Lagrangian approach to origami vertex analysis: kinematics

Area of Science:

• Engineering
• Mathematics
• Robotics

Background:

• Origami principles are increasingly applied in engineering for deployable structures, robotics, and metamaterials.
• Kinematic analysis of origami is challenging due to nonlinear equations and multiple degrees of freedom.

Purpose of the Study:

• To derive reduced-order compatibility conditions for rigid-facet origami vertices.
• To obtain exact, multi-degree of freedom kinematic solutions for symmetric vertices.
• To develop a systematic method for constructing lower-symmetry solutions.

Main Methods:

• Leveraging a Lagrangian approach to derive compatibility conditions.
• Deriving exact kinematic solutions for degree 6 and degree 8 vertices.
• Investigating kinematic topology and self-contact constraints.
• Developing a symmetry-breaking procedure for kinematic solutions.

Main Results:

• Reduced-order compatibility conditions for symmetric origami vertices.
• Exact, multi-degree of freedom solutions for degree 6 and 8 vertices.
• Efficient investigation of kinematic topology and design parameter influence.
• A procedure to construct lower-symmetry kinematic solutions.

Conclusions:

• The derived solutions enhance understanding of origami vertex kinematics.
• This work facilitates the development of algorithmic design procedures for origami engineering.
• The findings support the creation of novel deployable structures and folding robots.