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Shape memory alloy- (SMA-) based actuators are widely applied in the compliant actuating systems. However, the measured data of the SMA-based compliant actuating system reveal the input-output hysteresis behavior, and the actuating precision of the compliant actuating system could be degraded by such hysteresis nonlinearities. To characterize such nonlinearities in the SMA-based compliant actuator precisely, a Jiles-Atherton model is adopted in this paper, and a modified particle swarm optimization (MPSO) algorithm is proposed to identify the parameters in the Jiles-Atherton model, which is a combination of several differential nonlinear equations. Compared with the basic PSO identification algorithm, the designed MPSO algorithm can reduce the local optimum problem so that the Jiles-Atherton model with the identified parameters can show good agreements with the measured experimental data. The good capture ability of the proposed identification algorithm is also examined through the comparisons with Jiles-Atherton model using the basic PSO identification algorithm.

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 [

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 [

However, one of the shortcomings of SMA-based compliant actuators is the hysteretic nonlinearity in the process of SMA phase transformation [

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 [

As introduced in [

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 [

The main content of the paper is as follows: Section

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 [

Schematic of the SMA compliant actuating system.

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 by the driven voltage. As shown in Figure

Experimental platform of the SMA-based compliant actuators.

The open-loop test at the PWM signal cycle was performed to show the input-output characteristics of the SMA-based compliant actuator used in this paper. A fixed duty cycle (50%) PWM signal under amplitude (5V) operation with 50s signal cycle was adopted to produce the temperature changing between 0°C and 60°C, and the angular position of the coupler was measured by a high precision potentiometer. The desired signal (Figure

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.

According to the experiment results of the open-loop 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 [

The Jiles-Atherton model [

Usually, magnetization

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 SMA-based 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

The classic method for the parameter identification of Jiles-Atherton model had been discussed in some literature [

For particle swarm optimization algorithm [

In the basic PSO algorithm, every candidate solution is treated as a particle point in a D-dimensional space [

The velocity vectors determine the moving direction of the particles which is described as

The purpose of the identification is to minimize the error between the measured system output

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

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 parameter-varying adjustment strategy is designed in this paper. The cognitive parameter

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

The flowchart of the modified PSO algorithm.

The initiation of each particle is defined for the identified parameters randomly.

Calculate the fitness function of each particle defined in (

For each particle

update

update velocity

check the bound of the particle velocity

if

if

and update position

check the bound of the position

if

if

End

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

Then

where

Evaluate the fitness function of the particle calculated in Step

If

If

If Fitness

If Fitness

Usually, the cognitive parameter

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

For the Jiles-Atherton model defined in Section

Using the experimental results for the PWM signal’s cycle as 60s with amplitude 5V and duty cycle as 50% given in Figure

Parameters search space.

Parameters | Min | Max |
---|---|---|

| 1e-14 | 1e-2 |

| 1e-2 | 1e4 |

| 1e-2 | 1e4 |

| 1e5 | 3e7 |

| 1e-2 | 3 |

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

Following the identification introduced in Section

Parameter identification results.

Parameter | Basic PSO | Modified PSO |
---|---|---|

| 1.000e-14 | 1.000e-14 |

| 7.982261 | 9.771736 |

| 58.11999 | 33.80507 |

| 2.2722e7 | 2.6155e7 |

| 1.552548 | 1.938721 |

| ||

| 3.611e-4 | 3.296e-4 |

To illustrate the performance of the proposed algorithm, the fitness functions of basic PSO and modified PSO algorithms are also given in Figure

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

In addition, the experiment is conducted by changing the PWM signal’s cycle as 60s with amplitude 5V and duty cycle is 50% (Figure

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

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

Comparisons of output-input responses between the Jiles-Atherton model with PSO identification algorithm and the measured data (red line: measured data; blue line: J-A model with basic PSO): (a) with basic PSO algorithm and (b) with modified PSO algorithm.

Comparison of modeling error between the Jiles-Atherton model with basic PSO algorithm and J-A model with modified PSO algorithm (red line: J-A model with BPSO; blue line: J-A model with MPSO).

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 novel identification algorithm. The results of the accurate modeling for SMA-based compliant actuating system can be furthered to conduct the proper controller cancelling the nonlinear effects and improve the driving performance, which would yield the high driving performance of compliant SMA actuators.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The work was partially supported by the Funds for National Key Research and Development Program of China (2017YFB1302302), Science Foundation of Science and Technology Planning Project of Guangdong Province, China (2017A010102004), the Natural Science Foundation of Guangdong Province (2018A030313331), and the Fundamental Research Funds for the Central Universities (2018MS71).