Active Origami: Modeling, Design, and Applications

Preface
Contents
List of Symbols
1 Introduction to Active Origami Structures
	1.1 Origami Structures
	1.2 Active Origami Structures
		1.2.1 Active Materials
		1.2.2 Review of Active Origami Structures
			1.2.2.1 Thermally Activated Origami Structures
			1.2.2.2 Chemically Activated Origami Structures
			1.2.2.3 Electromagnetically Activated Origami Structures
	1.3 Origami Design
	1.4 Simulation and Visualization of Origami Structures
	Chapter Summary
	Problems
	References
2 Kinematics of Origami Structures with Creased Folds
	2.1 Introduction
	2.2 Fundamental Concepts
	2.3 Fold Pattern Description
	2.4 Kinematic Constraints for Origami with Creased Folds
		2.4.1 Developability Constraint
		2.4.2 Loop Closure Constraint
	2.5 Folding Map Formulation
		2.5.1 Parameters Required to Derive the Folding Map
		2.5.2 Folding Map Formulation
	2.6 Computational Implementation of the Model
	2.7 Simulation Examples of the Kinematic Model
	Chapter Summary
	Problems
	References
3 Unfolding Polyhedra Method for the Design of Origami Structures with Creased Folds
	3.1 Introduction
	3.2 Unfolding Polyhedra Method Considering Creased Folds
		3.2.1 Problem Definition
		3.2.2 Goal Mesh Description
		3.2.3 Determination of Spanning Trees
		3.2.4 Formulation of the Unfolding Map
		3.2.5 Determination of Folding Motion
		3.2.6 Limitations of the Unfolding Polyhedra Method
	3.3 Examples of the Unfolding Polyhedra Method
	Chapter Summary
	Problems
	References
4 Tuck-Folding Method for the Design of Origami Structures with Creased Folds
	4.1 Introduction
	4.2 Tuck-Folding Method Considering Creased Folds
		4.2.1 Problem Definition
		4.2.2 Goal Mesh Description
		4.2.3 Edge Module Parameterization and Constraints
			4.2.3.1 Loop Closure Constraints
			4.2.3.2 Constraints for Valid Edge Module Geometry
			4.2.3.3 Constraints to Prevent Intersections Among Tuck-Folded Edge Modules
			4.2.3.4 Summary of Design Constraints
		4.2.4 Edge Module Trimming
		4.2.5 Determination of Design Variables
		4.2.6 Determination of Folding Motion
		4.2.7 Design Requirements of the Tuck-Folding Method
	4.3 Examples of the Tuck-Folding Method
	Chapter Summary
	Problems
	References
5 Kinematics of Origami Structures with Smooth Folds
	5.1 Introduction
	5.2 Fundamental Concepts
	5.3 Shape Formulation of Smooth Folds
		5.3.1 Continuity Conditions for Smooth Folds
		5.3.2 Fold Parameterization Examples
	5.4 Fold Pattern Description
	5.5 Kinematic Constraints for Origami with Smooth Folds
		5.5.1 Developability Constraint
		5.5.2 Loop Closure Constraints
	5.6 Folding Map Formulation
		5.6.1 Parameters Required to Derive the Folding Map
		5.6.2 Folding Map Formulation
	5.7 Computational Implementation of the Model
	5.8 Simulation Examples of the Kinematic Model
	Chapter Summary
	Problems
	References
6 Unfolding Polyhedra Method for the Design of Origami Structures with Smooth Folds
	6.1 Introduction
	6.2 Unfolding Polyhedra Method Considering Smooth Folds
		6.2.1 Problem Definition
		6.2.2 Face Trimming Step
	6.3 Examples of the Unfolding Polyhedra Method
	Chapter Summary
	Problems
	References
7 Tuck-Folding Method for the Design of Origami Structures with Smooth Folds
	7.1 Introduction
	7.2 Tuck-Folding Method Considering Smooth Folds
		7.2.1 Problem Definition
		7.2.2 Face Trimming Step
		7.2.3 Edge Module Parameterization and Constraints
			7.2.3.1 Loop Closure Constraints
			7.2.3.2 Constraints for Valid Edge Module Geometry
			7.2.3.3 Constraints to Prevent Intersections Among Tuck-Folded Edge Modules
			7.2.3.4 Summary of Design Constraints
	7.3 Examples of the Tuck-Folding Method
		7.3.1 Design and Fabrication of Shape Memory Polymer Self-Folding Sheets
			7.3.1.1 Fabrication of Shape Memory Polymer Active Folds
			7.3.1.2 Self-Folding Behavior of Shape Memory Polymer Sheets
	Chapter Summary
	Problems
	References
8 Structural Mechanics and Design of Active Origami Structures
	8.1 Introduction
	8.2 Kinematics of Origami Structures with Smooth Folds of Non-Zero Thickness
	8.3 Structural Mechanics Modeling Approach
		8.3.1 Conservation of Linear and Angular Momentum
		8.3.2 Constitutive Equations
		8.3.3 Boundary Value Problem
		8.3.4 Variational Formulation
	8.4 Structural Mechanics Model Formulation
		8.4.1 Model Development
		8.4.2 Numerical Implementation
			8.4.2.1 Quadrature Rules for Numerical Integration
	8.5 Examples of the Implemented Model
		8.5.1 Examples of Structures Having One Fold
		8.5.2 Examples of Structures Having One Fold Intersection
		8.5.3 Examples of Structures Having Multiple Fold Intersections
		8.5.4 Computational Efficiency Comparison
	8.6 Unfolding Polyhedra Method for the Design of Self-Folding Structures
	8.7 Tuck-Folding Method for the Design of Self-Folding Structures
		8.7.1 Design of a Self-Folding Parabolic Antenna Using the Tuck-Folding Method
			8.7.1.1 Antenna Design Problem
			8.7.1.2 Results of the Antenna Design Exploration Study
	Chapter Summary
	Problems
	References
Appendix A Notation and Useful Formulas
	A.1 Vectors in Three-Dimensional Space
	A.2 Vectors of Arbitrary Dimensions
	A.3 Matrices of Arbitrary Dimensions
	A.4 Block Matrices
	A.5 Systems of Linear Equations
	Problems
Appendix B Examples of Implementation Codes
	B.1 Implementation of the Kinematic Model for Origami with Creased Folds (Chap.2)
	B.2 Implementation of the Unfolding Polyhedra Method for Origami with Creased Folds (Chap.3)
	B.3 Implementation of the Tuck-Folding Method for Origami with Creased Folds (Chap.4)
	B.4 Implementation of the Kinematic Model for Origami with Smooth Folds (Chap.5)
	B.5 Implementation of the Unfolding Polyhedra Method for Origami with Smooth Folds (Chap.6)
	B.6 Implementation of the Tuck-Folding Method for Origami with Smooth Folds (Chap.7)
Appendix C Constitutive Models
	C.1 Linear Elastic Materials
	C.2 Thermoelastic Materials
	C.3 Piezoelectric Materials
	C.4 Phase Transforming Materials
	References
Index