Aerospace represents the development of national science and technology. It is an important foundation for exploring space and an important guarantee for the construction of aerospace power. There are many large workpieces in the aerospace field. The box insulation layer of large workpieces is an important processing problem. A new thick processing equipment is proposed to process the box insulation layer of large workpieces. The thick processing equipment consists of the XYZ shaft long guide rail and five degrees of freedom (5-DOF) RAPA. The mechanical structure of the 5-DOF RAPA is a redundantly actuated parallel mechanism (RAPM). Meanwhile, this paper proposes a new method to design 5-DOF redundantly actuated parallel mechanisms (RAPMs) with large output rotational angles. Based on configuration evolution and Li group, two articulated moving platforms (AMPs) and four kinds of limbs are designed, and a series of 3T2R (T represents translation, R represents rotation) RAPMs and 2T3R RAPMs are synthesized. To verify the designed RAPMs with large angle, an example of RAPMs, 4UPS-{2UPR}-R is analyzed. To ensure that the RAPM has no mechanism vibration impact in movement, this paper represents the RAPM adopts a newly proposed trajectory planning method. The results show that the 4SPU-(2UPR)R mechanism possesses large angles and verifies the efficiency of the new proposed trajectory planning method in simplified trajectories. This work lays the foundation for processing the box insulation layer of large workpieces with straight lines and arcs paths.