Jiles-Atherton Based Hysteresis Identification of Shape Memory Alloy-Actuating Compliant Mechanism via Modified Particle Swarm Optimization Algorithm

1College of Automation Science and Engineering, South China University of Technology, Guangzhou, Guangdong 510641, China 2Key Laboratory of Autonomous Systems and Networked Control, Ministry of Education, South China University of Technology, Guangzhou 510640, China 3Department of Electronic and Information Systems, Shibaura Institute of Technology, Saitama 337-8570, Japan 4School of Electronic Engineering and Automation, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China


Introduction
Shape memory alloys (SMA) are a class of smart-materials which can be utilized as "artificial muscle" actuating device, imitating the performance of natural muscles [1][2][3].Compared with the conventional actuator devices, such as hydraulics and DC motor, the special feature of SMA materials can work around the limitation of low power density and large size in these conventional actuators [4,5].Currently, there are several artificial muscle actuators, such as pneumatic artificial muscles [6], electroactive polymers (EAP), and SMA actuators [7,8].For the pneumatic systems, they require several peripheral types of equipment such as air compressors, valves, and pressure sensors, and these components make the system bulky and noisy.As for SMAbased actuators, they can provide silent operating condition and do not need the separate pump to provide the energy, which can save the actuating device size and provide better operation environment [8,9].Different from EAP actuators, SMA-based actuators with the low-voltage driving mode can be used easily as "muscle" devices.Besides, the flexible shape, wide operation range, and high power to weight ratio are also the good features of the SMA-based compliant actuators and have been applied in microactuating systems, miniature robots, compliant mechanism, and biomedical instruments [10,11].
The working principle of SMA-based actuators is the energy transformation from thermal energy to mechanical energy with the change of the temperature or vice versa [1].The heating or cooling of SMA-based actuators will cause the shape deformation of the SMA materials, which creates the smooth stress or strain as the actuating force to the actuated plants, and this output performance can be used as compliant actuators well [12].In this paper, a different 2 Complexity SMA-based actuator structure introduced in [5] is adopted as the testified platform, and the connecting way of the SMA wires can decrease the stiffness and increase the compliance, showing the muscle-like actuating performance [13].
However, one of the shortcomings of SMA-based compliant actuators is the hysteretic nonlinearity in the process of SMA phase transformation [4].The thermodynamical characteristics of SMA materials are not reversible, which lead to strong saturated hysteretic behavior.Besides, the delay between the input driving signal and the measured temperature can couple with the internal hysteresis in the SMA materials, causing complex coupling hysteresis nonlinear characteristics in the output of SMA-based compliant actuators [14].The strong hysteresis nonlinearities decrease the feasibility to implement the SMA-based compliant actuators as muscle-like operation devices.Therefore, the modeling of hysteresis becomes the primary work to improve the actuating precision of SMA-based compliant actuators [14,15].
Generally, there are two types of hysteresis modeling methods.One is to establish the model based on the physical equations or parameters of the mechanism, such as Stoner-Wohlfarth model and Globus model [16]; another is to build the model based on the input-output relationship of the mechanism, which does not need the physical parameters of the system and can be extended to the different systems easily, for example, Preisach model [17], Prandtl-Ishlinskii model [18,19], Bouc-Wen model [20], and Jiles-Atherton (J-A) model [21].Therein, the Jiles-Atherton modeling method originates from the hysteresis description in the magnetic field, and it can provide an accurate description for the strong saturated hysteresis in the SMA-based compliant actuators [22].Moreover, the Jiles-Atherton model needs less storage compared with other hysteresis modeling methods [23], which is easier for the real-time application.In this paper, the Jiles-Atherton model is adopted to describe the saturated hysteresis in the SMA-based compliant actuators to show the output characteristics of the compliant actuators.
As introduced in [21], the structure of the Jiles-Atherton model is combined with several differential equations, and the output of Jiles-Atherton model is to obtain the analytical solution of these differential equations, so the parameters in the Jiles-Atherton model are not easy to be identified directly by using the common identification method which limits the implementation of the Jiles-Atherton model.Therefore, the parameter identification method of the Jiles-Atherton model is addressed as the primary goal of this study to show the output performance of SMA-based compliant actuators.
To solve the difficulties of the parameter identification in the SMA-based compliant actuators, a modified particle swarm optimization (PSO) algorithm is proposed in this paper.The PSO algorithm is a new computation technique proposed in 1995 [24], which is a class of swarmbased algorithms, starting with a population of random solutions and continuing by improving population using a cost function.Different from the conventional identification algorithms, PSO algorithm can obtain the optimal solution of the parameters in a multidimensional space [25,26], and it should obtain better solutions of the parameters in the Jiles-Atherton model effectively.Compared with the basic PSO algorithm, the proposed PSO algorithm is designed to reduce the chance of getting trapped in a local optimum.By using the random mutation step for the updating of the parameters in the location and velocity, the parameters of the Jiles-Atherton hysteresis model identified using the modified PSO algorithm can describe the hysteresis nonlinearities in the SMA-based compliant actuators more accurately.To verify the accuracy of the proposed identification method, the experiment is conducted by choosing the PWM signal's cycles as 60s with amplitude 5V and duty cycle as 50%.The experimental comparison results with the basic PSO algorithm with random mutation step illustrate that the proposed identification algorithm behaves better for the SMA-based compliant actuators.
The main content of the paper is as follows: Section 2 introduced the SMA-based compliant actuators discussed in this paper.The Jiles-Atherton hysteresis model adopted to describe the hysteretic nonlinearities in the SMA compliant actuating mechanism is introduced in Section 3. In Section 4, the structures of the basic PSO algorithm and the proposed modified PSO algorithm are presented.The experimental output and the model accuracy evaluation test are given in Section 5 to verify the effectiveness of the proposed identification method.Finally, the conclusion about the proposed identification method for the Jiles-Atherton model is presented in Section 6.

SMA-Based Compliant Actuating Platform
In this section, the compliant actuating platform based on SMA actuators is introduced.By changing the temperature, the phase transformation of shape memory alloys from the low-temperature state (martensite phase) into the high-temperature state (austenite phase) or vice versa will stimulate the mechanical force (strain or stress) with the high strength and large recovery strain properties during the activation process (heating process) or the deactivation process (cooling process).
Usually, there are three typical structures: one-way SMA actuator, bias-spring SMA actuators, and different (antagonistic) actuators.One-way SMA actuator and bias-spring SMA actuators have slow response since the deactivation process is determined by the cooling process or the passive spring which leads to the uncontrollable deformation speed.To solve this problem, a different SMA wires structure [5] is used in this study, which consists of two complementary SMA wires, two couplers and a torsion spring.As shown in Figure 1, the antagonist wires can produce the active force to imitate the rotational motion of the human joint in bidirection, and the motion speed can be adjusted by the active contraction force between the SMA wires and the couplers connected by the spring.The advantage of this structure is to get the muscle-like actuating output with functions of high flexibility, lightweight, and noiselessness in the human motor systems.
To show the actuating output of the SMA-based compliant actuating system, the input-output relationship was testified in the experiment device, where the change of temperature acting on the SMA wires could be obtained Remark 1.According to the experiment results of the openloop test, the hysteresis in the SMA-based compliant actuator with strong saturation property and some hysteresis modeling methods cannot cover this special characteristic, for example, conventional Prandtl-Ishlinskii model [19] and Backlash-like model [27].The existence of the saturated hysteresis will degrade the output performance of the SMAbased compliant actuator, and the modeling of hysteresis becomes a primary step to improve the actuating performance.In this paper, a Jiles-Atherton model is employed as an illustration to show the way of dealing with the strong hysteresis nonlinearities.

Hysteresis Modeling with Jiles-Atherton Model
The Jiles-Atherton model [21,23] is a typical hysteresis model, which is derived from ferromagnetic magnetization theory to show the relationship between applied magnetic field  and magnetization  by considering the nonmagnetic impurity, grain boundary, residual stress, and the hysteresis.In this section, the Jiles-Atherton model is introduced briefly.It enables the mathematical description of hysteresis curves between magnetic flux intensity  and magnetic field intensity  with five parameters [28].Usually, magnetization  includes two components, reversible magnetization  V and irreversible magnetization   : where  V and   can be obtained by the anhysteretic magnetization   which is produced by the rotation of the magnetic domain. is the magnetization weighting factor.The anhysteretic magnetization can be provided by the Langevin function: where   is the saturation magnetization,  is the anhysteretic form factor, and   is the Weiss effective field defined as Regarding the energy dissipation, the differential expression of the irreversible magnetization   is given as where  is a direction parameter to show the ascending and descending of the hysteresis loop as The anhysteretic magnetization   is calculated by considering the thermodynamics of the magnetostrictive material.
The reversible magnetization  V calculates the extent of bulging of domain walls before they acquire the energy required to escape the inclusions.It is defined as Then it can be deduced since the amount of bending of the domain Summation of the reversible and irreversible components of the differential components leads to the total differential susceptibility of /: where , , , , and   are the parameters of Jiles-Atherton hysteresis model. is the coefficient of reversibility of the movement of the walls,  is the form factor,   is the saturation magnetization, and  and  represent the interaction between the domains and the hysteresis losses, respectively.The hysteresis output which is defined in the magnetic field is the magnetic flux intensity , and it can be obtained using the relationship between  and  as  =  0 ( + ) with  0 as  ⋅ 4 ⋅ 10 −7 .Finally, the hysteresis curve is represented by the mathematical expression of  − .
Remark 2. Compared with other hysteresis models, Jiles-Atherton model can formulate the hysteresis by several differential equations related to the behavior of ferromagnetic materials, so it is easy to describe the hysteresis of the SMAbased compliant actuators with strong saturation.Besides, the structure of the Jiles-Atherton model is based on the differential equations requiring less memory storage, which makes it convenient for the real-time application.
As introduced, the parameter identification is significant to make good use of Jiles-Atherton model describing hysteresis in SMA materials.For such purpose, the identification process is discussed in Section 4 in detail.

Parameter Identification Based Modified PSO Algorithm
The classic method for the parameter identification of Jiles-Atherton model had been discussed in some literature [22,26].Considering the differential-equation based structure of Jiles-Atherton model, the iterative identification algorithm causes the problems of being sensitive to the initial values of the model parameters, and the common identification algorithm may not converge or get trapped in local optimum in some special cases [25].To solve the existing identification problems in Jiles-Atherton model, an artificial intelligence method is implemented in this paper.In this section, a modified particle swarm optimization (MPSO) is proposed.
4.1.Basic PSO Algorithm.For particle swarm optimization algorithm [25,26,29], each particle is associated with a position, a velocity, and a fitness value which depends on the optimization function.The particle updates itself by following two extreme values in every iteration.One is the optimal solution found by the particle itself, called personal extreme point (Pbest).The other one is called global extreme point (Gbest) which is the overall optimal solution.The basic PSO method is introduced briefly in this section.
In the basic PSO algorithm, every candidate solution is treated as a particle point in a D-dimensional space [24].Accordingly, the position of -th particle can be represented by a D-dimensional vector and the population of  candidate solutions constructs the vector X which is treated as a swarm To obtain the optimal solution, the trajectories of particles in the search space can follow the motion equation as where  and  + 1 are the algorithm iterations, and the velocities of the particle k  along the D-dimensions are defined as which can be treated as bounded in the rage k  ∈ [−V max , V max ], V max is the given maximum velocity, and  = 1, 2, ..., .
The velocity vectors determine the moving direction of the particles which is described as where p  is the personal extreme point (Pbest) previously visited of the -th particle which is defined as and p  defined in (19) is the global extreme point (Gbest) of the particle swarm is the size of the swarm, and Λ is the inertia weight, which is generally between 0.8 and 1.2. 1 and  2 are the cognitive and the social parameters, which are nonnegative constants and are chosen in the range 0 ≤  1 ,  2 ≤ 4. Usually,  1 =  2 = 2 can satisfy most of the applications [24].R 1 and R 2 are diagonal matrices of random numbers being generated in [0, 1].The purpose of the identification is to minimize the error between the measured system output y  () and the calculated output y().A fitness function of the particle  is defined as where E is the number of measurement points used for the parameter identification.  max is the maximum of the measured  sets of system output data.

Modified PSO Algorithm.
In practical applications, the basic PSO algorithm shows some problems, such as premature convergence and poor local search ability.It is necessary to do some improvements for this identification method.In this paper, a modified PSO algorithm is proposed to reduce the local optimum problem in the basic PSO algorithm.The details are given as follows.
In the modified PSO algorithm, an iteration-varying inertia weight Λ() is adopted for the velocity term of update 6 Complexity equations, and the velocity and position of the -th particle are defined as where  and  + 1 are the algorithm iterations and  = 1, 2, ..., .The value of the inertia weight Λ() in ( 21) is updated using the rule defined in (23) to improve the accuracy.
where Ξ denotes the maximum iteration number, and Λ min and Λ max represent the selected minimum and maximum values for the inertia weight, respectively, and satisfy Λ max , Λ min ∈ (0, 1] and Λ max > Λ min .
Considering the structure of the Jiles-Atherton model, which is combined by several differential equations, the analytical expression of Jiles-Atherton model cannot be obtained directly.To match the nonlinear feature of the identified model during the calculation process, a novel parametervarying adjustment strategy is designed in this paper.The cognitive parameter  1 and the social parameter  2 both nonlinearly will be changed with the change of the number of iterations: where  1ini ,  2ini are positive constants for cognitive parameters  1 and  1end ,  2end are positive parameters for the social parameter  2 in the modified iteration-varying adjustment strategies in (24) and ( 25), satisfying  1ini ,  1end ,  2ini ,  2end ∈ (0, 4] and  1ini ≥  2end >  2ini ≥  1end , which is related to the inertia weight Λ and design parameters Λ min and Λ max .

MPSO Identification Procedure.
In this subsection, the parameter identification process for the Jiles-Atherton hysteresis model based on the modified PSO algorithm is introduced.The optimization process includes 6 steps and the flowchart of the modified PSO algorithm is given in Figure 4.
Step 1.The initiation of each particle is defined for the identified parameters randomly.
where  ini  and V ini  are the initial values of   and V  ,  max  and  min  are the maximum and minimum values of the jth parameter, and V max  and V min  are the maximum and minimum values of the j-th parameter update speed.rand 1 and rand 2 are two random numbers with a range of [0, 1].
Step 2. Calculate the fitness function of each particle defined in (20), where   is the calculated output and   is the measured system output.
Initialize the values of personal best p  of each particle and global best p  .If Fitness(p  ) > 10 −7 , go to Step 6.
For each particle  do update Λ,  1 and  2 using the equations in ( 23), ( 24) and ( 25) update velocity k  using the equations in ( 21) check the bound of the particle velocity and update position x  using equations ( 22) check the bound of the position if   >  max  , then   =  max  , and set V  = 0 if   <  min  , then   =  min  , and set V  = 0

End
Step 4. To reduce the chance of getting trapped in a local optimum, the random mutation step for the parameters of each particle is conducted according to the following rules.
If rand 3 > 0.95 where ⌈⌉ is a roundup function and N is the identified parameter number.When the random number rand 3 > 0.8,   , the j-th parameter of the i-th particle, can be updated randomly at each step.rand 3 , rand 4 , and rand 5 are the random numbers between [0, 1].
Step 5. Evaluate the fitness function of the particle calculated in Step 2.

Start
Update the velocity of each particle using (21), check the bound of the particle velocity Update the position of each particle using (22), check the bound of the position Calculate the fitness function of each particle using (20) Update the values of Pbest and Gbest Output the identification results Calculate the fitness function of each particle using (20).Initialize the values of Pbest and Gbest End Fitness (P g ) < 10 −7 or n ≥ Ξ?
Calculate the inertia weight Λ in (23), the cognitive parameter c 1 and the social parameter c 2 in ( 24), ( 25) Fitness(P g ) < 10 −7 or n ≥ Ξ? Step 6.If Fitness (p  ) < 10 −7 or  ≥ Ξ, the global solution p  = ( 1 ,  2 ,  3 ,  4 ,  5 ) is obtained, stop.Remark 3. Usually, the cognitive parameter  1 and social parameter  2 satisfy  1 ≫  2 , and it can improve the capability of searching the global optimal value.Since the cognitive parameter  1 and social parameter  2 are related to the number of iterations, the parameters  1 and  2 show the flexibility with the change of the number of iterations.

Experiment Verification
In this section, the results of applying the proposed modified PSO identification algorithm for the Jiles-Atherton model are given.The results are illustrated by comparing the modeling results with experimental data acquired for the SMA-based compliant actuator platform introduced in Section 2.
For the Jiles-Atherton model defined in Section 5, there are 5 parameters to be identified, so the position vector x of each particle can be defined as x = [, , ,   , ].In addition, to verify the validity of the proposed algorithm, the basic PSO algorithm is also applied for the parameter identification of Jiles-Atherton model in the test.
Using the experimental results for the PWM signal's cycle as 60s with amplitude 5V and duty cycle as 50% given in Figure 3, the search space of basic PSO and modified PSO algorithms for 5 parameters is given in Table 1.The maximum of search velocity is also set as 15% of the search space for basic PSO and modified PSO (MPSO) algorithms.
In the basic PSO and the modified PSO algorithms, the size of the swarm is selected as 50 and the maximum iteration number as 300.For the basic PSO algorithm, the inertia weight Λ is set as 0.9 and the cognitive parameter  1 and social parameter  2 are both selected as 0.5.For the modified PSO algorithm, the initial value and the final value for the inertia weight, cognitive parameter, and social parameter are chosen as Λ max = 0.9,  1ini = 2.5,  2ini = 0.5 and Λ min = 0.4,  1end = 1.3,  2end = 1.7.Using the parameter initialization method defined in (26), the initial value of the identified parameters can be set randomly in search space, and the particle with the minimum value of fitness function can be chosen as the initial position of p  .
Following the identification introduced in Section 4, the identification results for 5 parameters in the Jiles-Atherton model are obtained, and the results are given in Table 2. 1210 To illustrate the performance of the proposed algorithm, the fitness functions of basic PSO and modified PSO algorithms are also given in Figure 5.The modified PSO algorithm with the dynamic adjustment for the parameters in the process shows some advantages in the accuracy and the convergence rate compared with the basic PSO algorithm.
In addition, the experiment is conducted by changing the PWM signal's cycle as 60s with amplitude 5V and duty cycle is 50% (Figure 6(a)) to show the accuracy of the proposed identification method.As shown in Figures 6(c)-6(d), the input-output hysteresis loop of the SMA-based compliant actuators between the temperature and the rotation angle is used to verify the modeling performance.
The result of using basic PSO algorithm for Jiles-Atherton model is also provided to show the accuracy of the identification performance.The comparison of hysteresis loops between the experimental data and the Jiles-Atherton model calculated using different identification results are given in Figure 7 with the absolute errors in Figure 8.It can be observed that the maximum errors of the radiation angle response are about 30.84% of BPSO case and 22.53 % of MPSO case, respectively.The parameters using MPSO method are more effective than the basic PSO method for the Jiles-Atherton model describing the output characteristics of the SMA-based compliant actuator.

Discussion
In this paper, the parameter identification of SMA-based compliant actuating system described by Jiles-Atherton model is addressed.Jiles-Atherton hysteresis model is a differential-equation based hysteresis model, which can describe the saturated hysteresis between the input (voltage or temperature) and output (rotation angle).The accurate identification of these parameters in the Jiles-Atherton model is helpful to estimate the actuator output in the application.For such purpose, a modified PSO identification algorithm is proposed to obtain the Jiles-Atherton hysteresis model parameters.To avoid the local optimum problem in the basic PSO algorithm, the proposed modified PSO algorithm can adjust the location and velocity of each particle by using the random mutation to extend the search space of the solution in the iteration step so that the global optimum can be obtained effectively.The experiment platform is conducted to verify the effectiveness of the proposed identification algorithm, and the comparison results with the basic PSO algorithms are given to illustrate the accuracy of the proposed method.
For this study, some future work to improve the compliant SMA driving performance can be considered.The thermal dynamics between the input current and the temperature is not considered which may cause the system response delay.Besides, the rate-dependent hysteresis properties of the SMA materials also degrade the driving performance when the input signal frequency changes.These issues yield a direction for extension of the hysteresis modeling method with the

Figure 3 :
Figure 3: Input-output characteristics of the SMA-based compliant actuators under a fixed duty cycle (50%) 5v PWM signal operation with 50s signal cycle producing the temperature changing between 0 ∘ C and 60 ∘ C. (a) Input desired signal (voltage).(b) Input temperature.(c) Output rotation angle.(d) Hysteresis loop.

Figure 4 :
Figure 4: The flowchart of the modified PSO algorithm.

Figure 5 :
Figure 5: Comparison of fitness function between the basic PSO algorithm and the modified PSO algorithm (red line: basic PSO; blue line: modified PSO).

Table 1 :
Parameters search space.