To solve the limitations in the abilities of aerial robots with legs/claws in grasping and perching on complex objects, a bionic raptor robot that can be integrated with a robot body and has the ability to reliably grasp and maintain stable perching in unstructured environments was presented in this paper. In this study, a steady-state perching model of a robot was established, and the optimal leg sizes with the minimum overturning moment were obtained. Angular momentum and linear momentum models of the robot were established, and a successful perching parameter domain satisfying the overturning moment and momentum constraints, which were generated by a combination of the three-dimensional parameters of the centroid velocity angle, centroid velocity, and leg angle, was obtained. A pair of under-actuated grasping legs/claws was designed to passively transform instantaneous impact energy into a strong grasping force and passively maintain a reliable grasping state by self-locking. In the perching experiments and perching motion simulations of this robot prototype, the kinematic parameters changed consistently and steadily. The above achievements confirm the rationality of the steady-state perching model, momentum model, and robot structural design scheme proposed in this study and lay theoretical and methodological foundations for the subsequent realization of the real-time position control of a bionic raptor perching robot and its wide applications in remote sensing, search and rescue, and environmental monitoring in unstructured environments.