Self-sensing shape memory alloy wire actuators using Kalman filters

The voltage divider circuit was developed to determine the electrical resistance of the SMA wire during drive. Here, the UKF model is developed from the system model to estimate the displacement of the spring from the electrical resistance variation of the SMA wire actuator.

Introduction

To move the system, the temperature of the SMA rises above a certain temperature, causing the SMA wire to contract and pull the load (see Figure 1.3). In this way, in addition to the actuator, the SMA wire can also be used as a sensor.

Figure 1.1: Phase transformation in SMA materials.

Literature Review

History and Application of SMA
Modeling of SMA Behavior
Feedback Control
SMA as Self-sensing Actuator
Kalman Filter (KF)

Frustet al[64] proposed the use of the self-sensing capability of SMA in controlling the flexible nozzle of a smart inhaler system. Cui et al[71] proposed a mathematical model to obtain a relationship between the ratio of electrical resistance change over initial resistance (∆RR) and strain ( ) of SMA.

Motivation and Objective

Organization of the Present Work

The model consists of heat balance equation and the constitutive relationship between the SMA wire coupled with the stiffness of the system. The displacement sensor measures the response of the system actuated by the SMA wire.

Constituent phases in SMA materials

Martensite Phase

Austenite Phase

SMA Wire Behavior

Shape Memory Effect (SME)

This temperature-induced transformation, often called reverse transformation, continues until all of the martensite has been transformed into austenite at a temperature above the final austenite temperature (Af). One must cool the SMA from above the austenite finish temperature to below the martensite finish temperature and then deform it by applying an external load to introduce a residual stress into it.

Pseudo-elasticity

After unloading, a large residual strain, typically 4 - 6%, can be observed after a small recovery of elastic strain, as shown in Figure. It continues until the stress value reaches σfM, where austenite is completely converted into detwined martensite.

Phase Diagram

As the load is removed, the material behaves elastically until the stress decreases to σAs; after which the reverse transformation begins and the SMA begins to recover the strain at a faster rate. The two straight horizontal lines represent the stress-induced transformation zone, where the cross-linking transformation starts and ends at σscr and σfcr, respectively.

SMA Actuators

Both linear and rotary movements can be produced with the SMA wire actuators. The presence of a preload spring is extremely necessary for the SMA wire actuator to work in the next cycle.

Modeling of SMA

One-Dimensional Constitutive Behavior of SMA

Following the approach proposed by Liang and Rogers [82] and Brinson [29], the 1-D constitutive model of SMA can be derived from. Here (.)0 represents the initial state of the parameter in parentheses, and ξ represents the total volume fraction of martensite.

Phase Kinetics

Simple Loading
Arbitrary Thermo-mechanical Loading

The normalized form of the distance traveled between Sj and Sj+1 within the loading path can be expressed as,. In the following, the phase kinetics in each of the transformation zones is defined in detail.

Figure 2.7: Simplified phase diagram of SMA.

Implementation of Phase Kinetics

Algorithm for Zone Selection

So for a given point in the phase diagram, defined as (T, σ), iffL1 ≤0 andfL2 ≥0, the point lies in the forward transformation zone and the equations defined in section 2.6.2.2.1 should be used to model the phase kinetics. Therefore, for a given (T, σ) in the phase diagram, iffL3 ≤ 0 andfL4 ≥ 0, the point is currently in the reverse transformation zone and the phase kinetics defined in section 2.6.2.2.2 are used.

Algorithm for Updating the Memory Parameters

Zone of inverse transformation: Similarly, the equation of two straight lines describing the zone of inverse transformation can be obtained as,. Now, in an active segment, the memory parameters must be known, and therefore they must be updated as soon as there is any change in the state of the transformation.

Heat Balance Equation

The second term in the RHS represents the convective heat loss, where Asurf is the total surface area of the wire and h represents the coefficient of convective heat transfer between the SMA wire and the environment. The third term in the RHS of Eqn. 2.39) is used to provide the latent heat of absorption and emission per unit volume of the wire during reverse and forward transformations, respectively.

Summary

In this chapter, extended Kalman filter models of two SMA wire-driven systems are developed to estimate the voltage and temperature of the wire from its measured electrical resistance. Then, some implementation issues are discussed, followed by the simulation results, which validate the developed models.

Self-sensing SMA Wire Actuator

To avoid this, the electrical resistance variation in SMA wires during phase transformation has been exploited as a measure of the system output. If the change in electrical resistance information is used to measure the recovery voltage or the output of the system being activated, then one can get rid of the feedback sensors; makes SMA wire a potential candidate as actuators for microscale applications.

SMA Wire Actuated Systems

The other ends of the SMA wire and the spring are attached to two rigid walls. Note that during this process, the electrical resistance of the SMA wire drops due to the formation of austenite and the change in wire geometry.

Figure 3.2: Schematic of a linear spring biased SMA wire actuator.

Extended Kalman Filter (EKF)

Here {w} represents the process noise vector, which is used to take care of the following errors. 3.9) is called residual covariance, which represents the uncertainty in the estimated output of the system.

Figure 3.4: Flow diagram of the EKF model.

Modeling

Force Equilibrium and Kinematic Constraint

Linear system
Nonlinear system

Here, Ks represents the spring stiffness and A is the cross-sectional area of the SMA wire. Here, 0 and L0 represent the amount of prestress and the length of the SMA wire corresponding to the undeformed state of the beam.

Figure 3.5a represents the undeformed beam (AB) attached with the SMA wire (CD) at an offset e y

Relation between Electrical Resistance and Strain

Here (σ) represents the stress of the SMA wire as a function of voltage, which as Eq. be suggested. The change in cross-sectional area of the SMA wire was neglected in the analysis.

EKF for SMA Actuated System

Linear System

Implicit Method
Explicit Method
Comparison between the EKF Models
Validation of EKF Model
Comparison between EKF Estimation and SMA Model

Thus, the electrical resistance of the SMA wire was taken as an observation or measurement. 3.3 the state vector consists of the voltage (σ) and temperature (T) of the SMA wire; as represented by Eq

Table 3.1: Properties assumed for the linear spring biased SMA wire actuated system [3].

Summary

In the previous chapter, an Extended Kalman Filter model is developed to estimate the output of the SMA actuated system from the change in the electrical resistance of the SMA wire. Both explicit and implicit schemes are adopted to obtain the a-priori estimation of the stress and temperature of the SMA wire and are found to provide results with the same accuracy.

Experimental Set-up and Procedure

Experimental Procedure

This voltage signal is then converted to analog form using the dSPACE - DS1006 and sent through the analog output port on the I/O board of the DS1006. The output from the displacement sensor Vout is recorded through the I/O card, from which the displacement is calculated according to Eqn.

Figure 4.6: (a) Schematic of experimental set-up.

EKF Model for Linear Spring Biased Wire Actuator

Linear Spring biased SMA Wire Actuator
Determination of Spring Stiffness
Determination of Process and Measurement Noise Covariance
Results and Discussion

EKF model, the displacement of the block is also estimated from the measured resistance and the applied voltage. The change in electrical resistance of the SMA wire is measured using the voltage divider circuit.

Figure 4.8: (a) Top and (b) isometric view of linear spring biased SMA wire actuator.

EKF for SMA Actuated Cantilever Beam

SMA Actuated Nonlinear System
Modulus of Elasticity
Determination of Process and Measurement Noise Covariance
Determination of Convective Heat Transfer Coefficient

Here, the offset is kept so as to produce large end displacement of the beam. The objective is to minimize the difference between the estimated and measured beam displacement for a given input voltage.

Figure 4.12: (a) Applied voltage across SMA wire, (b) measured electrical resistance of SMA, (c) variation of measured displacement with applied voltage, (d) change in measured displacement with electrical resistance of SMA, and (e) comparison between meas

Summary

From the change in electrical resistance behavior of the SMA and the equations of state, the state and response of the system are estimated and compared with the measured responses. In the following, the detailed steps of UKF and the development of the same for SMA wire actuator are discussed, followed by the experimental verification of the developed model.

Unscented Kalman Filter (UKF)

The Unscented Kalman Filter

The estimated measurement is obtained from Yˆ(i)k following,. iv) Similarly, the covariance of the estimated measurement is obtained by using, . Here, the first term {Xˆ−tk} represents the a-priori estimated state of the system, calculated using Eqn. 5.7), and the second term denotes the product of Kalman gain (Gtk) and innovation (Ytk −Yˆtk).

Figure 5.1: Steps in Unscented Kalman Filter.

System Description

UKF for SMA Actuated System

Validation of the UKF Model

From the transformed sigma points, the system output(Yˆtk) is estimated according to Eqn. From the same sigma points and Yˆtk, the required covariance matrices PYtk and PXYtk are determined according to Eq.

Results and Discussion

This shows that the accuracy of the developed UKF model is almost the same as that of the EKF model. It can be observed that the computation time of the UKF model is much less compared to that of the EKF model.

Figure 5.3: Comparison between experimental response and UKF estimation for continuous periodic signal of time period, (a) 10 sec, and (b) 5sec.

Summary

Furthermore, the phase kinetics followed by the model may not be accurate enough, especially in cases of partial transformation. In addition, the material properties of the SMA wire are taken from the literature and are not necessarily the same as the properties of the wire used in the experiments.

Differential Scanning Calorimetry (DSC)

Determination of Transformation Current Zone

In the previous case, to identify the transformation status at the current point (T, σ), the relative location of the point with respect to the start and end boundary of a given transformation zone was used. In the posterior step, if Pstatus∼= 3, then update the memory parameters as the final value of stress, temperature and martensite volume fractions obtained in the last time step.

Artificial Neural Network (ANN)

Description of the System
EKF based Artificial Neural Network (ANN)
Experimental Details

To measure the electrical resistance of the SMA and the actual displacement of the SMA-enabled system, the experimental setup presented in section 4.2 is used. The outputs of the ANN models for a similar voltage signal with a frequency of 0.2 Hz rad/sec. is shown in fig.

Figure 6.3: (a) Comparison between EKF estimated and experimental response, (b) schematic of the ANN models, (c) details of ANN-I and (d) details of ANN-II.

EKF model with Varying Process Noise

Modified Extended Kalman Filter (EKF)
Varying Process Noise Covariance
Determination of Process Noise Covariance Q

Here, Jtk−1 represents the Jacobian of the process vector function (η) evaluated attk−1, and is calculated using the state transition matrix (A) after,. Furthermore, the Jacobian of the process function with respect to the parameter (p) can be expressed as,.

Parameter Estimation

Modified Extended Kalman Filter

Modified System Model

Here, the Jacobian of the process function with respect to the state, obtained using J =eA(t)∆t. The result illustrates that the modified approach is able to represent the electrical resistance behavior of the SMA wire satisfactorily.

Figure 6.7: (a) Comparison between estimated and measured electrical resistance of SMA wire (b) before and (c) after modification.

Modified EKF model for linear spring biased SMA wire

The rest of the steps remain the same as in the case of the EKF discussed in section 3.6.1.2. Since, in this case, the size of the state error covariance (P) and the process noise covariance (Q) are increased, the computational effort is also increased.

Results and Discussion

The maximum discrepancy between the EKF-III model estimate and the measured one is less than 1 mm. The performances of the EKF models are also evaluated for different step inputs and are shown in Figs.

Figure 6.9: Comparison between EKF-I, EKF-III and experimental response for continuous input voltage (a) amplitude = 6.2 V and time period =10 sec (b) amplitude = 6.2 V, time period = 5 sec and (c) amplitude = 4 V, time period = 10 sec.

Summary

The temperature and voltage of the SMA wire actuator during its limited recovery are estimated from its electrical resistance variation. In this chapter, the implementation of the developed EKF in Arduino Uno Atmega328 is discussed.

Figure 7.1: Arduino UNO development board.

Experimental Set-up

It implies that the output signal has reverse polarity than that of the input signal. This displacement of the SMA-activated linear spring is measured by the laser displacement sensor, discussed in section 4.2.