An Origami Continuum Manipulator with Modularized Design and Hybrid Actuation: Accurate Kinematic Modeling and Experiments

Herein, this study contributes significantly to the advancement of continuum manipulators in two main aspects. First, a modularization concept and a hybrid actuation scheme to create a novel origami continuum manipulator with exceptional deformability are introduced. Second, an accurate model and framework for the forward and inverse kinematic analysis of origami manipulators are proposed. Specifically, each origami manipulator module can achieve axial extension and bending deformation by coordinated actuation of shape memory alloy (SMA) and pneumatic muscles, and the manipulator's end is equipped with a deformable gripper based on waterbomb origami and actuated by SMA. Through careful consideration of the self‐weight and torque balance, an accurate kinematic model based on the Denavit–Hartenberg method is established, which enables one to effectively predict the reachable extreme positions and spatial poses of the manipulator and solve the inverse kinematics using a genetic algorithm. Comprehensive experiments are conducted to validate the design's rationality and model's accuracy . In these tests, the rich spatial configurations are not only demonstrated that can be achieved by integrating hybrid actuators with origami modules but also the accuracy and reliability of the kinematic model are confirmed, opening up possibilities for the advancement and application of origami‐inspired robotics in various fields.


Introduction
Conventional robotic manipulators, characterized by rigid links interconnected by actuated joints, have found extensive use in various industrial applications due to their high stiffness and fast dynamic performance.However, their range of motion is significantly limited due to the finite degrees of freedom (DOFs) and joint deformations. [1]In recent years, the development of continuum manipulators, inspired by the natural movements of animals such as elephants' trunks, octopus tentacles, and snakes, has gained momentum due to the growing demand for applications in fields such as medical operations, [2] space exploration, [3] and humanmachine interaction. [4]Compared to conventional rigid manipulators, continuum manipulators offer distinct advantages, including motion flexibility, smoothness, and compliance. [5,6]Typically, a continuum manipulator is composed of a series of interconnected segments or modules, each capable of achieving planar or spatial deformations and possessing infinite passive DOFs. [7,8]umerous attempts have been made to design and fabricate continuum manipulators, showcasing the versatility and potential of this field.Reviewing the existing designs, the actuation of continuum manipulators can be broadly categorized into cable-driven, [9] pneumatic-driven, [10] shape-memorymaterial-driven, [11] and electromagnetic-driven types. [12]Each actuation method has its own merits and demerits.Cable-driven systems offer high driving force but require additional motors and devices for cable retraction and release, limiting miniaturization. [13]Pneumatic actuation enables compact robot structures Herein, this study contributes significantly to the advancement of continuum manipulators in two main aspects.First, a modularization concept and a hybrid actuation scheme to create a novel origami continuum manipulator with exceptional deformability are introduced.Second, an accurate model and framework for the forward and inverse kinematic analysis of origami manipulators are proposed.Specifically, each origami manipulator module can achieve axial extension and bending deformation by coordinated actuation of shape memory alloy (SMA) and pneumatic muscles, and the manipulator's end is equipped with a deformable gripper based on waterbomb origami and actuated by SMA.Through careful consideration of the self-weight and torque balance, an accurate kinematic model based on the Denavit-Hartenberg method is established, which enables one to effectively predict the reachable extreme positions and spatial poses of the manipulator and solve the inverse kinematics using a genetic algorithm.Comprehensive experiments are conducted to validate the design's rationality and model's accuracy .In these tests, the rich spatial configurations are not only demonstrated that can be achieved by integrating hybrid actuators with origami modules but also the accuracy and reliability of the kinematic model are confirmed, opening up possibilities for the advancement and application of origami-inspired robotics in various fields.
but demands high air tightness and control accuracy. [14]hape-memory-material-driven actuators provide sufficient force and facilitate compact design, but have a long time to restore the original length. [13,15]Electromagnetic-driven manipulators can be miniaturized to a millimeter scale, but necessitate additional equipment for generating the required magnet fields. [16]dditionally, dielectric elastomers can actuate manipulators with large deformations but require high driving voltages. [3]The aforementioned analysis of actuators suggests that it is possible to improve the performance of continuous manipulators by adopting a combination of different actuators to leverage their respective advantages.
Moreover, it is worth noting that certain continuum manipulators, particularly those of the cable-driven type, usually employ a centralized arrangement of motors. [17,18]While this configuration simplifies the driving control of the robot, it limits the DOFs and the overall deformability of the continuum manipulator.In contrast, a modular design approach allows for independent actuation of each module, thereby enhancing the manipulator's deformability.For instance, in the design of a multi-module pneumatic manipulator, each module is controlled by an individual pneumatic circuit, resulting in a significantly large workspace. [19]In another study, a rigid-flexible coupling manipulator consisting of three segments with independent bending capabilities is designed and fabricated; [20] simulation and experimental results demonstrate that the end effector of this manipulator can effectively track circular, triangular, and rectangular trajectories.An additional advantage of the modular design is the ease with which functional redundancy can be expanded by adding modules, without requiring modifications to the overall system architecture.
Origami, an ancient art of paper folding, has garnered great attention due to its boundless design potential, exceptional deformability, and distinctive reconfigurability.The essence of origami involves transforming 2D flat sheets into various 3D structures based on elaborate crease patterns.As a result, origami has found wide-range applications in aerospace engineering, [21] biomedical engineering, [22] material science, [23] and robotics. [24]Notably, by utilizing folding techniques, origami robots could be constructed with thin materials, resulting in remarkably lightweight robot prototypes.Rather than the conventional process of "3D component design, 3D component fabrication, and final assembly," origami revolutionizes robot development by adopting a "2D crease design, 2D crease pattern fabrication, and folding" approach. [24,25]This innovative development process leverages the maturity of 2D fabrication techniques (e.g., lithography, laser machining, etc.) to significantly reduce costs, enhance accuracy, improve efficiency, and enable flexible design modifications. [25,26]By combining crease pattern design with nonlinear folding kinematics, origami structures can exhibit unconventional mechanical properties such as negative Poisson's ratio, [27] shape morphing, [28] multistability, [29][30][31] and large deformation, [32,33] etc.Moreover, by integrating origami structures with various actuation methods mentioned earlier, large deformation can be achieved in an active manner.Indeed, these advantages have opened up significant opportunities for the advancement of soft manipulators.Several origami manipulators have already been designed and prototyped based on twisted towers [34] and bistable Kresling, [35] showcasing their excellent deformability.
Despite demonstrating the feasibility and unique advantages of origami technology in manipulator development, there are pressing challenges that require immediate attention.First, most origami manipulators still adopt the same type of actuators, limiting the overall deformability and spatial accessibility of the manipulator.Second, existing studies mainly demonstrate the reachable workspace and spatial posture of origami manipulators through proof-of-concept experiments, whereas systematic kinematic modeling and analysis methods are lacking, the primary reason is that the conventional kinematic method struggles to precisely predict the position of the manipulator with a payload.While existing studies have employed mechanics-based methods to investigate origami manipulators, such as treating them as elastic beams [36] or directly applying finite-element methods, [37] but these methods face three problems: 1) treating a module as a continuum poses difficulties in accurately measuring the requisite crease and facet stiffness for this modeling process.Additionally, when there are changes in the module's material or an evaluation of the manipulator's overall performance is necessary, the physical parameters must be reevaluated.This process, aside from being cumbersome, can introduce inaccuracies.2) The mechanics-based approach for hybrid-actuated modules demands not only modeling the origami structure and the various types of actuators, but also considering the coupling between them.This complexity makes the modeling process challenging and may result in models that are overly intricate for practical application.3) The primary application of these methods lies in characterizing the deformation capacity or load-bearing capability of individual modules of the manipulator, without providing an analysis of the overall workspace and spatial postures.Moreover, there is a lack of comparative studies between simulation and experimental results to verify the accuracy of the model, thereby impeding the systematization and applicability of origami manipulators.
This research aims to contribute to the current state of the art in three main aspects.First, it proposes a hybrid actuation method for an origami continuum manipulator (referred to as "origami manipulator" for simplicity) based on the modularization concept.This involves integrating the Yoshimura origami structure with shape memory alloy (SMA) and pneumatic muscle (PM) actuators, as well as incorporating a printed circuit board (PCB).Unlike the pneumatic-only and SMA-only solution, in the hybrid actuation scheme, the airbags in each module need to be controlled by only one airway, and the SMA can be controlled by circuitry on the PCB, thus greatly simplifying the airway system and the entire control system.Second, systematic methods are developed for modeling, calibrating, and analyzing the origami manipulator.The manipulator module, viewed as a complex with multiple rigid joints, is analyzed using the homogeneous transformation matrix, and the Denavit-Hartenberg (D-H) kinematic model of the entire manipulator is derived.To address the effect of self-weight on axial elongation and bending deformations, we conduct static correction experiments, which allow us to determine the kinematic parameters of the D-H model by considering torque balance.With each module capable of exhibiting 31 deformation states, a four-module manipulator provides a wide range of configuration states, resulting in numerous feasible spatial poses.Based on the developed model, an inverse kinematics scheme for the origami manipulator is proposed, utilizing a genetic algorithm (GA)-based method.Finally, comprehensive experiments are carried out to quantitatively verify the effectiveness and accuracy of the modeling and analysis methods.These experiments showcase the rich achievable spatial configurations and grasping capabilities of the origami manipulator, while also validating the kinematic model in predicting the reachable extreme positions and spatial poses.The design and kinematic model of the origami manipulator presented in this study would contribute valuable insights for the development of flexible or soft manipulators, while also laying a solid foundation for the kinematic modeling and analysis of continuum manipulators.

Modularized Design and Hybrid Actuation Scheme
The origami continuum manipulator (referred to as "origami manipulator" for simplicity) is designed and prototyped based on the idea of modularity and hybrid actuation.Each module of the manipulator (Figure 1a) is made up of a Yoshimuraori structure (the detailed design is introduced in Section S1 and Figure S1a,b, Supporting Information, and the fabrication can be found in Experimental Section.), three PMs actuators (detailed in Experimental Section and Figure S1c, Supporting Information), four SMA actuators, and two PCB plates (detailed in Experimental Section).The Yoshimura-ori structure is selected as the foundation due to its excellent axial and bending deformability and strong resistance to torsional deformation.This is manifested by the large order differences among the axial stiffness (0.095 N mm À1 ), bending stiffness (12.711N mm rad À1 ), and torsional stiffness (1120.422N mm rad À1 ), see Experimental Section in Section S2, Supporting Information.40] Each module is equipped with three PM actuators arranged in parallel to provide sufficient thrust to extend the Yoshimuraori structure and prevent the tube from buckling.To ensure efficient energy supply, actuator control, and posture sensing of the manipulator, two PCB plates are specifically designed and fabricated (Figure 1b).The two PCB plates are also used to connect the Yoshimura-ori structure with the PM and the SMA spring actuators.The photo of the manipulator module is shown in Figure S1a, Supporting Information.
To endow the origami manipulator with grasping capability, an origami gripper is designed (Figure 1c).It is composed of a waterbomb origami structure (detailed in Experimental Section and Figure S1e, Supporting Information) and a 3D-printed sliding track, they are both attached to a PCB board.A circular slider is capable of sliding vertically along the 3D-printed track.The four fingers are connected to the slider via thin cords, while the slider itself is connected to the PCB plate using two SMA spring actuators.The photo of the origami gripper is shown in Figure S1f, Supporting Information.When the SMA springs are heated to induce contraction, the slider is pulled downward, resulting in the inward folding of the four fingers and the execution of the gripping action.Conversely, when the SMA is de-energized, the elasticity of the origami structure allows the fingers to reset to their original position.
Connecting the four modules and the origami gripper in sequence, a four-module origami manipulator is prototyped in Figure 1d.The length of the prototype in its initial configuration is 430.7 mm, and the weight is 355.2 g.From top to bottom, the four modules are labeled "1," "2," "3," and "4" in order.The control architecture of the origami manipulator is presented in Figure 1e and is detailed in Section S3, Supporting Information.
Each module of the origami manipulator could exhibit six deformation modes, including idle, axial elongation, and bending deformations (leftward, rightward, forward, or backward), operating under specific actuation patterns as outlined in Table 1.By inflating the PM actuators only, axial elongation can be achieved; by inflating the PM actuators and energizing the adjacent pair of SMA springs, the module undergoes bending deformation as the SMA springs contract.In addition, due to the large deformation capabilities and rates of the SMA and PM, the module could possess relatively high speeds of axial elongation and bending deformation (detailed in Section S4, Support information).For the axial elongation, the actuation time of pressure 0.125 and 0.225 Mpa are 2.2 and 0.8 s, respectively.For the bending deformation, the actuation time of pressure 0.125 and 0.225 Mpa are 9.3 s (i.e., SMA heating time is 8 s, PM inflation time is 3.5 s) and 11.5 s (i.e., SMA heating time is 8 s, PM inflation time is 1.3 s), respectively.For different deformation modes and different air pressures, the resetting time is 1.2 s.For each non-idle mode, six different pressures can be applied to the module: 0.1, 0.125, 0.15, 0.175, 0.2, and 0.225 MPa.The gripper module solely utilizes SMA actuators for achieving the gripping state by energizing the SMA springs.

The D-H Kinematic Model
This section studies the kinematic modeling of the origami manipulator, whose homogeneous transformation matrix is constructed according to the D-H method.

Kinematic Parameters and the Set of Actuation Parameters
We first determine the kinematic parameters needed to describe the deformation of an individual module.In the axial elongation mode, j i L axial is used to characterize the length of the module (the superscript "j" represents the applied pressure state, i.e., "0" for 0 MPa, "1" for 0.100 MPa, "2" for 0.125 MPa, "3" for 0.150 MPa, "4" for 0.175 MPa, "5" for 0.200 MPa, and "6" for 0.225 Mpa; the subscript "i" denotes the module number).In the bending modes, the bending deformations in all four directions are assumed to be the same and are characterized by three parameters: the angular deformation j i ψ, the bending angle j i β, and the length of the module j i L bending , which are described in Figure 2a,b.
Note that each module in this study is capable of exhibiting six distinct deformation modes, and for each mode, six different pressures can be applied to the module.As a consequence, each module is expected to exhibit 31 deformation states, including 1 idle state and 30 deformation states (i.e., 5 non-idle deformation modes Â 6 pressures).Since each module can deform independently, the four-module manipulator is expected to possess 31 4 ¼ 923521 configurations.To simplify subsequent descriptions, a set of actuation parameters V is defined to denote the configuration of the entire manipulator, expressed as where i V represents the deformation mode of module "i," and its value "2, 3, 4, 5, 6" corresponds to axial elongation, leftward, rightward, forward, and backward bending, respectively.i p denotes the internal pressure of module "i."An idle module is described by i V ¼ 1 and i p ¼ 0.

D-H Kinematic Model of the Manipulator
By mapping the origami manipulator module to a conceptual "virtual" rigid-link robot module, we can derive the corresponding D-H formulation.The utilization of the D-H formulation facilitates the establishment of geometric solutions that relate We arbitrarily chose a module "i" as an example to illustrate the modeling process.To represent the axial elongation and bending deformations (Figure 2a,b) accurately, we propose an equivalent rigid-joint model.This model composes four revolute joints and one prismatic joint, as shown in Figure 2c,d.The rotation of the revolute joint "1" determines the bending axis of the module.Since the module can only bend in four directions, the rotation angle of joint "1" can only take four discrete values: 0°for forward bending, 90°for rightward bending, 180°for backward bending, and 270°for leftward bending.The rotation angle of revolute joint "2" is used to equivalently describe the bending angle j i β of the module around the rotation axis defined by joint "1."The prismatic joint "3" is used to represent the length change during deformation, relating to the length parameter j i L bending or j i L axial of the module.The revolute joint "4" is employed to determine the orientation of the module's lower plate.In contrast, the revolute joint "5" does not contribute to describing the module's pose.Instead, its role is to rotate the local frame in such a way that its x axis aligns with the x axis of the global frame, ensuring the proper orientation of the subsequent module.Consequently, the variables of the first revolute joint and the last revolute joint are interdependent.
Based on this rigid-joint model, the link frames of module "i" can be established (Figure 2c,d).To simplify the representation, the local coordinate system of each link is named according to the following rule: fx i,k y i,k z i,k g represents the frame located on link "k" of module "i."For module "i," the base frame is the fx iÀ1,5 y iÀ1,5 z iÀ1,5 g, whose origin O iÀ1,5 locates at the center of the lower plate of module "i-1."The origin of joints "1" and "2" (i.e., O i,1 and O i,2 ) is located on the negative z iÀ1,5 axis at a distance of d from O iÀ1,5 .Here, d ¼ 35.3 mm is the distance between the upper plate of the origami manipulator module "i" and the lower plate of module "i-1" (Figure 2a).In the initial state, x i,1 , x i,2 , x i,3 , x i,4 , and x i,5 axes are all aligned with the x iÀ1,5 axis.The origins of frames fx i,4 y i,4 z i,4 g and fx i,5 y i,5 z i,5 g coincide and are located at the endpoint of the module, represented by the pentagram in Figure 2c,d.Particularly, for module "1," the base frame is fx 0 y 0 z 0 g, with its origin O 0 located at the center of the top fixed bracket (Figure 2e).
Using the established link frames, we can derive the D-H table of the constituent module in both scenarios of axial deformation and bending deformation.Table 2 and 3 list the D-H parameters (the link twist i α kÀ1 , link length i a kÀ1 , link offset i d k , and joint angle i θ k ) of module "i" (denoted by the left subscript) for bending and axial elongation, respectively.Specifically, i θ 1 ¼ ϕ is determined by the bending direction, whose value can take among 0°, 90°, 180°, and 270°, corresponding to forward, rightward, backward, and leftward bending, respectively); i θ 2 is equal to the negative value of bending angle, i.e., À j i β; i d 3 is equal to the length of the module in different deformation modes, i.e., À j i L axial or À j i L bending ; i θ 4 is determined by both j i β and j i ψ, which is ; and i θ 5 is coupled with i θ 4 , as mentioned earlier, and its value is Àϕ.
Particularly, in the case of axial elongation, where the module does not undergo bending, joint "1" does not require any rotation.Hence, the corresponding joint angle i θ 1 ¼ ϕ ¼ 0. In addition, since the module does not undergo any bending or angular deformation, the rotation angles of joints "1," "2," "4," and "5" are all zero, i.e., i θ Based on the D-H tables provided earlier, the homogeneous transformation matrix i P kÀ1 k that defines the frame fx i,k y i,k z i,k g relative to fx i,kÀ1 y i,kÀ1 z i,kÀ1 g can be derived: Particularly, for i ¼ 1, the frame fx iÀ1,5 y iÀ1,5 z iÀ1,5 g is essentially fx 0 y 0 z 0 g, and 1 P 0 1 actually defines the frame fx 1,1 y 1,1 z 1,1 g relative to fx 0 y 0 z 0 g.Then, the homogeneous transformation matrix iÀ1 i T that defines the frame fx i,5 y i,5 z i,5 g relative to the frame fx iÀ1,5 y iÀ1,5 z iÀ1,5 g yields Similarly, 0 1 T defines the frame fx 1,5 y 1,5 z 1,5 g relative to the frame fx 0 y 0 z 0 g.
With the derived homogeneous transformation matrix of the constituent module iÀ1 i T (i.e., Equation ( 2) and ( 3)), the kinematics can be extended to the entire four-module origami manipulator, and a mathematical expression for the complete model is derived.Figure 2e,f shows the equivalent rigid-joint model of the entire four-module manipulator as well as the associated link frames; the origin O 0 of the base frame fx 0 y 0 z 0 g remains located at the center of the top fixed bracket.The endpoint of each module "i" is represented by a position vector i o end ¼ i x end , i y end , i z end ½ T in the base frame fx 0 y 0 z 0 g, which can be used to characterize the reachable position of module "i" and can be calculated by Particularly, 4 o end represents the position of the endpoint of the entire origami manipulator (without the gripper).

Calibration of the Kinematic Parameters
It is worth pointing out that the value of the kinematic parameters j i L axial , j i L bending , j i β, and j i ψ in the D-H model need to be determined by static tests and torque balance analysis.When the module is in its idle and axial elongation mode, the internal PM actuators can effectively resist the torques applied by other modules, resulting in minimal bending deformations that can be ignored.As a result, the idle and axial elongated modules are treated as rigid bodies, and we only need to calibrate the length j i L axial by considering the self-weight.In contrast, when the module undergoes bending deformation, additional moments would be generated, which calls for torque balance analysis for calibrating the three kinematic parameters j i L bending , j i β, and Arbitrarily selecting a module as the test sample, the kinematic parameters are determined under six different pressures: 0.100, 0.125, 0.150, 0,175, 0.200, and 0.225 MPa.For the axial elongating module, it needs to bear the weight of the subsequent modules and the gripper (listed in Table S1, Supporting Information).The experiment setup is shown in Figure 3a, where the upper plate is fixed on the holder, and the additional weight is attached to the lower plate.A camera is used to record the length j i L axial of the module, and partial results are demonstrated in Figure 3b.When the pressure is 0.1 MPa, the module with different loads exhibits almost the same length, equaling to the initial length when no pressure is applied.At pressures above 0.1 MPa, the module length remains almost constant for different and increases nonlinearly with increasing pressure.The Table 2.The D-H table of the origami manipulator module "i" in bending.
For the bending module, the kinematic parameters j i ψ and j i L bending and the output torque j i τ of module "i" are assumed to be functions of the bending angle j i β ("j" represents the applied pressure state).The specific expressions are determined by fitting the data obtained from static correction experiments.The experiment setup is shown in Figure 3c, and the details are given in Experimental Section and Section S5, Supporting Information.The module is initially bent by the SMA and PM actuators to a bending angle j i β ("j" represents the applied pressure state) and then straightened by an inextensible string.During the test, the moveable platform rises at a speed of 0.5 mm s À1 until j i β reduces to 0°, effectively straightening the module.The deformation is captured using a high-definition camera, and the positions of the marked points are analyzed using Kinovea.This analysis allows for the determination of the angular deformation j i ψ, the length j i L bending , the bending angle j i β, the vector of the applied force F, the position vector O i,1 A i !, and the vector O i,1 O i,5 ! with respect to time.By considering the torque equilibrium, the restoring torque of the bending module can be derived as follows: where i G (0.72 N) is the gravitational force acting on module "i," which is assumed to act at the geometric center of the module (i.e. the midpoint of the vector O i,1 O i,5 ! ).As an example, when the applied pressure is 0.225 MPa (i.e., j ¼ 6), Figure 3d-f respectively illustrates the relationship of 6 i ψ, 6 i L bending , and 6 i τ with respect to 6 i β (the hat indicates the measured value of β).For the other pressures, the measured relations of j i ψ, j i L bending , and j i τ about j i β are plotted in Figure S2, Supporting Information.They can be fitted by the following fifth-order polynomials, with the polynomial coefficients a k , b k , and c k , as well as the coefficient of determination r 2 listed in Table S3-S8, Supporting Information.
Based on the aforementioned relations, we can now conduct the torque balance analysis.If all four modules are in idle or axial elongation mode, the effect of gravity on torque can be neglected.However, when at least one module undergoes bending deformation (e.g., as shown in Figure 3g where only module "1" is bent, and in Figure 3h where all four modules are bending), the torque balance equation can be derived based on the current configuration where "i" represents the number of the module, M s represents the geometric center of module "s" (i.e. the midpoint of the vector O s,1 O s,5 ! ), and the vector e i is the unit vector representing the normal direction of the bending plane of module "i."The gravitational force of the gripper is assumed to act at the center of the lower plate of module "4."Here, O i,1 M s ! and e i can be derived from the kinematic model constructed based on the D-H table (Section 2.2.3).Considering that the kinematic parameters j i ψ and j i L bending and the torque j i τ are now functions of j i β according to Equation ( 6), the torque-balance Equation ( 7) becomes dependent on the variable When the set of actuation parameters V is given, by numerically solving Equation ( 7) for each module, the bending angle j i β can be uniquely determined, which in turn allows the determination of j i ψ and j i L bending .Consequently, the overall configuration of the origami manipulator can be determined, taking into consideration the torque balance.Particularly, when a module is in idle or axial elongation mode, its minimal bending deformations can be neglected, treating it as a rigid body.As a result, the torque balance for this module does not need to be considered, leading to a reduction in the number of torque balance equations.

Forward and Inverse Kinematics
With the established kinematic model and the calibrated kinematic parameters, the forward and kinematics of the origami manipulator be analyzed.For forward kinematic, we can calculate the spatial position of each module's endpoint i o end and spatial pose of the whole manipulator under any given actuation parameters; for inverse kinematic, we can determine the optimal actuation parameters for a given position of the manipulator's endpoint 4 o end .

Forward Kinematics
We first examine the reachable extreme positions of the constituent modules.Typically, the extreme positions are reached when the inflation pressure of the PM actuators reaches the maximum.Hence, the kinematic parameters under the maximum pressure 0.225 MPa, i.e., 6 i ψ, 6 i β, 6 i L bending , and 6 i L axial , are calculated based on Equation ( 6) and (7) and the data listed in Table S8, Supporting Information, which are then used to determine the reachable extreme positions.
In addition to the extreme positions of the modules, the spatial poses of the origami manipulator can also be predicted by the kinematic model.As examples, eight characteristic configurations are examined: including the configurations in which the endpoint of module "4" reaches the lowest position (Figure 5a), the farthest position (Figure 5b), the highest position (Figure 5c), and an end-bending configuration (Figure 5d), a J-shape configuration (Figure 5e), an upper-bending configuration (Figure 5f ), an S-shape configuration (Figure 5g), and a 3D space-curve configuration (Figure 5h).The corresponding sets of actuation parameters are listed in Table S10, Supporting Information.

Inverse Kinematic Analysis
The joint variables of conventional rigid manipulators can be determined geometrically or analytically based on the spatial pose of the endpoint, resulting in a continuous solution space of inverse kinematics.However, origami manipulators differ in this aspect, as each origami module's deformation state corresponds to a distinct combination of pneumatic pressure and heated SMA spring, yielding a discrete solution set of inverse kinematics.In this research, the inverse kinematic problem is solved based on the established kinematic model and using a GA.
Specifically, the inverse kinematics is solved by determining the optimal configurations of all modules that correspond to the minimal distance between the endpoint (described by the position vector 4 o end ) and the target point (described by the position vector o Ã ).The optimization problem is mathematically described by where f V ð Þ quantifies the Euclidian distance between 4 o end and o Ã , V is the set of actuation parameters to be optimized.This single-objective optimization problem is solved via the GA, whose setting is detailed in Section S7, Supporting Information.At the conclusion of the simulation, the configurations (described by V) with f < 3 are the output.In case no configuration satisfies the condition of f < 3, the configuration with the shortest distance between the endpoint 4 o end and the target point o Ã is selected as the output.
Note that the GA's ability to identify all solutions may be limited in cases when multiple solutions exist.Despite being capable of finding at least one solution, it might not discover all configurations satisfying the condition f < 3. The effectiveness of obtaining all configurations relies significantly on the size and distribution of the initial population.Increasing the initial population size would enhance the chance of capturing all solutions that meet the condition, but this comes at the cost of reduced computational efficiency, resulting in a trade-off.

Experiments on Reachable Extreme Positions
To verify the accuracy of the kinematic model, experiments on the reachable extreme positions are carried out using the fourmodule origami manipulator prototype and the corresponding control architecture (Figure 1e).The experiment setup is introduced in Experimental Section and Section S6, Supporting Information.The position of the endpoint 4 o end of each module is obtained by calculating the average spatial coordinates of the four markers, which are measured via the Vicon motion capture system.
By applying the actuation parameters listed in Table S9, the Supporting Information, Figure 4a-d presents the experimentally measured extreme positions of modules "1," "2," "3," and "4," which are compared with the predicted extreme positions based on the kinematic model.The left two columns show the distribution of the extreme positions on the xoz plane and yoz plane, and the right three columns show photos of the actual configuration of the origami manipulator when the extreme positions are reached (i.e., the lowest, farthest, and highest points).The results demonstrate a favorable agreement between the experimentally achieved extreme positions and the predicted ones.Furthermore, we quantitatively access the errors between the experimental and the predicted results by calculating the radius and z-coordinate values of the endpoints, as depicted in Figure 4e.The findings indicate that the errors are generally below 8% for all metrics, except for the radius of the lowest points (the radius is zero in predictions).This confirms the efficacy of the kinematic model, which considers self-weight and torque balance, in accurately predicting the extreme positions attainable by the origami manipulator.It should be noted that the errors in the z-axis position are relatively smaller compared to the errors in the radius (e.g., the maximum error is 7.94% for the radius and 5.67% for the z-axis position).This discrepancy is attributed to the fact that the moments of gravity mainly affect the horizontal component of the bending modules' deformations.

Experiments on Spatial Configurations
Experiments on spatial configurations are also carried out to provide further validation of the accuracy of the kinematic model.Eight characteristic configurations are tested, and the obtained results are compared with the configurations predicted by the kinematic model, as depicted in Figure 5a-g (the corresponding actuation parameters are listed in Table S10, Supporting Information).The experiment results and the theoretical predictions are represented by purple and black lines, respectively.To quantitatively assess the error between the experimental results and the predictions, the distance between the experimentally measured position i o $ end and the predicted position i o end is computed for the endpoint of each module where i is the number of the module.The results are presented in Figure 5i, and the detailed data are listed in Table S11, Supporting Information.Overall, the spatial poses calculated by the model agree well with the experimentally measured spatial poses.Among them, the configuration of "lowest position" exhibits the best agreement (Figure 5a), as in this configuration, all modules are axially elongated, and the gravitational forces do not produce significant moments.In contrast, the errors for the J-shape configuration and the 3D configuration are relatively larger (Figure 5e,h).For the J shape configuration, although bending occurs in the last two modules near the free end, the axially elongated modules in experiments are not entirely rigid and tend to deflect due to the effect of the gravitational moments, resulting in relatively large errors in the endpoint positions.The errors between experimental and predicted results for the 3D configuration are the largest.This is because the model only considers the component of the gravitational moment in the plane where the bending module is located.This model performs well when all bending modules are situated in the same plane, but it may lead to relatively large errors when bending modules are located in different planes.Furthermore, we conducted an analysis of the effects of end payload on the prediction accuracy of the kinematic model (detailed in Section S9, Supporting Information), revealing that a substantial increase in end payload led to a decrease in model accuracy.The main source of this deviation stems from the fact that, under a significant end payload, the larger moment is generated, inducing a bending deformation in the axially elongated module.To rectify this error, we must refrain from merely regarding the axially elongated module as a rigid body.In future research, we plan to calibrate the deformation of the axially elongated module under bending moments, account for the moment of the axially elongated module in the moment balance equation, and ultimately construct a kinematic model of the manipulator with enhanced accuracy.Future work will be dedicated to addressing this issue to further improve prediction accuracy.

Versatile Origami Gripper
To demonstrate the application potential of the origami manipulator and to correspond to the gripper's weight that was taken into account in kinematic modeling and calibration, the experiments also include testing the origami gripper's grasping capability.Four typical objects with different sizes, shapes, materials, and weights are selected for this purpose, including an orange peel (30.2 g), a cubic sponge block (1.3 g), a table tennis ball (2.8 g), and a shuttlecock (5.1 g), as shown in Figure 6a.The origami manipulator performs four steps to grasp each object: 1) it starts from an initial state, preparing for grasping; 2) the SMA and PM actuators of the four modules are energized to position the origami gripper at the object's location; 3) the SMA of the origami gripper is heated to complete the grasp of the object; 4) each module is retracted back to the initial state while lifting the object.
The testing results shown in Figure 6b indicate that the origami gripper can reliably lift a wide range of objects with varying sizes (width up to 81.1 mm and height up to 80.5 mm), smooth or rough surfaces, spherical or irregular shapes, and weights ranging from 1 g to over 30 g.This demonstrates the versatility and effectiveness of the origami gripper.

Conclusion and Discussion
Compared to conventional continuum manipulators, origamiinspired manipulators possess characteristics such as lightweight design and a wide range of deformation modes.However, they also face certain challenges that need to be addressed urgently.First, the single-actuation method limits the overall deformability and spatial accessibility of the origami manipulator.Second, the lack of systematic kinematic modeling and analysis methods hinders the development and applicability of origami manipulators.This study aims to address these challenges by introducing a hybrid actuation method for a Yoshimura-ori-based modularized continuum manipulator.Additionally, a systematic framework is developed for modeling, calibrating, and analyzing the origami manipulator.Going a step further than previous studies, we conduct comprehensive experiments to demonstrate the rich achievable space postures of the origami manipulator and validate the accuracy of the proposed kinematic model.By tackling these key issues, we aim to advance the state of the art of origami manipulators pave the way for their practical applications in various industries.
The novel design and prototype of the origami manipulator is composed of four origami manipulator modules and an origami gripper module.Each independently working origami manipulator module is developed by integrating the Yoshimura origami structure with hybrid actuations of SMA and PM.This integration, along with the inclusion of a PCB with an embedded inertial measurement unit (IMU), achieves a balance between maintaining the deformability of the manipulator and significantly simplifying its control system.In contrast, the gripper module is designed based on the waterbomb origami structure and is actuated by SMA actuators.The combination of these elements results in a flexible origami manipulator with promising potential for various applications.
Based on the developed prototype, this research focuses on the systematic development of methods for modeling, calibrating, and analyzing the origami manipulator.First, treating each module as an equivalent rigid link system with five joints, the D-H kinematic model of the entire manipulator is derived.Then, to account for the influences of self-weight and on axial elongation and torque balance on bending deformations, static experiments are performed on a single module to calibrate the kinematic parameters in the D-H model.Such calibration enhances the accuracy of predicting the reachable extreme positions and spatial poses of the origami manipulator.Moreover, with a vast library of possible configurations, the inverse kinematic problem is tackled based on the GA algorithm.The GA algorithm, along with the torque balance equations, efficiently identifies the optimal actuation parameters for scenarios involving a single solution, multiple solutions, or no solution, aiming to approach the target position.In fact, the modeling framework presented in this article can be applied to other origami manipulators with different structures and the number of modules.For different prototypes, we only need to change the number of the torque balance equation, and the corresponding parameters, and obtain the forward and inverse kinematic solutions.
Finally, comprehensive experiments are conducted to evaluate the reachable extreme positions, spatial poses, and grasping capabilities of the origami manipulator.These experiments showcase the excellent deformation and grasping capabilities of the origami manipulator.Comparing to the existing prototypes in other research, the advantage of the origami continuum manipulator presented in this article lies in its comprehensive performance, which is able to achieve a certain axial deformation ratio and reachable space while realizing complex 3D spatial configurations.In contrast, the experiment also validates the accuracy of the kinematic model.The results provide strong evidence of the kinematic model's capability to accurately predict the reachable extreme positions and spatial poses, exhibiting low quantitative errors.
Moving forward, there are several promising research directions to further enhance the capabilities of origami manipulators.One key area for improvement is the refinement of the kinematic model to achieve more accurate analyses of both forward and inverse kinematics.For instance, it is imperative to delve deeper into the influence of substantial end loads during the modeling process, aiming to refine the model and elevate its accuracy.Concurrently, there is room for enhancement in the mechanical modeling method to enable its application to hybrid-actuation origami manipulators.Additionally, the development of a dynamic model that considers the effects of inertial forces and external excitations would enable more realistic simulations and predictions of the manipulator's behavior in dynamic scenarios.Furthermore, the integration of embedded IMUs for online pose recognition would enhance the manipulator's autonomy and adaptability in various tasks.

Experimental Section
Fabrication of the Yoshimura Origami Structure: The Yoshimura origami sheet was laser cut from 0.15 mm thickness polyethylene terephthalate (PETE) film based on the crease pattern (Figure S1a,b, Supporting Information).The creases were perforated to some extent to weaken the bending stiffness, and small holes were cut at the intersection of creases to prevent stress concentration.Additional parts were added to the sheet to facilitate connection.The sheet was then manually folded and pasted into a 3D Yoshimura-ori structure.
Fabrication of the Pneumatic Muscle Actuator: The PM actuator was made up of an inner silicone tube, a 3D-printed end cap, and an outer nylon-braided sheath (Figure S1c).One end of the silicone tube was fixed to the cap, and the other end was connected to the air tube; a nylonbraided sheath was wrapped around the silicone tube to limit its radial expansion.
Fabrication of the Waterbomb Origami Structure: The origami gripper was developed based on waterbomb origami crease pattern with 3 Â 8 units (detailed design parameters are given in Figure S1e).This 2D sheet incorporated finger components as well as additional parts for connection and pasting, and it was also laser cut from 0.15 mm PETE film.The sheet was then manually folded and pasted into a 3D waterbomb origami structure.
PCB Plates: The lower PCB plate incorporated electronic components including an IMU module for accurate posture sensing, a microprocessor unit (MPU) module for precise actuator signal control, a Wi-Fi module for seamless communication with the host computer, and two voltage regulator modules for circuit voltage reduction and power supply to the MPU (Figure 1b).The PCB plates also featured four power supply holes designated as "1," "2," "3," and "4," providing current to the SMA springs.
Calibration Tests: A universal testing machine (Instron 5965, 2 kN force sensor) was used to measure the output force of the module during bending.The experiment setup was shown in Figure 3c and detailed in Section S5, Supporting Information.

Figure 1 .
Figure 1.Design, fabrication, and control architecture of the origami manipulator.a) The CAD (Computer Aided Design) design of the origami manipulator module.b) The lower and upper PCB plates, where the main components are indicated.c) The CAD design of the origami gripper module.d) The CAD design and the photo of the four-module origami manipulator.e) Control architecture.

Figure 2 .
Figure 2. Equivalent rigid-joint modeling of the origami manipulator.a) The photo of module "i" in its initial state.b) The photo of module "i" in leftward-bending deformation.c) The equivalent rigid-joint model of the module in the initial state, where the joints "1," "2," "3," "4," and "5" are colored in grey, blue, green, red, and purple, respectively.d) The equivalent rigid-joint model of the module in leftward-bending deformation.e) The photo of the origami manipulator.f ) The equivalent rigid-link model of the origami manipulator.

Figure 3 .
Figure 3.The forward and inverse kinematic analysis of the origami manipulator.a) Experiment setup of the axial elongation test, showing the initial and elongating state.b) The length of the module under axial elongation.c) Experiment setup of the bending elongation test, showing the initially bent state and the final straightened state.d-f ) The obtained relations of 6i ψ, 6 i L bending , and 6 i τ with respect to 6 i β when the applied pressure is 0.225 MPa.g,h) The kinematic models of the origami manipulator corresponding to the configurations that the first module is bent (V ¼ f4, 1, 1, 1j0.225, 0, 0, 0g) and all modules are bent (V ¼ f4, 4, 4, 4j0.225, 0.225, 0.225, 0.225g).i-k) The results of the inverse kinematic with target points locating at ½À190; À 105; À 233, ½60; À 290; À 420, and ½117, À 117, À 276, respectively.

Figure 4 .
Figure 4. Experiments on the extreme positions of the origami manipulator.a-d) The comparison between the experimental results and the predictions from the kinematic model for modules "1," "2," "3," and "4," respectively.The left two columns show the extreme positions in the xoz and yoz planes, respectively; the right three columns display the photos of the actual configurations when the origami manipulator reaches the extreme positions.e) The radius and the z position of the endpoint of the origami manipulator when the extreme positions are reached.Both the experimentally measured and model-predicted results are given, and the relative errors are indicated.

Figure 5 .
Figure 5.The analytically obtained space curve of the origami manipulator measured through experiments.a-c) The configurations in which the endpoint of module "4" reaches the lowest, farthest, and highest position, respectively.d) End-bending configuration, e) J-shape configuration, f ) upper-bending configuration, g) S-shape configuration, and h) 3D space-curve configuration.i) The distance deviation between the experimentally measured position and the predicted position.

Figure 6 .
Figure 6.Experimental tests of the origami gripper.a) The four objects (an orange peel, a cubic sponge block, a table tennis ball, and a shuttlecock) to be gripped.b) Video snapshots of the origami manipulator when the four objects are grasped.

Table 1 .
Deformation modes and actuation patterns of an origami manipulator module.
Energize SMA "1" and "2"þ inflate PM Rightward bending Energize SMA "3" and "4"þ inflate PM Forward bending Energize SMA "2" and "3"þ inflate PM Backward bending Energize SMA "4" and "1"þ inflate PM Gripping objects (gripper) Energize SMA of the gripper the D-H model to the shape of a continuum module.It is worth noting that the D-H table provided in this subsection is specific to an individual module.