## 1 Introduction
Besides, some modified APF methods were proposed to give more precise constraints of trajectory of proximity, such as sigmoid function-based APF [18], Cissoid-based APF [19; 20], and harmonic functions based APF [21]. This paper proposed Gaussian Mixture model (GMM)-based APF, which combines the characteristics of the two methods. GMM has been widely used in estimation of space object collision probability [22], uncertainty propagation [23; 24], trajectory optimization [25], model construction [26], and image reconstruction [27]. Based on GMM, an equal-collision-probability-curve method is proposed for safe spacecraft close-range proximity maneuvers, which uses the collision-probability to describe the space around the target spacecraft for safety problem [28; 29]. In this paper, the GMM is used to reconstruct the complex shape of spacecraft, and a novel APF is built to constrain the motion trajectory. Compared with existing modified APF, the GMM-based APF can depict obstacle constraints of arbitrary targets with complex shape. Despite the fact that proximity operation task is performed with the aforementioned control schemes, the convergence or finite-time convergence can only be guaranteed asymptotically. Time for stabilization is seriously affected by the initial conditions of the system [30]. Nevertheless, time constraints are often needed to be considered in on-orbit service. Therefore, a prior precise estimation of the settling-time is necessary. The notion of fixed-time stability (FTC) has been proposed to provide a desired finite time convergence regardless of initial condition, which features that the bound of settling-time is just related to the control gains [31]. The characteristics of fixed-time stability have attracted many studies and are widely used in different control scenarios. For example, [PERSON] et al. [32] summarized the latest progress in applying fixed-time cooperative control to multi-agent systems. [PERSON] et al. [33] discussed the local and global fixed-time stability respectively for a port-Hamiltonian system based on interconnected and damped distributed passive control technology, and defined the fixed-time stable area and stable boundary. In terms of spacecraft on-orbit control applications, [PERSON], [PERSON], [PERSON] et al. [34; 35; 36] designed an adaptive nonsingular fixed-time sliding mode controller with the consideration of system uncertainty, external interference, actuator failure, and input saturation. Fixed-time convergence of the controller can be guaranteed. [PERSON] et al. [37] designed a sliding mode controller for spacecraft system with flexible attachments, which can restrain disturbance and make the attitude-tracking error converge to origin in a fixed time. [PERSON] et al. [38] designed a closed-loop control law based on non-linear feedback FTC for spacecraft fixed-point hovering, and the principle of fixed-time stabilization had been described and proved. Considering the control problem of presupposed time transfer operation with tumbling target, [PERSON] et al. [39] designed a time-varying sliding mode surface, and a control scheme with sliding manifold was proposed to achieve fixed-time convergence of relative parameter tracking error. [PERSON] et al. [40] designed a non-singular terminal sliding mode controller with fixed-time convergence for approaching non-cooperative spacecraft with disturbance and parameter uncertainty, which can achieve timing tracking control, as well as avoid singularity and chattering. The study of this paper focuses on control design for safe proximity to a rotating spacecraft with complex shape so that superior features including collision avoidance and fixed-time convergence property. Motivated by the discussion above, the principal contributions of this paper can be outlined as follows:  
1.