The deployment of large space structures is a key issue in the future development of space technology. On-orbit assembly using modular spacecraft is one of the mainstream solutions. Ensuring the safety of this mission requires precise collision avoidance control during the close-proximity approach of multiple spacecraft. Most of the existing research on spacecraft on-orbit assembly uses spherical envelopes to approximate the shape of the spacecraft, which cannot accurately describe the geometric features of the spacecraft. To meet the demand for precise collision avoidance in motion planning, a control method based on the distance gradient of superquadrics is proposed. Compared to spherical envelopes, superquadrics provide a more accurate convex representation of a spacecraft's shape. The closest distance between spacecraft is determined by solving an optimization problem, enabling accurate collision detection. Furthermore, a sensitivity analysis of the optimization problem is carried out to obtain the gradient of the closest distance with respect to the spacecraft's pose. This gradient is then introduced into an artificial potential function to establish the collision avoidance and assembly control law. Finally, the effectiveness of the proposed method is verified through numerical simulations of multi-spacecraft collision avoidance and assembly control.