## 3 Guidance Algorithms  
The rendezvous maneuver requires, in the presence of obstacles, to be performed completely autonomously with only sensors and on-board GC algorithms. Moreover, the GC algorithms must be able to simultaneously achieve a series of translational maneuvers in the presence of external disturbances and uncertainties. The potential field method can produce low impulsive fuel and can be implemented online with a low computational effort. At each time step, the artificial potential field algorithm generates the desired velocity and orientation required for reaching the Target. The desired vectors are generated considering the end of each maneuver as a minimum and the obstacle as a maximum and are not known a priori (no predetermined trajectory is considered). A paraboloid artificial potential field is considered because it is a stabilizing function, and we impose that, as the Chaser approaches the goal, its speed decreases. Two different artificial potential fields are considered: (i) one is related to the position dynamics and (ii) the second is related to the attitude dynamics. For the position dynamics, the artificial potential field is attractive to the goal and repulsive with respect to the obstacles. As in Figure 5. The attractive potential field is  
\[U_{a}(\mathrm{x})=\tfrac{1}{2}\mathrm{k_{a}}||\mathrm{e}( \mathrm{x})||^{2} \tag{10}\] \[f_{a}(\mathrm{x})=-\bigtriangledown U_{a}(\mathrm{x}), \tag{11}\]where \(k_{a}\) defines how fast the attractive gradient goes to the goal and \(e(x)\) is the error in position in which \(x=(x,y,z)^{T}\in\mathbb{R}^{3}\). The attractive force is due to the gradient of the artificial potential field. To assign the direction of the desired speed, a unit vector of the potential field is evaluated  
\[E_{U}=\frac{f_{a}(x)}{||\bigtriangledown\mathcal{U}_{a}(x)||},\]  
thus the desired speed \(\hat{x}_{d}\in\mathbb{R}^{3}\) is evaluated as  
\[\hat{x}_{d}=\hat{x}_{d,max}E_{U},\]  
where \(\hat{x}_{d,max}\) is the maximum speed to perform the maneuver, which is scalar and equal along the three axes. To avoid the obstacles, a repulsive potential field is defined, one for each obstacle (\(i=1,\ldots N_{obs}\) with \(N_{obs}\) number of obstacles)  
\[U_{r},i(x)= \begin{cases}\frac{k_{r,j}}{\gamma}(\frac{1}{\eta_{i}(x)}-\frac{1} {\eta_{0,i}(x)})^{\gamma}&\text{if }\eta_{i}(x)\leq\eta_{0,i}(x)\\ 0&\text{if }\eta_{i}(x)>\eta_{0,i}(x)\end{cases} , \tag{12}\]  
where \(k_{r,i}\) is the gain related to the repulsive field, \(\gamma=2\) is defined for hyperbolic field, \(\eta_{i}(x)=\min_{\varsigma_{obs}\in\text{CO}^{i}}||x-x_{obs}||,x_{obs}\in \mathbb{R}^{3}\) is the obstacle position, and \(\eta_{0,i}(x)\) is the safety radius. \(CO^{i}\) is the convex set of obstacles. The repulsive field is defined for each obstacles, which are assumed convex. As before, the repulsive force is  
\[f_{r,i}(x)=-\bigtriangledown\text{U}_{r},i(x)= \begin{cases}\frac{k_{r,j}}{ \eta_{i}^{*}(x)}(\frac{1}{\eta_{i}(x)}-\frac{1}{\eta_{0,i}(x)})^{\gamma-1} \bigtriangledown\eta_{i}(x)&\text{if }\eta_{i}(x)\leq\eta_{0,i}(x)\\ 0&\text{if }\eta_{i}(x)>\eta_{0,i}(x)\end{cases} . \tag{13}\]  
The radius \(\eta_{0,i}\) for \(i=1,\ldots N_{obs}\) is the safety radius and it means that the Chaser \"senses\" the obstacle when it is \(\eta_{0,i}\) m away from the obstacle. The applied artificial potential field is the sum of the attractive and repulsive part  
\[U_{t}(x)=\text{U}_{a}(x)+\sum_{i=1}^{N_{obs}}\text{U}_{r;i},\]  
and the total vector indicating the motion direction, opposite of the artificial potential field, is  
\[E(x)=-\bigtriangledown U_{t}(x),\]  
with \(E(x)\in\mathbb{R}^{3}\). Figure 5: Attractive and repulsive potential field. As explained before, the desired behavior is reached when the speed vector \(\dot{x}\in\mathbb{R}^{3}\) is collinear with \(E(x)\).