The soft pneumatic actuator (SPA) has strong flexibility and good compliance, allowing its assembly into a multi-functional soft gripper that can be applied to object grasping and handling, post-disaster rescue, rehabilitation training, and in other fields. However, many actuators have limited grasping modes and can only bend in a single direction. And most of the soft grippers are studied in free space for grasping. Herein, a novel 3D-SPA with internal bellows chambers, inspired by the structure of the bird claws and the human fingers, is presented. Based on Yeoh hyperelasticity theory and virtual work principle, the mechanical theoretical model between different input air pressure Pin and the bending angle θ of the actuator is established, and the spatial conversion relationship of obstacle avoidance motion is given. The influence of the length and number of the chamber, the radius of the circular section, the shape of the cross section and the length of the joint on the bending performance and stress distribution of the actuator are studied in detail, respectively. Two soft grippers with different structures, composed of 3D-SPA, have a variety of bending configurations on different planes, which improves the inclusiveness of grasping objects with different shapes and sizes. Finally, the process of object grasping in free space and obstacle avoidance grasping and handling in obstacle space are studied respectively. The validity of the proposed mechanical model and spatial transformation relationship are verified. The investigation results reveal that the developed scheme assists the robot’s obstacle avoidance and grasping, which can be integrated into industrial robotic arms and AGVs for mobile operations and applied to rehabilitation training to help patients gradually recover hand joint function.