Kanty Rabenorosoa

Kejun Hu
email: kejun.hu@femto-st.fr

Morvan Ouisse
email: morvan.ouisse@femto-st.fr

Design of a Bidirectional Rotational Motion actuator by SMA with geometrico-static requirements

Keywords: shape memory alloy, origami robot, self-folding, modeling, bi-directional rotational motion

published or not. The documents may come

Design of a bidirectional rotational motion actuator by

SMA with geometrico-static requirements

INTRODUCTION

SMAs made of nickel and titanium (Nitinol) have attracted wide interest in both research [START_REF] Lester | Review and perspectives: shape memory alloy composite systems[END_REF][START_REF] Rodrigue | An overview of shape memory alloycoupled actuators and robots[END_REF] and industry [START_REF] Benafan | Recent advancements in rotary shape memory alloy actuators for aeronautics[END_REF][START_REF] Stroud | Shape memory alloy torsional actuators: a review of applications, experimental investigations, modeling, and design[END_REF] thanks to their outstanding properties such as high energy density, [START_REF] Lee | Shape memory alloy (sma)-based microscale actuators with 60% deformation rate and 1.6 khz actuation speed[END_REF] reasonable operational strain relying on shape memory effect, mechanical simplicity, and long work life. Integrated into smart composites, these properties can be used to actuate folding. [START_REF] Rus | Design, fabrication and control of origami robots[END_REF] Origami provides a means of transforming thin tessellated panels into 3D form with many unique and desirable structural properties such as auxetics, tunable stiffness, and multistability. [START_REF] Li | Architected origami materials: How folding creates sophisticated mechanical properties[END_REF][START_REF]An Origami-Based Tunable Helmholtz Resonator for Noise Control: Introduction of the Concept and Preliminary Results ], Smart Materials, Adaptive Structures and Intelligent Systems Volume 1: Development and Characterization of Multifunctional Materials; Mechanics and Behavior of Active Materials; Bioinspired Smart Materials and Systems; Energy Harvesting[END_REF] Moreover, robots inspired by folding mechanical structures known as 'Origami robots' gained much attention recently. As for the robotics field, the introduction of origami engineering enables 'semi-rigid' properties. [START_REF] Benouhiba | Origami-based auxetic tunable helmholtz resonator for noise control[END_REF] By means of actuation of origami robots using SMA, variety of actuators were introduced in last decades: unidirectional rotational motion [START_REF] Hawkes | Programmable matter by folding[END_REF][START_REF] Nayakanti | Twist-coupled kirigami cells and mechanisms[END_REF] and bi-directional rotational motion (BRM). [START_REF] Zhakypov | Designing minimal and scalable insect-inspired multilocomotion millirobots[END_REF][START_REF] Firouzeh | Robogami: A fully integrated low-profile robotic origami[END_REF][START_REF] Kim | Morphing origami block for lightweight reconfigurable system[END_REF] Previous works [START_REF] Hu | A review of sma-based actuators for bidirectional rotational motion: Application to origami robots[END_REF] provided a review of SMA-based actuation of state of the art in SMA-based actuators for BRM. Additionally, the torsional-SMA-based actuators (T-SMAs) showed an exciting potential for the BRM actuation of meso-scale origami robotics. However, due to the SMA's highly nonlinear hysteretic and temperature-depended behavior, the BRM actuators designing and characterizations are proven to be challenging. Concerning the SMA-based design, the stress-strain/force-displacement diagram on which the austenite curve of the active element and the hosting structure is frequently mentioned as a valuable and effective tool. [START_REF] Ameduri | Chapter 9 -design of sma-based structural actuators[END_REF] Previous works have shown that this method is acceptable not only for a single SMA device but also for antagonist designing. [START_REF] Hu | A review of sma-based actuators for bidirectional rotational motion: Application to origami robots[END_REF] However, the method is based on a fixed dimension of the SMA element. Thus studies usually focus on designing the bias element * . In fact, the ability due to shape memory effect (SME) of an SMA according to a certain temperature is well described by its 'M→A curve' so-call 'Austenite curve', which can be identified and modeled using the method described in. [START_REF] Hu | A review of sma-based actuators for bidirectional rotational motion: Application to origami robots[END_REF] Regarding the experimental characterization of torsional SMAs, two characterization approaches are frequently mentioned in the literature: the isobaric test and isotherm tests. The former is performed by holding constant stress and sweeping temperature up and down while measuring the strain response, relying on the shape memory effect of SMAs. The latter is performed by holding constant temperature and sweeping the strain up and down while measuring the axial load response, relying on the pseudo-elastic (pure Austenite phase) and inelastic behavior (partial or pure martensite phase). [START_REF] Churchill | Tips and tricks for characterizing shape memory alloy wire: Part 2-fundamental isothermal responses[END_REF] However, these experimental approaches require precise multi-physical controlling for thermal and mechanical conditions. Consequently, very limited experimental results of millimeter-sized actuator using T-SMA can be found in the literature. This work aims to develop a design model of a BRM actuator using SMA constitutive laws according to a geometrico-static demand. The main contributions of this research are: 1) development of a direct designing model using four design parameters; 2) experimental characterizations of the millimeter-scale T-SMAs actuator; 3) an inverse design methodology for a BRM actuator using T-SMAs for origami robots.

METHODS

Constitutive model

The three-dimensional phenomenological Lagoudas constitutive model [START_REF] Lagoudas | Shape Memory Alloys: Modeling and Engineering Applications[END_REF] is utilized in this work. This constitutive model starts with the definition of the Gibbs energy G. The energy function G is described by the set of independent state variables (σ, T, ξ, ε t ), which describe the stress, the temperature, the martensitic volume faction, and the transformation strain. The explicit form of the Gibbs free energy is given by,

G (σ, T, ξ, ε t ) = - 1 2ρ σ : S : σ - 1 ρ σ : α (T -T 0 ) + ε t + c (T -T 0 ) -T ln T T 0 -s 0 T + u 0 + 1 ρ f (ξ) ( 1
)
where ρ is the mass density, S the effective compliance tensor, α the effective thermal expansion coefficient, c the effective specific heat, s 0 the effective specific entropy, and u 0 the effective internal energy reference state. The effective compliance tensor S is determined using a parallel Voigt model of properties according to two pure phases and the martensitic volume fraction ξ via the following expression:

S(ξ) = ξS M + (1 -ξ)S A = S A + ξ∆S (2)
where the subscripts A and M denote the austenitic and martensitic phases, respectively. The symbol ∆(•) indicates the difference between the pure martensitic and austenitic phases. Also, other material parameters α; c; s 0 ; u 0 are averaged this way. To describe the interaction kinematic between the martensitic and austenitic phases, a hardening function f (ξ) is introduced. As indicated in, [START_REF] Lagoudas | Shape Memory Alloys: Modeling and Engineering Applications[END_REF] the hardening function can be written into different form according to the same state variable set: 1) exponential form, [START_REF] Tanaka | A thermomechanical sketch of shape memory effect: one-dimensional tensile behavior[END_REF] 2) cosine form, [START_REF] Liang | One-dimensional thermomechanical constitutive relations for shape memory materials[END_REF] 3) polynomial form. [START_REF] Lagoudas | Thermomechanical modeling of shape memory alloys and composites[END_REF] For example, a hardening function in cosine form is given by,

f (ξ) = ξ 0 -ρ∆s0 a M c π -cos -1 (2ξ -1) dξ + (µ c 1 + µ c 2 ) ξ; ξ > 0 ξ 0 -ρ∆s0 a A c π -cos -1 (2ξ -1) dξ + (µ c 1 -µ c 2 ) ξ; ξ < 0 (3)
where a M c , a A c , µ c 1 , µ c 2 are transformation strain hardening model parameters. The inequality of martensitic volume fraction rate ξ denote the phase transformation direction. For example, ξ > 0 presents a forward phase transformation (A → M ), and vice versa. The constitutive relation is determined using Gibbs free energy as

ε = -ρ ∂G ∂σ = S : σ + α (T -T 0 ) + ε t , (4)
based on the assumption that any change in the material's current state relies on a change of the martensite volume fraction. The evolution equation of transformation strain and the martensitic volume fraction is obtained as, ˙ = Λ ξ (5) where Λ is the normalized transformation tensor that changes principal directions depending on the direction of the martensitic transformation. The expression is given as

Λ = 3 2 H σ σ ; ξ > 0 H ε t-r εt-r ; ξ < 0 (6)
where H, σ , σ , ε tr , εtr are the maximum transformation strain, the deviatoric stress tensor, the von Mises equivalent stress, the transformation strain tensor at at the reversal point and the effective transformation strain, respectively.

Numerical model

An FEA-based geometrico-quasi-static model is constructed using an SMA torsional rod in the commercial software COMSOL. The aforementioned constitutive laws using the cosine form hardening function is implemented.

As the diameter of the rod is small with respect to its length, only a representative (unit) segment of an SMA rod is developed. [START_REF] Lagoudas | Shape Memory Alloys: Modeling and Engineering Applications[END_REF] Model features including geometry parameters (diameter and length) and material properties are chosen to compare those reported in experimental studies performed by, [START_REF] Chapman | Torsional behavior of niti wires and tubes: Modeling and experimentation[END_REF] thus providing an experimental baseline for this study. The rods were characterized in that work had three different diameters measuring around 0.5 mm and a constant length of 10.16 mm. The pseudo-elastic behavior of each sample at room temperature was characterized. It is worth mentioning that, due to the non-uniform shear stress distribution, the angle versus torque relationship relies on the material properties and the geometry. Compared to an SMA with a tensile test, the return of elastic behavior after a complete phase transformation is less obvious. The comparison between experimental results and simulation results is shown in Figure 1 B). The FE model can capture the influence of sample dimension on the angle versus torque pseudo-elastic relationship. Moreover, the criteria points such as the start and finish points according to both M → A, and A → M transformation are accurately captured. Therefore, this model is consequently utilized for the designing of the BRM actuation using antagonist arrangement in the following.

Forward model of BRM actuator for designing

A system consisting of two antagonists, so-called 'active-biased' SMA elements, is implemented in this work. Regarding the bi-directional rotational motion, an SMA-based actuator has to meet two requirements: the output torque and the motion range. In this case, four design parameters are taken into account: 1) The dimension of elementary SMA: radius r and length L of an SMA element.

2) The position angle of actuator θ: regarding the antagonist design, the maximal folding angle that the actuator can realize depends on the prestrain/pre-twisting SMA elements. This prestrain presents the strain that the SMA torsional rod undergoes when the folding hinge to be actuated is in its original position and the elementary SMA in the passive state. For a prescribed prestrain of both SMA elements, the motion range of the actuator is determined by the intersection of the austenite characteristics of the activated SMA element and the total payload torque.

3) The maximum activated temperature T : conventional SMA designing usually uses a straight form curve with Young's modulus corresponding to the pure austenitic phase. [START_REF] Ameduri | Chapter 9 -design of sma-based structural actuators[END_REF][START_REF] Koh | Self-folding origami using torsion shape memory alloy wire actuators[END_REF] However, the SMAs' phase transition temperature highly depends on the applied stress. Thus a biased SMA requires a higher temperature to achieve the transition. The overheating is indicated in the literature as a risk of degradation of SMA components and structure material. [START_REF] Mohd | A review of shape memory alloy research, applications and opportunities[END_REF] In this work, the structure to be activated utilizes polyester material, exhibiting relatively low thermal stability regarding the functional origami structure. The maximum temperature should have be taken into account. The actuation performance based on SME of an SMA at a specific temperature is accurately described by its 'Austenite curve, which can be modeled using the method described in 2.1. Here, a torque-angle-radius † iso-temperature surface is introduced in order to address the influence of the parameters as mentioned above on actuator performance. A parametric analysis is performed via the unit length SMA model based on variable radius and temperature. Twelve different radii between 0.25mm to 0.5mm and three temperatures above the finish transition temperature A f are considered.

According to the average martensite volume fraction rate ξ, thirty-six M→A curves are obtained. Those curves are then interpolated non-linearly using mathematical tools ‡ . It is worthing to remind that the implemented constitutive model does not consider the stress-induced reorientation (detwinning) of the martensite phase at low temperatures. Consequently, a virtual passive surface with modified material properties is realized to provide a similar mechanical response (see figure 5) to characterization results. We use the intersection curve between the active iso-temperature surfaces and virtual passive surfaces to predict actuator performance relying on actuator dimension and positioning angle at a specific temperature. Assuming that the antagonistic SMA are identical, only one active surface in the clockwise direction and a passive surface in the anti-clockwise direction is sufficient. The working flow is given in the following: 1) The active curves are placed at the origin point, and the passive curves are placed with a particular position angle. The two surfaces are then implemented using a grid density of 1000×1000 points. 2) The primary zone around the intersection curve is then determined using the mesh of the previous step. 3) The two localized surfaces are interpolated using mathematical tools in the primary zone. A local nonlinear solver is then used to detect the intersection point analytically according to every radius increment. Figure 3a shows an example of the intersection curve between an iso-surface at a temperature of T = A f + 70 • C with a position angle of 14 degrees/mm. The solid white line presents the intersection curve between two surfaces, and the solid black line illustrates the maximum output torque during the whole stroke. It is trivial to observe that the radius and the motion range at this position angle exhibit a nonlinear relationship.

Experimental characterization of T-SMAs

Actuator modelling requires a precise understanding of mechanical performance. The results of SMAs depend on test thermomechanical history due to their high hysteresis. In order to test the static response of the material, torque-angle measurements were performed with the electrothermal controller that maintained at a constant temperature. The objective of the test is to identify the material parameter using the FEA mentioned earlier.

As shown in figure 4 5.4mm length are tested in this case. The specimen is held using a drill chuck with split collet grips. Therefore the slippage during the isotherm test is eliminated. The samples have a coil 'shape' as initial memory. Thus all the samples are annealed in a mold at 430 degrees Celsius for 30 mins, then quenched in water to memorize a straight and untwisted shape. Concerning the isotherm test, seven temperatures between room temperature and higher temperatures are tested with five cycles at a constant rotation rate of 3 degrees/s. Both torque captured by the sensor and rotation angle given by the servomotor are then recorded to obtain the curve torque versus angle of the wires for different temperatures (see figure 5).

RESULTS

Characterization results

Figure 5 shows the experimental characteristics of SME-based T-SMA at different temperatures show the inelastic and pseudoelastic characteristics. The test results at room temperature are shown using the black dash line. It is trivial that the material yields an elastic stiffness until it achieves a 4Nmm torque, then detwinning starts and finishes at a torque of 10Nmm. A considerable amount of twisting angle is observed after an elastic release (80 degrees). Subsequently, load provides an elastic response with similar stiffness; thus, a typical elastoplastic behavior with small hysteresis is observed at room temperature. On the other hand, complete PE recovery is The isotherm test results: Torsion test of two 0.5mm diameter and 5.4mm length wires undergoing five cycles at seven temperatures. Ramp form loading/unloading torque with a maximum value of 26Nmm is cyclically applied at a rotation rate of 3 degree/s. observed experimentally at a temperature higher than 35 • C. The variations induced thermal stimulus on the start and finish criteria point of both M→A and A→M transformation are observed. Regarding the 26Nmm maximum torque, the increasing temperature reduces rotation stroke from 156 degrees to 81 degrees. Similar observation at a larger scale was reported in. [START_REF] Prahlad | Modeling and experimental characterization of sma torsional actuators[END_REF] One important difference is that in the SME T-SMAs comparing the PE T-SMAs (see figure 1 b)), the PE behaviors showed a smaller hysteresis compared to existing data of PE T-SMAs in the literature. [START_REF] Chapman | Torsional behavior of niti wires and tubes: Modeling and experimentation[END_REF][START_REF] Rao | Internal loops in superelastic shape memory alloy wires under torsion-experiments and simulations/predictions[END_REF] Additionally, a significant deviation of the start criteria of M→A transformation (beginning point of PE recovery) induced by a small difference of temperature is observed (see curves according to 35, 40 and 42 • C). The identified model parameters are shown in table 1.

Parametric analysis results

Figure 6 and figure 7 show the parametric analysis results via the direct model using the identified material properties. The motion range and the maximum output torque are computed versus the actuator dimension the motion range, the actuator radius, and the large position angles yields a highly nonlinear relationship. For a position angle less than 12 degrees/mm, the motion range is dominated by the position angle. However, for a BRM actuator with a position angle larger than 12 degrees/mm (equivalent to 180 degrees with a 15mm length actuator), local optimum of motion range is obtained with a radius equal to 0.3mm for all three cases. Regarding the technical demand for self-folding application, the folding stiffness belongs from 0.106Nmm/mm/rad to 2.1Nmm/mm/rad 28 (for example the stiffness are [0.037,0.7330]Nmm/deg for 20mm length hinge). Even with a small radius, a BRM actuator is capable of providing a typical 'open-close' self-folding motion. However, origami structures that induce geometrical nonlinearity (aka 'non-rigid' origami), or material nonlinearity (for example, 'thick-panel-origamis' and 'polymer-based origamis') require higher actuation performance. [START_REF] Meloni | Engineering origami: A comprehensive review of recent applications, design methods, and tools[END_REF] Consequently, an inverse design methodology is introduced to design a BRM actuator in a general manner.

Inverse design model

A methodology for designing the BRM actuator is developed regarding the technical requirement according to origami robots. This method couples the aforementioned direct design model and loop-form mechanical feature to identify the actuator dimensions for an origami structure that capable of 'shape-changing'. The proposed design methodology is shown in figure 8: 1. In the system requirement block, a structure's single or multiple shape morphing are defined first, locally or globally [START_REF] Nayakanti | Twist-coupled kirigami cells and mechanisms[END_REF][START_REF] Jianguo | Folding behavior of a foldable prismatic mast with kresling origami pattern[END_REF] ). According to each shape morphing strategy, the necessary performance can be determined using experimental characterization of prototype or modelling.

2. In the choice of the material bloc, a proper SMA adapted to an application should be determined. This selection should be based on system constraints and requirements, which include the torque and stroke, surrounding and actuator temperatures, fatigue, energy consumption, and the working environment conditions.

3. After the selection of SMA material, the properties mentioned in 2.1 are mandatory for the performance prediction.

4. In the thermochemical modelling block, the torque-angle-radius iso-temperature surface presented in 2.3 is etablished to identify the dimension of SMA element and the position angle that answers the system requirement.

5. A characterization on the whole system to validate the designing of the actuator, the design process will be finished if the technical requirements are attained. If it is not the case, the design loop will re-begin at actuator dimensions for another iteration with modification.

These are the indications for the design process for the dimensioning of a antagonistic BRM actuator using T-SMA; all the considerations are valid for both commercial torsional wires or home made samples.

CONCLUSION

A design model has been established for the design of a BRM actuator activated by antagonist T-SMAs. The model is constructed via a unit-length T-SMA FEA model using Lagoudas three-dimension constitutive law. Four design parameters are considered to meet the actuator requirements (motion range and output torque). Two design approaches are introduced according to the different design problems. Regarding the direct design method, a parametric analysis is performed to understand the influence of each parameter on actuator performance. The results showed that a tuning of elements' radius and position angle induces a performance modification. Regarding the inverse design model, a loop-form mechanical design feature is introduced for prototype fabrication and characterization of the BRM actuator for specific application requirements.

Figure 1 :

 1 Figure 1: Simulation results of a pseudo-elastic SMA-based torsional element: a) boundary conditions for the rod, red arrows represents the applied torque, blue arrows represent the fixed constraint boundary condition; b) comparison between experimental results and simulation results, the experimental data is from.23

Figure 2 :

 2 Figure 2: Examples for iso-temperature surfaces using identified parameters. The image on the left presents the virtual passive surface at room temperature, and the image on the right shows the unit length actuator performance at A f + 70 • C.

Figure 3 :

 3 Figure 3: An example of forward model for designing: a) an example for intersection curve assuming identical antagonistic T-SMAs activated at A f + 70 • pre-twisted at a unit rotation angle of 14 degrees; b) an example of maximum output torque surface for different radius and different position angle; and c) an example of motion range surface for different radius and different position angle.

Figure 4 :

 4 Figure 4: Experimental setup for charaterization of T-SMA actuator

Figure 5 :

 5 Figure5: The isotherm test results: Torsion test of two 0.5mm diameter and 5.4mm length wires undergoing five cycles at seven temperatures. Ramp form loading/unloading torque with a maximum value of 26Nmm is cyclically applied at a rotation rate of 3 degree/s.

  the position angle according to three activation temperature. As shown in the figure6, an increase of elements' radius and a decrease of position angle induce the increase of output torque. Additionally, an increase of output torque around 100% can be obtained by increasing the activation temperature from A f + 20 • C to A f + 70 • C thanks to the high stress influence coefficient of austenite. On the other hand, the relation between

Figure 6 :

 6 Figure 6: An example of BRM actuator output torque prediction using the inverse design model for temperature, the corresponding activation temperatures from left to right are: A f + 20 • C, A f + 50 • C and A f + 70 • C

Figure 7 :

 7 Figure 7: An example of BRM actuator motion range prediction using the inverse design model for temperature, the corresponding activation temperatures from left to right are: A f + 20 • C, A f + 50 • C and A f + 70 • C

Figure 8 :

 8 Figure 8: A loop form inverse design method of SMA-based BRM actuator

Table 1 :

 1 Identified Model parameters of tested T-SMA samples: M: PE represents pseudo-elastic behaviors at temperatures above A f , and VP represents the virtual passive response at room temperature.

		Model parameter
	Symbols	Description	Values	Units
			PE	VP
	E M	Elastic Modulus of Martensite	41	38	GPa
	E A	Elastic Modulus of Austenite	65	38	GPa
	M f	Martensite Finish Transformation Temperature	0	16	• C
	M s	Martensite Start Transformation Temperature	25	26	• C
	A s	Austenite Start Transformation Temperature	8	-	• C
	A f	Austenite Finish Transformation Temperature	33	-	• C
	C M	Stress Influence Coefficient of Martensite	31	25.5	MPa/ • C
	C A	Stress Influence Coefficient of austenite	30.

† The angle here is a unit length rotation angle ‡ in this case, the data are fitted using 'griddata' code via Matlab

ACKNOWLEDGMENTS

The authors would like to acknowledge the financial support provided by the French National Agency for Research (OrigaBot ANR-18-CE33-0008), and EUR EIPHI (Contract No. ANR-17-EURE-0002).


Umar Saeed Malik
email: umarsaeed.malik@edu.univ-fcomte.fr

Kejun Hu
email: kejun.hu@femto-st.fr

Morvan Ouisse
email: morvan.ouisse@femto-st.fr

Kanty Rabenorosoa

Modeling of a

Keywords: Shape Memory Alloy, Shape Memory Polymer, Origami robots, Torsional Actuator, Shape Changing, Shape Blocking, Finite Element Analysis

published or not. The documents may come

INTRODUCTION

Origami is the art of paper folding which dates back to the 16 th century and is known to be discovered by the Japanese. Traditionally, origami involves the manual manipulation of paper to transform it into complex 3D shapes. There are many avenues where origami can be used ? . For example, its excellent morphing abilities makes it an ideal candidate in space, ? biomedical, ? or acoustic ? applications.

Integration of origami in the field of robotics is a relatively new and fascinating field where 'Origami Robots' are designed to have extraordinary capabilities ? . An origami robot is a moving device whose morphology and functions are based on self deployment/folding. Ref. ? used the waterbomb origami structure to build deformable wheels of a mobile robot. Robogami is a robot developed by ref. ? demonstrating multi DOF motion using (soft) sheet SMA actuators.

In recent times, the introduction of smart materials in origami robots has enhanced their performance to new levels ? . The mechanical properties of smart materials can vary with a physical property like temperature and manipulating it can impart fascinating behaviours in the structure. For example, figure 1 (adapted from ref. ?) shows a comparison of the elastic modulus vs. temperature for SMA, SMP and conventional metallic materials. Metallic materials have a nearly constant elastic modulus, thus having similar properties at all temperatures, however, SMA exhibits properties such as Shape Memory Effect (SME) and Pseudo Elasticity (PE). Similarly, based on the temperature-dependent elastic modulus, SMP becomes soft or rigid, thus acquiring the ability of blocking or enabling the structure. Ref. ? demonstrated the self-folding capability of origami using shape memory polymers. Owing to a hybrid approach (using torsional actuators) dynamic shape changing and (utilizing

Nearly Constant Elastic Modulus

Figure 1: Variation of elastic modulus with temperature for different classes of materials. A s and A f define the start and finish temperatures for free SME in the SMA, respectively. T g is called the glass transition temperature and it defines the transition between glass and rubbery states for polymers. Shape changing and shape blocking are behaviours exhibited by SMP. Adapted from ref. ?. supplementary mechanism) shape blocking has been demonstrated by ref. ? and ?. While the former possesses reversible actuation and uses the buckling effect in a modified version of the kresling tower based structure, the latter is irreversible and uses a rigid mechanism to block its shape after being fully deployed.

Torsional SMA actuators are still a relatively emerging field and not much work has been done related to them. The main advantages of using torsional SMA actuators are their design simplicity, less space requirements and their ability to undergo large shear strains ? . Ref. ? fabricated and analytically studied a torsional SMA actuator which is embedded in origami structure and is responsible for folding the structure when heated. Torsional SMA actuated structures can be integrated with SMP to impart tunable stiffness to structures controlled by the temperature. Ref. ? presents a weakly coupled FE model for a rigid host structure-SMA structure where first, a comparison between linear circular wire and torsional tubes is done based on FEA and then, their implementation on a morphing wing is presented. Although, a few works have been carried out to study the performance of millimeter-sized torsional SMA actuators using FEA, largely the works are based on the fabrication and realization of such folding structures. Moreover, works on FE studies having SMA actuators and host-structure/SMP coupling exist but are still less in number. This work contributes to the following:

1. Development of a FE based thermomechanical model which couples a torsional SMA actuator and a rigid hosting structure.

2. Influence of temperature-dependent tunable stiffness of SMPs for shape changing and shape blocking in torsional SMA actuated millimeter-sized origami robot applications.

In section 2, modelling is discussed laying out the conceptual and mathematical models, study cases and the constitutive behaviours of SMMs. Section 3 presents simulation results showing the influence of SMP stiffness for shape changing and shape blocking scenarios. Lastly, section 4 presents the outcome of the work and provides perspectives regarding the FEA based approach for designing of SMM based reconfigurable origami robots.

MODELLING

Conceptual Model

The complete structure is built as a millimeter-sized geometry (32 mm x 20 mm x 4 mm). Figure 2 shows the geometric model of the structure identifying the three material domains. The SMA actuator is represented by the 17 mm long cylinder in the middle having a diameter of 1 mm. The actuator is connected to the rigid structure on the ends of the cylinder. The rigid structure is built as a symmetric structure about the ZY plane with a distance of 10 mm between the two rigid parts. It consists of a connecting cubic block (3 x 6.5 x 3 mm) which connects the actuator to the main geometry having a cross section of 4 mm x 5 mm and a length of 32 mm. The rigid part is modelled with a very high stiffness. To have tunable stiffness, two SMP stripes are placed at the opposite ends of the structure. The thickness of each stripe is 1.5 mm while it is 0.4 mm in the middle to have a rotational hinge. Each SMP stripe is 4 mm wide.

Constitutive Behaviour

Shape Memory Alloy (SMA)

SMAs are alloys made of metals that are capable of undergoing large deformation without permanent change in shape thanks to their two distinct phases, martensite and austenite. Over the years, many efforts have been made to develop constitutive laws for SMAs ? . These models can be mainly divided into 3 types which are the microscopic models, micro-macroscopic models and the phenomenological models ? . The first two types provide more details than the third one but are complex to apply at device scale. Phenomenological models represent the average macroscopic behavior of SMAs. They take into account a limited number of internal variables and assume the material to be homogeneous which makes their implementation relatively easier. These models capture the behavior at macro and millimeter-sized level; therefore, they are a good choice for application level modelling. In the present work, 3D Lagoudas ? constitutive phenomenological law has been used. In this law, to define the thermodynamic potential, the Gibbs free energy is used. It has the advantage of the custom definition of thermomechanical loading path for SMAs in the stress-temperature space. The internal variables that best represent the thermomechanical response of the SMAs at the micro-structure level are chosen. Thus, the constitutive model is represented by a set of internal state variables (σ, T, ξ, ε t ), which represent the stress, the temperature, the martensitic volume fraction and the transformation strain, respectively. To precisely define the transformation between the two phases, a hardening function f (ξ) is introduced. According to ref. ?, this hardening function can be written in terms of the same state variable set and in different forms such as the exponential form, cosine form, and polynomial form. Mathematically, the Gibbs free energy can be written as given in equation 1, where ρ is the mass density, S is the effective compliance tensor, α is the effective thermal expansion coefficient, c is the effective specific heat, s 0 is the effective specific entropy and u 0 is the effective internal energy reference state.

G(σ, T, ξ, ε t ) = -1 2ρ σ : S : σ -1 ρ σ : [α (T -T 0 ) + ε t ] + c (T -T 0 ) -T ln T T 0 -s 0 T + u 0 + 1 ρ f (ξ) (1)
Since the SMA has two distinct phases, martensite and austenite, its effective material properties are written in terms of the properties for the pure phases and martensitic volume fraction ξ, which can be written as:

Υ(ξ) = ξΥ M + (1 -ξ)Υ A = Υ A + ξ∆Υ Υ = S; α; c; s 0 ; u 0 (2)
where the superscripts A and M denote the austenitic and martensitic phases, respectively. The symbol ∆ indicates the difference between the pure martensitic and austenitic phases. Using the Gibbs free energy formulation, the constitutive relation for the infinitesimal strain tensor can be written as:

ε = -ρ ∂G ∂σ = S : σ + α (T -T 0 ) + ε t (3)
There is a key assumption in this model wherin the martensitic variant reorientation is not taken into account during the phase transformation and thus, any change in the current state of the material is directly linked to a change of the martensitic volume fraction ξ. Consequently, the evolution of transformation strain and the evolution of martensitic volume fraction can be related by the following equation:

˙ = Λ ξ (4)
where Λ is the normalized transformation tensor that explains the direction of the martensitic transformation.

It can be written as:

Λ = 3 2 H σ σ ; ξ > 0 H ε t-r εt-r ; ξ < 0 (5)
where H is a material parameter associated with the maximum transformation strain, σ , σ , ε t-r , εt-r are the deviatoric stress tensor, the effective von Mises equivalent stress, the transformation strain tensor at the reversal point and the effective transformation strain at the reversal of the phase transformation, respectively. The inequality of martensitic volume fraction rate ξ indicates the phase transformation direction. ξ > 0, represents a forward phase transformation (A → M ), and vice versa.

For the present work, the mechanical, thermal and transformation properties are provided in table 1. SMPs are viscoelastic materials whose response varies over time even if applied load remains constant. Their properties depend on the frequency of loading and also on the temperature. The stiffness of SMP is generally increasing with frequency and decreasing with temperature (see figure 1).

The material used for this study has been already studied in ref. ? in the frequency domain. In that work, using a DMA, experimental data was acquired (for a cylindrical specimen with a diameter of 1.75 mm and length of 6 mm) from 33 o C to 83 o C with an increment of 5 o C at a heating rate of 2 o C/min. This experimental data is used for the simulation reported in this work.

For the present work, the interest is to show the influence of the SMP stiffness according to the temperature on the motion range of an active hinge (shape changing) and also on the global rotational stiffness (shape blocking). Neglecting the dynamic experimental data, only the storage modulus E at the minimum frequency (0.1 Hz) is extracted from the data and used as a function of temperature in this work. The storage modulus varies from 1455 MPa at 33 o C to 3.26 MPa at 83 o C. For the shape changing case, several values in between these two values are considered and a parametric study is performed. For the shape blocking study, only the the stiffness at 33 o C (1455 MPa) and 53 o C (18.2 MPa) is considered. The SMP is assumed as an isotropic linear elastic material with a constant elastic modulus in each study and a Poisson's ratio of 0.37.

Finite Element Model Configurations/Settings

Stationary analysis is performed neglecting the dynamic effects due to inertia. The domain mesh consists of mainly tetrahedra elements having quadratic shape functions, though other types of elements are also present but small in number. It is important to mention here that to achieve a large folding angle, the small strain assumption will not be valid in this study and large deformation theory (geometric nonlinearity) is used. With the SMA having a nonlinear constitutive behaviour and also the geometric nonlinearity, specific techniques are used to ensure convergence. A direct linear PARDISO solver is used along with nonlinear solver based on Newton Method. The technique of continuation method is used. It divides the total load/displacement in several steps thus making the load start from 0 and end at the desired final load. At each step, a study is performed and the initial guess is taken from the solution of the preceding step which ensures better convergence. To implement this, a continuation parameter (cont. para.) is used. The applied displacements/load and temperature (for SMA) are a function of this parameter. The solution is obtained at each parameter value. The continuation parameter is given as a number starting from 0 to a specific final value which is different in each of the two study cases.

Study Scenarios and Boundary Conditions

Two different studies are performed in order to represent the shape changing and shape blocking capabilities. Both studies are performed as a 3 step procedure.

In the first step (cont. para. = 0 -1), the structure is loaded with an initial angular rotation of 45 on one side, while the other side of the structure is fixed. This is called as the 'Pre Strain' step. In the second step (cont. para. = 1 -2), the structure is set free from the loaded side (while the other side is still fixed) and allowed to elastically release (called 'Elastic Release').

The aforementioned first two steps are common to both studies while the third step for each study is explained in subsequent subsections.

Case 1: Shape Changing

The main purpose of performing this study is to present shape changing/recovery (after an initial rotation) because of SME of the SMA actuator and also to see the influence of SMP stiffness on the motion and hence the final state of the structure.

After the two initial steps, in the third step (cont. para. = 2 -4), the SMA is subjected to a temperature increase, thus activating the SME. The evolution of the thermal stimuli and applied displacements will be discussed in section 3.

Practically, this study represents the case when it is desired to achieve a partial or full unfolding/folding of the structure which can be controlled by controlling the temperature of the SMP.

Case 2: Shape Blocking

In real life, origami structures are light weight and soft. While it is not desired, a small disturbance can be enough to change their shape. SMP can be used in such scenarios to block the shape of the structure when desired. They can be cooled or left at room temperature to contribute to higher global stiffness and hence ensure the locking of the shape. Following the step 2, in the third step (cont. para = 2 -3), one side of the rigid part is subjected to a small normal force causing a very small rotation about the SMA axis, while the other side stays fixed. This study is performed to quantify the augmented rotational stiffness of the complete structure due to SMP for shape blocking application.

RESULTS AND DISCUSSION

Pre Strain

Figure 3 shows the spatial evolution of von Mises stress and the martensitic volume fraction ξ in the cross section of SMA actuator at the pre strained stage. As also observed by ref. ?, because of the 45 rotation, the torsional SMA actuator undergoes large shear strains at the outer ends while it tends to zero at the center. This explains the increase of von Mises stress through the radial direction. Additionally, as the martensitic volume fraction is directly linked to the stress levels, the contours represent that there is more transformation at the ends and the martensitic volume fraction ξ is about 16%, while the austenite phase persists in the centre. In fact, even if the applied rotation tended to infinity, the volume fraction ξ would still not completely reach 100%.

Shape Changing

Input parameters

Figure 4a shows the evolution of inputs during the three steps of the shape changing study. During the prestrain (cont. para. = 0 -1), the structure is folded to an angle of 45, while it is fixed from the other end. This is followed by an elastic release (cont. para. = 1 -2). In the shape changing step, (cont. para. = 2 -4), the temperature of SMA increases from M s + 5 to A f at the end to demonstrate SME. As a result, the SMA actuator starts to experience reverse transformation from martensite to austenite, however the presence of SMP becomes a hindrance to that and depending on its stiffness, different final folding angles described as shape changing are achieved. This is further described in the subsequent subsections.

Influence of SMP Stiffness on Total Displacement

As a result of the SME of the SMA actuator, it is observed that the structure unfolds/deploys back to its initial unfolded state. However, this is the case only when there is no SMP patches. In the case when SMP patches are present, during reverse transformation, the structure faces some resistance depending on the stiffness of the SMP patches which limits the rotation angle travelled by the structure and thus, shape changing is achieved. Figure 5 shows the total displacement of the structure at the final state of simulation (after the SME). The displacements shown are relative to the unfolded initial state of the structure as shown in figure 2. Two cases are shown. Both cases are achieved after a similar prestrain of 45 and elastic release. However, the two cases show the displacements for the two extreme cases of SMP stiffness. On the left, the case with the lowest SMP stiffness (E SM P = 3.2 MPa) is shown, which occurs at a temperature of 83 o C, while shown on the right is the case with the highest SMP stiffness (E SM P = 1455 MPa), that occurs at a temperature of 33 o C. It is evident from the figure that at the final instance, for the lowest stiffness case (left), the structure nearly gets restored to its original unfolded state with a maximum displacement of just 0.08 mm. On the contrary, for the highest stiffness case (right), the SMA actuator faces a large resistance from the SMP during SME and ends up unfolding the structure by a very small angle as a result of which there is a total maximum displacement of 6 mm. Furthermore, it indicates that at ambient SMP temperature (high SMP stiffness), the SMP is able to block/limit the SME of the SMA.

For better quantification of the motion range, a parametric study was performed with 9 cases of SMP stiffness in between the two aforementioned extreme values.

Influence of SMP Stiffness on Rotation Angle: Parametric Results

Figure 6a represents the effect of SMP stiffness on the total angular rotation restored back by SME, as well as the percentage of shape changing (red text) in each case. The shown values of rotation angle are relative to the state of the structure after the elastic release. The percentage values are computed as the ratio of the achieved rotation angle and the initial prestrain angle of 45. The results represent an asymptotic trend with respect to the two extreme values. With an initial prestrain of 45 in all the cases, 99% shape changing is observed when the SMP offers very small stiffness, however, as the stiffness of SMP increases, the motion range of the structure becomes more limited and asymptotically reaches a minimal value of 7.1 resulting in just 16% shape changing. This demonstrates the capability of shape changing due to temperature-dependent tunable stiffness of the SMP material. By controlling the temperature, any desired angular rotation can be achieved within the bounds of the structure's maximum motion range.

Influence of SMP Stiffness on Average Martensitic Volume Fraction: Parametric Results

As the SMA actuator undergoes reverse transformation (SME), the volume fraction is bound to decrease, however, its final value depends on the strains at the final rotation angle with respect to the initial (prestrain) one. Figure 6b represents the average volume fraction of martensite ξ inside the SMA actuator at the final state (after SME). ξ, at the end of elastic release has a value of 10.4% which is shown by the blue straight dashed line. The orange columns represent the volume fraction (in %) at the final step for each case of SMP stiffness while the red dashed line represents the trend of the volume fraction. Similar to the case for rotation angle, an asymptotic trend is observed. For the case with the lowest SMP stiffness (E SM P = 3.2 MPa), figure 6b shows that the martensitic volume fraction ξ is 0.1% which highlights that the SMA has gone through the reverse transformation almost completely. On the contrary, for the case with the highest SMP stiffness (E SM P = 1455 MPa), the martensitic volume fraction is 9.2% which is very close to the initial value (10.4%, after elastic release), thus showing that there is very less reverse transformation.

Conclusively, it can be said that the motion range of the structure varies distinctively with the different values of SMP modulus and by controlling the temperature of the SMP stripes, shape changing can be achieved.

Shape Blocking

Another objective to have a SMA-SMP coupled model is the capability of resisting external disturbances from the environment. This can also be referred to as shape blocking. Since origami structures are light weight and soft, they have less inherent blocking abilities, it is therefore important to artificially enhance shape blocking abilities. In this section it is presented, that how by using some SMP patches, whenever desired, the rotational stiffness of the structure can be enhanced.

Input Parameters

Figure 4b presents an evolution of inputs for the three steps of the shape blocking study with the continuation parameter. The first two steps are the same as in the case of shape changing and after the elastic release, while in the third step (cont. para. = 2 -3), a force of 0.15 N is exerted on one edge resulting in a small angular rotation towards the initial unfolded state.

Rotational Stiffness

This section provides the quantification of the rotational stiffness offered by the folded structure in resistance to an external load. The objective is to enhance the shape blocking capabilities, not shape changing. For this, SME should not be activated (because it helps in shape changing), therefore the temperature in the SMA during this study is constant. It is important to mention here that since the SMP material will intuitively provide more stiffness at lower temperatures and will offer less stiffness at higher temperatures (higher than its glass transition temperature), so only three cases are considered: first without SMP, second with normal SMP (33 o C, E SM P = 1455 MPa), and third with hot SMP (53 o C, E SM P = 18.2 MPa).

Figure 7 presents the linear evolution of reaction torque vs rotation angle for the complete structure caused as a result of the applied force for the three cases. The red line represents the case when no SMP material is used whereas the blue and yellow lines show the cases when two stripes of SMP material are used at low/normal (33 o C) and high (53 o C) temperatures respectively. The initial value of the rotation angle is taken from the end of elastic release stage and starts from 0. The reaction torque is computed about the center of the structure (in the middle of the SMA axis). The markers on the figure correspond to the results at each continuation parameter. Initially, there is no rotation and no resulting reaction torque.

For the applied load, in the case without the SMP, the structure rotates approx. 0.75 and generates a reaction torque of approx. 1.5 N.mm which results in a rotational stiffness (slope of red curve) of K θ ≈ 1.97 N.mm/ • . For the case with the (normal) SMP at 33 o C, the structure rotates approx. 0.6 and generates a reaction torque of approx. 1.5 N.mm which results in a rotational stiffness (slope of blue curve) of K θ ≈ 2.5 N.mm/ • . Lastly, for the case with (hot) SMP at 53 o C, the rotational stiffness (slope of yellow curve) is K θ ≈ 2 N.mm/ • , which is almost the same as in the case without the SMP (1.97 N.mm/ • ).

Conclusively, it can be deduced that because of the use of SMP material (at low temperature (33 o C)), the rotational stiffness of the structure increases by approximately 25% as compared to the 'without SMP' case, thus resulting in enhanced shape blocking (higher resistance to external disturbances). However, when the SMP is at a high temperature (53 o C), the increase in the global rotational stiffness is negligible and thus, it becomes favorable to change the shape. Furthermore, increasing the thickness and width of the SMP stripes increases the rotational stiffness significantly and a parametric study can be performed to quantify this increase in the global rotational stiffness.

Figure 7: Variation of rotation angle after elastic release state vs reaction torque about SMA axis. The red curve shows the case without SMP, the blue curve shows the case with the normal SMP (E SM P = 1455 MPa) and the yellow curve shows the case with the hot SMP (E SM P = 18.2 MPa). An increase of 25% in the rotational stiffness is observed by using the SMP at low temperature, while, with the SMP at high temperature, the increase in stiffness is negligible. K θ represents the rotational stiffness.

CONCLUSIONS AND PERSPECTIVES

The objective of this work was to build and simulate a strongly coupled shape memory material based self deploying thermomechanical model of an origami hinge, capable of both, shape changing and shape blocking. A torsional SMA actuator mounted on a rigid structure with patches of SMP material was presented. The SMA was represented by the 3D Lagoudas constitutive law and the SMP was assumed as a linear isotropic material. The technique of continuation method was implemented along with a nonlinear solver to resolve a model consisting of nonlinear constitutive behaviour and geometric nonlinearity. While the SMA actuator imparts the capability of self deployment because of the SME, the SMP (thanks to its temperature-dependent tunable stiffness) provides the advantage of controlling the unfolding angle (shape changing) and enhancing the ability to block the shape when desired.

For the given geometric configuration, two study cases were presented namely, the shape changing and shape blocking. For shape changing, it was concluded that the folding of the structure can be controlled and can undergo a percentage shape changing from 16% to 99% thanks to the tunable stiffness of SMP. For shape blocking, it was deduced that by using SMP at a lower temperature (33 o C), the global rotational stiffness can be increased by approximately 25%, therefore enhancing the shape blocking capabilities. However, when the SMP is heated and used at a higher temperature (53 o C), there is a very small added stiffness and it becomes favorable to fold/unfold the structure (change the shape).

Furthermore, as a future perspective, the geometry can be topologically optimized to obtain the desired shape changing and shape blocking properties. Naturally, the proposed hinge will be integrated on a physical prototype for experimental validation and performance assessment.

Figure 2 :

 2 Figure 2: 3D geometry with different material domains. Small figure shows the folded geometry after pre strain (45). All dimensions are in mm.

( a )

 a Case 1 -shape changing: Evolution of applied rotation angle and temperature in SMA vs continuation parameter.(b) Case 2 -shape blocking: Evolution of applied rotation angle and force vs continuation parameter. Temperature in SMA is constant at (Ms + 5).

Figure 4 :

 4 Figure 4: Input parameters for both studies.

Figure 5 :

 5 Figure 5: Influence of SMP modulus on shape changing after an initial rotation of 45. At low SMP stiffness (left), maximum shape changing from elastic release stage is observed (d max = 0.08 mm), while at high modulus (right), just a slight restoration of rotation from elastic release stage is observed (d max = 6 mm). The displacements shown here are relative to the unfolded initial state of the structure as shown in figure 2.

  SME Before SME Percent (b) Martensitic volume fraction ξ after SME at different values of SMP modulus. An increasing trend is seen with increasing stiffness.

Figure 6 :

 6 Figure 6: Influence of SMP on rotation angle (6a) and martensitic volume fraction (6b).

Table 1 :

 1 Parameters for SMA model

	Parameter	Value
	Mechanical Properties
	E A , E M	75 GPa, 28 GPa
	ν A , ν M	0.33, 0.33
	Phase Properties
	M s , M f	325.15 K, 315.15 K
	A s , A f	341.15 K, 351.15 K
	C A , C M	7.4 MPa/K, 7.4MPa/K
	Transformation Strain Properties
	tr max	8%

ACKNOWLEDGMENTS

This work was supported by the French National Agency for Research (OrigaBot ANR-18-CE33-0008), and EUR EIPHI (Contract No. ANR-17-EURE-0002).