We propose a novel singularity-avoidance method for continuous path planning that maintains zero path-tracking error of a non-redundant free-floating space robot (FFSR). Singularities are a major challenge in the path planning of space robots and may lead to the failure of end-effector tracking tasks. First, the singularity characteristics of FFSRs are analyzed, and a numerical method for calculating singularity surfaces is developed. The symmetry properties of the singular surface are further investigated. Subsequently, based on the nonholonomic characteristics of FFSRs, the existence of quasi-self-motion is demonstrated. A quasi-self-motion-based singularity-avoidance method is then proposed for continuous end-effector pose tracking. In addition, the workspace regions where quasi-self-motion does not exist are analytically derived. Numerical simulations verify the effectiveness of the proposed method. Compared with existing singularity-avoidance approaches, the proposed method achieves accurate path tracking while enabling both active singularity avoidance and singularity escape.