg \\ Initial inertia tensor & \(l_{0}\) & 14413 kgm\({}^{2}\) \\ Zero shoot time of thrusters & \(t_{sp0}\) & 0.02 s \\ Maximum thrust & \(T_{max}\) & 1 N \\ Specific impulse of thrusters & \(l_{sp}\) & 220 s \\ RW Maximum torque & \(g_{max}\) & 1 Nm \\ RW Inertial tensor & \(l_{RW}\) & 0.113 kgm\({}^{2}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Chaser and Thruster characteristics. RW: reaction wheel. \begin{table}
\begin{tabular}{c l l} \hline
**Waypoint** & **Description** & \multicolumn{1}{c}{**Position (m)**} \\  & & \([X,Y,Z]_{LVLH}\) \\ \hline S0 & Initial simulation waypoint & \([-16,100,\ 0,\ 3000]\) \\ S1 & Initial waypoint for Hohmann & \([-16,600,\ 0,\ 3000]\) \\ S2 & Terminal waypoint for Hohmann/Initial waypoint for Radial Boost & \([-3000,\ 0,\ 0]\) \\ S3 & Terminal waypoint for Radial Boost/Initial waypoint for Straight Line & \([-500,\ 0,\ 0]\) \\ S4 & Terminal position & \([x_{target},y_{target},z_{target}]\) \\ \hline \end{tabular}
\end{table} 
Table 2: Waypoint LVLH position. Figure 6: First maneuver highlights for the LQR (Linear Quadratic Regulator) controller. (**a**) First Maneuver: \(V_{BAR}-R_{BAR}\) plane. The red circle represents the obstacle (dotted line = safety radius, red circle = obstacle center). (**b**) Relative velocity. The simulated maneuver is depicted in Figure 1. The resulting maneuver is substantially different from the nominal one (Figure 1) because the Chaser deviates its path from the ideal one to avoid obstacles along the nominal path. The ellipsoidal shape of the radius of influence of the obstacles is due to the axis scaling. The three phases of the maneuver are separately discussed.