## 1 Introduction
Rendezvous and docking could be seen as a planned collision of two spacecraft, controlled considering the geometric location of the contact points on the two vehicles, and the linear velocities and angular rates at contact. Most of the literature assumes that the Chaser may access any region of the maneuver in the space. However, many scenarios involve operations near large space structures or, due to the increase of space debris, in proximity to obstacles [12]. Consequently, one of the essential requirements for automated rendezvous operations is the ability to maneuver in proximity to them, without collision. The motivation of this research is to safely reach the Target without collisions, designing an analytical and computationally efficient (real time) method. For close range maneuvers, the guidance and control (GC) algorithms must be able to handle actuator constraints as well as dynamic and obstacle constraints. For this reason, the guidance algorithms for the complete rendezvous maneuver are based on the theory of artificial potential fields. Artificial potential fields (APFs) have evolved in the past years for guiding the motion of mobile robots. The main idea of the artificial potential field theory is to construct a potential field with a gradient acting attractively toward the goal and repellently from obstacles. Different approaches using artificial potential fields are proposed in literature. An obstacle avoidance algorithm for a rendezvous maneuver is proposed in [13]. Two limitations of this approach are: (i) an \"ad hoc\" method for guidance is proposed and (ii) the control algorithm works at high frequency. Starting from this work in [14], a method for spacecraft maneuver with obstacles is proposed, but no control algorithms are considered and a simple dynamics is analyzed. In a similar way, real-time on-board execution of APF as guidance scheme is analyzed in [15]. More recently, interesting applications of APF as guidance algorithms are proposed in [16; 17]. In [16], an adaptive artificial field for proximity operations is considered and the computational efficiency is proven in an experimental testbed. In [17], a near optimal hybrid guidance method using APF is proposed, combined with a sample based path planning method. In both cases, the APF algorithm is used as a GNC system and an \"ad hoc\" field is generated. Moreover, only a forced motion is considered. In our work, feedback control based on SMC theory is designed for tracking the gradient, generated by the APF algorithm. Our idea is to propose a method for an orbital spacecraft maneuver in which obstacles and external disturbances are considered. Moreover, the proposed algorithm is able to guarantee real time applications, considering the requirements of current space hardware (limitations of the sample frequency). In [PERSON] et al. [18] an optimal sliding mode combined with an APF algorithm is considered. In this research, only proximity maneuver and orbital dynamics are analyzed, even if external disturbances are included. In detail, the desired velocity and the desired attitude are defined by the APF algorithm to track the desired trajectory and avoid obstacles. This interpretation of the gradient was proposed in [19], in which a sliding mode control strategy for tracking the gradient due to a artificial potential field is described. In [20], the problem of controlling an autonomous wheeled vehicle with a SMC is proposed, including collision avoidance. The use of SMC in this context proves to be appropriate because of its robustness features. Usually, pseudo-sliding mode control, as in [21], based on continuous approximations, are proposed for mechanical systems. For spacecraft maneuver and, in particular, for position dynamics, a conventional first order SMC is the most suitable, due to the thrusters, for which the designed control signals are discontinuous.