Mathematical Evolution of Origami Structures and Their Applications in Soft Robotics

Origami, as an ancient art form, has become a critical design paradigm in science and engineering. While numerous studies have explored various origami structures and materials, they often overlook the fundamental principles that unite them. This review seeks to bridge that gap by examining origami engineering through the perspective of mathematical evolution. We outline a continuous developmental process, from kinematics-driven planar patterns to innovative non-rigid, energy-responsive systems and mechanics of active materials. A thorough comparison of key origami families reveals the essential balance between geometric simplicity and mechanical complexity. Furthermore, we introduce advanced modeling and design techniques, including numerical simulation, inverse design, and active material mechanics, particularly in the application of soft robotics. This study establishes a foundational framework for future research and inspires the design of sophisticated, functional systems. Note to Practitioners—Origami structures, particularly those exhibiting bistable properties, exhibit potential in diverse fields such as soft robotics, deployable systems, and flexible electronics. However, the design and optimization of these structures for real-world applications are still challenging. This paper transcends the categorization of origami patterns by uncovering the evolutionary connections among various origami families, providing a more in-depth theoretical foundation for selecting appropriate structures. Another focus of this work is a critical analysis of engineering trade-offs, particularly between deployability and stiffness, which are essential for optimizing applications in dynamic environments like soft robotics. We also provide a comprehensive overview of state-of-the-art modeling tools and introduce innovative reverse design strategies that streamline the transition from performance specifications to conceptual prototypes. For practitioners in the field of robotics, this review investigates the integration of stimulus-responsive materials and the associated modeling challenges, aiming to facilitate a deeper incorporation of origami technology into engineering practice.