In recent years, the field of robotics has seen a surge in interest for deployable and reconfigurable systems, particularly those inspired by origami. Origami robots, which utilize folding principles to achieve complex morphologies and functions, offer significant advantages in terms of compactness, lightweight design, and adaptability. Among these, the end effector—a critical component responsible for interaction with the environment—benefits greatly from origami-based approaches. Traditional end effectors often face limitations in multi-degree-of-freedom control, assembly complexity, and spatial efficiency. To address these challenges, we propose a novel space-foldable end effector that leverages rigid origami mechanisms combined with single-loop closed-chain structures. This design enables single-degree-of-freedom actuation to control multiple folding angles, simplifying control systems while maintaining high functionality. In this article, we delve into the design, analysis, and validation of this end effector, emphasizing kinematic and dynamic aspects through mathematical formulations, tables, and simulations. The key innovation lies in using external linkage mechanisms to indirectly control rigid origami folds, reducing the need for multiple actuators and enabling reversible folding and unfolding. Throughout this discussion, the term “end effector” will be frequently highlighted to underscore its central role in our design. Our work aims to contribute to advancements in robotic manipulation, especially in applications like space exploration, minimally invasive surgery, and adaptive grasping, where compact and versatile end effectors are paramount.
The concept of rigid origami involves treating foldable panels as rigid links and creases as revolute joints, allowing for precise kinematic modeling. This approach has been widely adopted in deployable structures, but controlling multiple folds simultaneously often requires complex actuation. Our end effector design circumvents this by integrating a single-loop closed-chain mechanism—specifically, a planar-symmetric Bricard mechanism—as an external linkage. This linkage acts as a passive controller, translating a single rotational input into coordinated folding motions across multiple creases. The end effector’s core structure is based on a triangular rigid origami pattern that folds into a tetrahedral shape, suitable for grasping or capturing tasks. By externalizing the control to these linkages, we minimize the number of actuators, reduce system complexity, and enhance reliability. The following sections detail the design methodology, starting with the configuration of the end effector, followed by freedom analysis, kinematic modeling, drive design, simulation experiments, and stress analysis. We incorporate mathematical equations using LaTeX syntax, such as $$ \cos \Phi \cos \theta + \cos \Phi + \cos \theta = 0 $$ for motion relationships, and tables to summarize parameters and results. This comprehensive analysis validates the feasibility of our origami-inspired end effector for practical robotic applications.
The configuration of our space-foldable end effector begins with a flat triangular sheet, analogous to a rigid origami pattern. As shown in the figure below, this sheet features pre-defined creases that enable folding into a tetrahedral structure. The design transforms a two-dimensional plane into a three-dimensional end effector capable of performing tasks like grasping or enveloping objects. The external linkages, composed of three fixed planar-symmetric Bricard mechanisms, are attached to the crease lines. These linkages serve as the actuation interface, where a single rotary driver controls their motion, thereby inducing folding in the origami structure. This configuration eliminates the need for direct actuation on each crease, streamlining the end effector’s mechanism. To quantify the geometry, we define parameters such as link lengths and angles, which are essential for subsequent analysis. For instance, the triangular sheet has side lengths denoted by \( L \), and the crease angles are represented by \( \gamma \). The integration of external linkages ensures that the end effector can transition between a fully flat state (for storage) and a fully folded state (for operation), making it highly adaptable. The following table summarizes key design parameters for the end effector configuration.
| Parameter | Symbol | Value | Description |
|---|---|---|---|
| Side Length | \( L \) | 100 mm | Length of each side of the triangular sheet |
| Crease Angle | \( \gamma \) | 0° to 60° | Folding angle range for operation |
| Link Length | \( l \) | 50 mm | Length of links in the Bricard mechanism |
| Offset Distance | \( h \) | 20 mm | Offset in the Bricard mechanism joints |
| Panel Thickness | \( t \) | 5 mm | Thickness of the rigid origami material |
The freedom analysis of the end effector is crucial to determine its mobility and actuation requirements. We focus on the planar-symmetric Bricard mechanism used as the external linkage. Using screw theory, we model each joint as a revolute pair with specific axes. For a single Bricard mechanism, the kinematic chain consists of six revolute joints. Denote the screw coordinates as \( \mathbf{S}_i \) for \( i = 1, 2, \dots, 6 \). The constraints are derived by finding the reciprocal screws. The public constraint number \( \lambda \) is calculated to be 1, indicating a single common constraint. According to the mobility formula for single-loop closed-chain mechanisms with public constraints:
$$ \lambda + j = 6 + M, \quad \text{for } j > 3 $$
where \( j = 6 \) is the number of joints, and \( M \) is the degree of freedom. Substituting values, we get:
$$ 1 + 6 = 6 + M \implies M = 1 $$
Thus, each Bricard mechanism has one degree of freedom. Since three such mechanisms are fixed together, the combined system retains a single degree of freedom, confirming that one actuator suffices to control the entire end effector. This analysis underpins the simplicity of our design, as the end effector’s multiple folds are driven by a minimal set of inputs. The screw coordinates for the Bricard mechanism are expressed as:
$$ \mathbf{S}_C = (0, 0, 1; 0, 2a, 0) $$
$$ \mathbf{S}_D = (\sqrt{3}, 1, 0; c, -\sqrt{3}c, \sqrt{3}b – a) $$
$$ \mathbf{S}_E = (0, 0, 1; -2b, 0, 0) $$
$$ \mathbf{S}_F = (-\sqrt{3}, 1, 0; c, \sqrt{3}c, a – \sqrt{3}b) $$
$$ \mathbf{S}_G = (0, 0, 1; 0, -2a, 0) $$
$$ \mathbf{S}_H = (0, 1, 0; 0, 0, 0) $$
where \( a \), \( b \), and \( c \) are constants related to link geometry. The reciprocal screws yield the constraint system, leading to the public constraint \( \mathbf{r} = (0, 0, \frac{\sqrt{3}}{a – \sqrt{3}b}; 1, 0, 0) \). This mathematical framework ensures that the end effector’s motion is predictable and controllable.
Kinematic analysis further elucidates the motion characteristics of the end effector. We employ the Denavit-Hartenberg (D-H) parameters to model the Bricard mechanism. The D-H parameters for the six joints are listed in the table below, with symmetry conditions \( \theta_1 = \theta_3 = \theta_5 = \theta \) and \( \theta_2 = \theta_4 = \theta_6 = \Phi \). The transformation matrix between consecutive links is given by:
$$ T_{i(i+1)} = \begin{bmatrix}
1 & 0 & 0 & 0 \\
-d_{i(i+1)} \cos \theta_i & \sin \theta_i & 0 & 0 \\
-R_i \sin \alpha_{i(i+1)} – \cos \alpha_{i(i+1)} \sin \theta_i & \cos \alpha_{i(i+1)} \cos \theta_i & \sin \alpha_{i(i+1)} & 0 \\
-R_i \cos \alpha_{i(i+1)} \sin \alpha_{i(i+1)} \sin \theta_i & -\sin \alpha_{i(i+1)} \cos \theta_i & \cos \alpha_{i(i+1)} & 0
\end{bmatrix} $$
For the closed-loop condition, the product of transformation matrices equals the identity matrix:
$$ T_{61} \cdot T_{56} \cdot T_{45} \cdot T_{34} \cdot T_{23} \cdot T_{12} = I $$
Substituting the D-H parameters and applying symmetry, we derive the kinematic equation:
$$ \cos \Phi \cos \theta + \cos \Phi + \cos \theta = 0 $$
This equation relates the input angle \( \Phi \) (from the actuator) to the output angle \( \theta \) (the folding angle of the end effector). Additionally, the output angle \( \theta \) depends on link lengths \( l \) and \( h \):
$$ h(l + \cos \theta) = (1 – l) \sin \theta \cos \theta $$
These equations allow us to predict the end effector’s configuration for any actuator input, facilitating precise control. The table summarizes the D-H parameters for clarity.
| Joint \( i \) | \( \alpha_{i(i+1)} \) | \( d_{i(i+1)} \) | \( R_i \) | \( \theta_i \) |
|---|---|---|---|---|
| 1 | \( \pi/2 \) | \( l \) | 0 | \( \theta \) |
| 2 | \( 3\pi/2 \) | \( l \) | \( h \) | \( \Phi \) |
| 3 | \( \pi/2 \) | \( l \) | 0 | \( \theta \) |
| 4 | \( 3\pi/2 \) | \( l \) | 0 | \( \Phi \) |
| 5 | \( \pi/2 \) | \( l \) | 0 | \( \theta \) |
| 6 | \( 3\pi/2 \) | \( l \) | \( h \) | \( \Phi \) |
The drive design of the end effector centers on the external linkage approach. By fixing three planar-symmetric Bricard mechanisms together at their input links, we create a unified drive system. A single rotary actuator is connected to the combined input link, which simultaneously drives all three mechanisms. This setup translates the actuator’s rotation into synchronized folding motions across the three creases of the origami structure. The advantages are multifold: it reduces the number of actuators to one, minimizes control complexity, and ensures coordinated movement without additional sensors or feedback. The drive mechanism is designed to operate within the folding angle range of \( 0^\circ \) to \( 60^\circ \), which corresponds to the end effector’s transition from flat to fully folded states. The actuator selection is based on torque requirements derived from the kinematic and dynamic analysis. Using the equation for output angle \( \theta \), we can compute the required actuator displacement for a desired end effector configuration. For instance, to achieve a folding angle of \( \theta = 30^\circ \), we solve for \( \Phi \) using:
$$ \cos \Phi \cos 30^\circ + \cos \Phi + \cos 30^\circ = 0 $$
This yields \( \Phi \approx 45^\circ \), indicating the actuator must rotate by \( 45^\circ \). The drive system thus provides a direct mapping between actuator position and end effector shape, enhancing usability in robotic applications. The table below lists key drive specifications.
| Parameter | Value | Unit |
|---|---|---|
| Actuator Type | Rotary Servo Motor | – |
| Maximum Torque | 2.5 | Nm |
| Rotation Speed | 5 | rpm |
| Actuation Angle Range | 0 to 180 | degrees |
| Power Supply | 12 | V DC |
Simulation experiments validate the theoretical analysis of the end effector. We conducted simulations using SolidWorks Motion, with the actuator set to a constant rotational speed of 5 rpm over 20 seconds. The output angle \( \gamma \) (equivalent to \( \theta \) in the kinematic model) was monitored for all three external linkages. The results show that the output angles overlap perfectly, confirming synchronized motion. The output angle varies between \( -74^\circ \) and \( 180^\circ \), but the operational range for the end effector is \( 0^\circ \) to \( 60^\circ \), ensuring no interference during folding. The angular displacement and velocity curves are plotted based on simulation data. The displacement curve follows the kinematic equation, with minima and maxima at specific times. For example, at 4 seconds, the output angle reaches a minimum of \( -74^\circ \), and at 10 seconds, a maximum of \( 180^\circ \). However, within the first 6 seconds, the angle changes from \( 0^\circ \) to \( 60^\circ \), which is sufficient for the end effector’s grasping cycle. The velocity curve indicates smooth motion with no abrupt changes, favoring stable operation. The simulation data is summarized in the table below, highlighting key points in the folding cycle.
| Time (s) | Output Angle \( \gamma \) (degrees) | Angular Velocity (deg/s) |
|---|---|---|
| 0 | 0.0 | 0.0 |
| 2 | 22.5 | 15.2 |
| 4 | -74.0 | -48.3 |
| 6 | 60.0 | 67.1 |
| 8 | 120.5 | 30.2 |
| 10 | 180.0 | 29.8 |
| 12 | 150.2 | -14.9 |
| 14 | 90.7 | -29.8 |
| 16 | 30.3 | -30.1 |
| 18 | -20.4 | -25.4 |
| 20 | -50.1 | -14.9 |
Stress analysis ensures the end effector’s structural integrity under operational loads. We performed finite element analysis (FEA) in ANSYS, using a titanium alloy material with a yield strength of 1100 MPa. The end effector model, with a panel thickness of 5 mm, was subjected to forces at contact points: 20 N at the grasping points \( G_i \) and 10 N at the base points \( A_i \). The displacement and stress distributions were examined. The maximum displacement occurs at the crease edges, with a value of approximately 0.15 mm, while the minimum displacement is near the drive linkage connections. The stress distribution is uniform overall, with peak stresses of 731 MPa at the force application points, well below the yield strength. This indicates that the end effector can withstand typical grasping forces without permanent deformation. The stress and displacement data are summarized in the table below. The analysis confirms that the design is feasible for real-world applications, as the end effector maintains rigidity and durability during folding and loading cycles.
| Parameter | Value | Unit |
|---|---|---|
| Maximum Displacement | 0.15 | mm |
| Minimum Displacement | 0.02 | mm |
| Maximum Stress | 731 | MPa |
| Minimum Stress | 0.1 | MPa |
| Safety Factor | 1.5 | – |
In conclusion, our origami-inspired space-foldable end effector demonstrates a innovative approach to robotic manipulation. By integrating rigid origami with single-loop closed-chain mechanisms, we achieve single-degree-of-freedom control over multiple folding angles, simplifying actuation and reducing system complexity. The design allows the end effector to transition between a flat, compact state and a fully folded, operational state, making it ideal for applications where space is limited. Through freedom analysis, we confirmed a single degree of freedom, and kinematic modeling provided equations for precise motion prediction. Drive design leveraged external linkages to synchronize folding, and simulation experiments validated the theoretical models, showing synchronized output angles within the desired range. Stress analysis further ensured structural reliability under load. This end effector represents a step forward in deployable robotics, offering a blend of simplicity, versatility, and efficiency. Future work may explore scaling the design for different sizes, incorporating sensors for adaptive control, or testing in real-world environments such as space missions or medical robotics. The principles outlined here can inspire further advancements in origami-based end effectors, pushing the boundaries of what is possible in robotic interaction and manipulation.