Optimum design is proven significant for improving task performances of robotic manipulators under certain constraints. However, when it is utilized for collaborative robots (Cobots), there are still many challenges such as complex smooth surface links, time-varying kinematic configurations, computational expensiveness, and nonstructural parameter optimization. Therefore, based on orthogonal design experiment (ODE) and finite element substructure method (FESM), a multi-objective optimum design method of Cobots is proposed with the structural dimensions and parameterized joint components as the optimization variables and the natural frequency, the Cartesian stiffness, and the mass of the robot as optimization objectives. Firstly, to obtain multiple global performance indexes (GPIs) of robots in real-time and efficiently, the FESM model of Cobots is established which can preserve the accuracy of the finite element method (FEM) while ensuring the computational efficiency. Then, the gray relational analysis method (GRAM) is used to construct the multi-objective optimization function which includes the global first-order natural frequency index (GFNFI), the global elastic deformation index (GEDI), and the mass of robots. The ODE is constructed, and the structural dimensions and parameterized joint components are taken as influencing factors. According to the orthogonal array (OA), the degree of gray incidence under different levels of influencing factors is solved. And the optimal combination of influencing factor levels is obtained by range analysis (RA), which is used to guide the design of Cobots. Finally, a Cobot SHIR5-I is taken as an illustrative example to perform optimum design in this paper.