Robots with openness and flexibility have attracted a large number of researchers to conduct in-depth studies in the field of surface machining. However, there is a redundant degree of freedom (DOF) in 6-DOF robot machining: when a ball end milling cutter is used to process curved parts, the tool point needs to strictly follow the planned milling trajectory, but the tool axis vector only needs to be within a certain range. During the machining process, the rotation of the tool around its axis is not constrained. Therefore, it is necessary to optimize the redundant DOF. Aiming at the redundant DOF of the tool axis vector in ball end milling for surface parts, a Redundancy Optimization strategy for Minimum Joint trajectory (ROMJ) is proposed. It takes the shortest trajectory of robot joints as the optimization objective, and the numerical optimization method is adopted to carry out the optimal design of tool axis vector trajectory in the milling process. Before optimization, to decrease the data volume, the number of track points is sampled and adjusted based on curve characterization errors. In the optimization process, considering the obvious difference in the load quality characteristics of the robot joints, a Redundancy Optimization strategy for Minimum Joint trajectory considering the different Load of joints (ROMJ-L) is proposed. The load difference coefficients of each joint are introduced into the optimization objective of the trajectory of robot joints. By using this method, the optimal design of each joint trajectory of the robot is realized. In order to verify the methods proposed in this paper, a comparison experiment is carried out. The results show that under the same tool point trajectory, the proposed methods can significantly reduce the robot joint trajectory, and the joint trajectory is influenced by the load difference of each joint. Finally, an Eflin-10 robot is used to process the butterfly trajectory tool path by the trajectory planned by the ROMJ-L method, and the results show that the method is practical.