Development of a cascade position control system for an SMA-actuated rotary actuator with improved experimental tracking results

Abstract

Hysteresis and significant nonlinearities in the stress–strain–temperature characteristics of shape memory alloy (SMA) actuators prevent effective utilization of these actuators and make them difficult to control. Due to these effects, the position control of SMA actuators has been a great challenge for many industrial applications. In this study, a cascade control for position tracking using torque and position measurement of an SMA-actuated rotary actuator is considered. The effects of the SMA characteristic model on control performance are also focused when simulation consequences are directly applied in practice. Based on nonlinear and unknown dynamics of an SMA actuator, a torque regulation adaptive controller using modified Brinson model is developed that requires to approximate the parameters of the system by only measuring SMA force and actuator rotation. Although it is shown that only an approximate model of the system is sufficient to develop the proposed controller, the system dynamic is simulated using MATLAB software, and the characteristics of SMA wire is considered by both modified Brinson constitutive model and Liang and Rogers constitutive model. The controller gains are regulated for both of the mentioned models using simulation. Finally, the performance of both controllers is further evaluated experimentally, and the accuracy of the proposed controller based on the modified Brinson model is compared to the controller based on the Liang and Rogers model. It is shown that since the Brinson model has a better prediction of the SMA wire behavior, the simulation results based on the modified Brinson model have better conformity to reality.

Appendix A: rearranging modified Brinson model

Equations (3)–(6) can be further written as follows by rearranging:

$$\xi = g_{1} \left( {T,\sigma } \right)$$

(47)

$$\xi_{\text{s}} = g_{2} \left( {T,\sigma } \right)$$

(48)

$$\dot{\xi } = g_{3} \left( {T,\sigma } \right)\dot{T} + g_{4} \left( {T,\sigma } \right)\dot{\sigma }$$

(49)

$$\dot{\xi }_{\text{s}} = g_{5} \left( {T,\sigma } \right)\dot{T} + g_{6} \left( {T,\sigma } \right)\dot{\sigma }$$

(50)

where the functions $g_{1} \left( {T,\sigma } \right)$ to $g_{6} \left( {T,\sigma } \right)$ are obtained from the phase transformation equations as follows:

$$\begin{aligned} & g_{1} \left( {T,\sigma } \right) \\ & \quad = \left\{ {\begin{array}{*{20}l} {\frac{{\xi_{{{\text{T}}0}} }}{{1 - \xi_{{{\text{s}}0}} }} + \frac{{1 - \xi_{{{\text{T}}0}} - \xi_{{{\text{s}}0}} }}{{1 - \xi_{{{\text{s}}0}} }}\left( {\frac{{1 - \xi_{{{\text{s}}0}} }}{2}{ \cos }\left( {\frac{\pi }{{\sigma_{\text{s}}^{\text{cr}} - \sigma_{\text{f}}^{\text{cr}} }}\left( {\sigma - \sigma_{\text{f}}^{\text{cr}} - C_{\text{M}} \left( {T - M_{\text{s}} } \right)} \right)} \right) + \frac{{1 + \xi_{{{\text{s}}0}} }}{2}} \right), \ldots } \hfill \\ {{\text{if}}\;T > M_{\text{s}} , \sigma_{\text{s}}^{\text{cr}} + C_{\text{M}} \left( {T - M_{\text{s}} } \right) < \sigma < \sigma_{\text{f}}^{\text{cr}} + C_{\text{M}} \left( {T - M_{\text{s}} } \right)} \hfill \\ {\frac{{2\xi_{{{\text{s}}0}} - 1}}{{\xi_{{{\text{s}}0}} - 1}}\Delta + \frac{{\xi_{{{\text{s}}0}} - 1 - \Delta }}{{\xi_{{{\text{s}}0}} - 1}}\left( {\frac{{1 - \xi_{{{\text{s}}0}} }}{2}{ \cos }\left( {\frac{\pi }{{\sigma_{\text{s}}^{\text{cr}} - \sigma_{\text{f}}^{\text{cr}} }}\left( {\sigma - \sigma_{\text{f}}^{\text{cr}} } \right)} \right) + \frac{{1 + \xi_{{{\text{s}}0}} }}{2}} \right), \ldots } \hfill \\ {{\text{if}}\;T < M_{\text{s}} , \sigma_{\text{s}}^{\text{cr}} < \sigma < \sigma_{\text{f}}^{\text{cr}} } \hfill \\ {\frac{{\xi_{0} }}{2}\left[ {{ \cos }\left( {a_{\text{A}} \left( {T - A_{\text{s}} - \frac{\sigma }{{C_{\text{A}} }}} \right)} \right) + 1} \right], \;{\text{if}}\; T > A_{\text{s}} , C_{\text{A}} \left( {T - A_{\text{f}} } \right) < \sigma < C_{\text{A}} \left( {T - A_{\text{s}} } \right)} \hfill \\ {\xi , \;{\text{otherwise}}} \hfill \\ \end{array} } \right. \\ \end{aligned}$$

(51)

$$\begin{aligned} & g_{2} \left( {T,\sigma } \right) \\ & \quad = \left\{ {\begin{array}{*{20}l} {\frac{{1 - \xi_{{{\text{s}}0}} }}{2}{ \cos }\left( {\frac{\pi }{{\sigma_{\text{s}}^{\text{cr}} - \sigma_{\text{f}}^{\text{cr}} }}\left( {\sigma - \sigma_{\text{f}}^{\text{cr}} - C_{\text{M}} \left( {T - M_{\text{s}} } \right)} \right)} \right) + \frac{{1 - \xi_{{{\text{s}}0}} }}{2}, \ldots } \hfill \\ {{\text{if}}\; T > M_{\text{s}} , \sigma_{\text{s}}^{\text{cr}} + C_{\text{M}} \left( {T - M_{\text{s}} } \right) < \sigma < \sigma_{\text{f}}^{\text{cr}} + C_{\text{M}} \left( {T - M_{\text{s}} } \right)} \hfill \\ {\frac{{1 - \xi_{{{\text{s}}0}} }}{2}{ \cos }\left( {\frac{\pi }{{\sigma_{\text{s}}^{\text{cr}} - \sigma_{\text{f}}^{\text{cr}} }}\left( {\sigma - \sigma_{\text{f}}^{\text{cr}} } \right)} \right) + \frac{{1 + \xi_{{{\text{s}}0}} }}{2},\; {\text{if}}\; T < M_{\text{s}} , \sigma_{\text{s}}^{\text{cr}} < \sigma < \sigma_{\text{f}}^{\text{cr}} } \hfill \\ {\xi_{{{\text{s}}0}} - \frac{{\xi_{{{\text{s}}0}} }}{{\xi_{0} }}\left( {\xi_{0} - \frac{{\xi_{0} }}{2}\left[ {{ \cos }\left( {a_{\text{A}} T - A_{\text{s}} - \frac{\sigma }{{C_{\text{A}} }}} \right) + 1} \right]} \right), \;{\text{if}}\; T > A_{\text{s}} , C_{\text{A}} \left( {T - A_{\text{f}} } \right) < \sigma < C_{\text{A}} \left( {T - A_{\text{s}} } \right)} \hfill \\ {\xi_{\text{s}} , \;{\text{otherwise}}} \hfill \\ \end{array} } \right. \\ \end{aligned}$$

(52)

$$\begin{aligned} & g_{3} \left( {T,\sigma } \right) \\ & \quad = \left\{ {\begin{array}{*{20}l} { - \frac{{\xi_{{{\text{T}}0}} - 1 + \xi_{{{\text{s}}0}} }}{2}{ \sin }\left( {\frac{\pi }{{\sigma_{\text{s}}^{\text{cr}} - \sigma_{\text{f}}^{\text{cr}} }}\left( {\sigma - \sigma_{\text{f}}^{\text{cr}} - C_{\text{M}} \left( {T - M_{\text{s}} } \right)} \right)} \right)\frac{{\pi C_{\text{M}} }}{{\sigma_{\text{s}}^{\text{cr}} - \sigma_{\text{f}}^{\text{cr}} }}, \ldots } \hfill \\ {{\text{if}}\; T > M_{\text{s}} , \sigma_{\text{s}}^{\text{cr}} + C_{\text{M}} \left( {T - M_{\text{s}} } \right) < \sigma < \sigma_{\text{f}}^{\text{cr}} + C_{\text{M}} \left( {T - M_{\text{s}} } \right)} \hfill \\ { - \frac{{1 - \xi_{\text{s}} \left( {T,\sigma } \right)}}{{1 - \xi_{{{\text{s}}0}} }}\frac{{1 - \xi_{{{\text{T}}0}} - \xi_{{{\text{s}}0}} }}{2}{ \sin }\left( {a_{\text{M}} \left( {T - M_{\text{f}} } \right)} \right)a_{\text{M}} , \;{\text{if}}\; M_{\text{f}} < T < M_{\text{s}} , T < T_{0} , \sigma_{\text{s}}^{\text{cr}} < \sigma < \sigma_{\text{f}}^{\text{cr}} } \hfill \\ { - \frac{{\xi_{0} }}{2}{ \sin }\left( {a_{\text{A}} \left( {T - A_{\text{s}} - \frac{\sigma }{{C_{\text{A}} }}} \right)} \right)a_{\text{A}} ,\;{\text{if}}\; T > A_{\text{s}} , C_{\text{A}} \left( {T - A_{\text{f}} } \right) < \sigma < C_{\text{A}} \left( {T - A_{\text{s}} } \right)} \hfill \\ {0, \;{\text{otherwise}}} \hfill \\ \end{array} } \right. \\ \end{aligned}$$

(53)

$$\begin{aligned} & g_{4} \left( {T,\sigma } \right) \\ & \quad = \left\{ {\begin{array}{*{20}l} {\frac{{\xi_{{{\text{T}}0}} - 1 + \xi_{{{\text{s}}0}} }}{2}{ \sin }\left( {\frac{\pi }{{\sigma_{\text{s}}^{\text{cr}} - \sigma_{\text{f}}^{\text{cr}} }}\left( {\sigma - \sigma_{\text{f}}^{\text{cr}} - C_{\text{M}} \left( {T - M_{\text{s}} } \right)} \right)} \right)\frac{\pi }{{\sigma_{\text{s}}^{\text{cr}} - \sigma_{\text{f}}^{\text{cr}} }}, \ldots } \hfill \\ {{\text{if}}\; T > M_{\text{s}} , \sigma_{\text{s}}^{\text{cr}} + C_{\text{M}} \left( {T - M_{\text{s}} } \right) < \sigma < \sigma_{\text{f}}^{\text{cr}} + C_{\text{M}} \left( {T - M_{\text{s}} } \right)} \hfill \\ {\left( {\frac{\Delta }{{1 - \xi_{{{\text{s}}0}} }} - 1} \right)\frac{{1 - \xi_{{{\text{s}}0}} }}{2}{ \sin }\left( {\frac{\pi }{{\sigma_{\text{s}}^{\text{cr}} - \sigma_{\text{f}}^{\text{cr}} }}\left( {\sigma - \sigma_{\text{f}}^{\text{cr}} } \right)} \right)\frac{\pi }{{\sigma_{\text{s}}^{\text{cr}} - \sigma_{\text{f}}^{\text{cr}} }},\;{\text{if}}\; T < M_{\text{s}} , \sigma_{\text{s}}^{\text{cr}} < \sigma < \sigma_{\text{f}}^{\text{cr}} } \hfill \\ {\frac{{\xi_{0} }}{{2C_{\text{A}} }}{ \sin }\left( {a_{\text{A}} \left( {T - A_{\text{s}} - \frac{\sigma }{{C_{\text{A}} }}} \right)} \right)a_{\text{A}} ,\;{\text{if}}\; T > A_{\text{s}} , C_{\text{A}} \left( {T - A_{\text{f}} } \right) < \sigma < C_{\text{A}} \left( {T - A_{\text{s}} } \right)} \hfill \\ {0,\;{\text{otherwise}}} \hfill \\ \end{array} } \right. \\ \end{aligned}$$

(54)

$$\begin{aligned} & g_{5} \left( {T,\sigma } \right) \\ & \quad = \left\{ {\begin{array}{*{20}l} {\frac{{1 - \xi_{{{\text{s}}0}} }}{2}{ \sin }\left( {\frac{\pi }{{\sigma_{\text{s}}^{\text{cr}} - \sigma_{\text{f}}^{\text{cr}} }}\left( {\sigma - \sigma_{\text{f}}^{\text{cr}} - C_{\text{M}} \left( {T - M_{\text{s}} } \right)} \right)} \right)\frac{{\pi C_{\text{M}} }}{{\sigma_{\text{s}}^{\text{cr}} - \sigma_{\text{f}}^{\text{cr}} }}, \ldots } \hfill \\ {{\text{if}}\; T > M_{\text{s}} , \sigma_{\text{s}}^{\text{cr}} + C_{\text{M}} \left( {T - M_{\text{s}} } \right) < \sigma < \sigma_{\text{f}}^{\text{cr}} + C_{\text{M}} \left( {T - M_{\text{s}} } \right)} \hfill \\ { - \frac{{\xi_{{{\text{s}}0}} }}{2}{ \sin }\left( {a_{\text{A}} \left( {T - A_{\text{s}} - \frac{\sigma }{{C_{\text{A}} }}} \right)} \right)a_{\text{A}} ,\;{\text{if}}\; T > A_{\text{s}} , C_{\text{A}} \left( {T - A_{\text{f}} } \right) < \sigma < C_{\text{A}} \left( {T - A_{\text{s}} } \right)} \hfill \\ {0, \;{\text{otherwise}}} \hfill \\ \end{array} } \right. \\ \end{aligned}$$

(55)

$$\begin{aligned} & g_{6} \left( {T,\sigma } \right) \\ & \quad = \left\{ {\begin{array}{*{20}l} { - \frac{{1 - \xi_{{{\text{s}}0}} }}{2}{ \sin }\left( {\frac{\pi }{{\sigma_{\text{s}}^{\text{cr}} - \sigma_{\text{f}}^{\text{cr}} }}\left( {\sigma - \sigma_{\text{f}}^{\text{cr}} - C_{\text{M}} \left( {T - M_{\text{s}} } \right)} \right)} \right)\frac{\pi }{{\sigma_{\text{s}}^{\text{cr}} - \sigma_{\text{f}}^{\text{cr}} }}, \ldots } \hfill \\ {{\text{if}}\; T > M_{\text{s}} , \;\sigma_{\text{s}}^{\text{cr}} + C_{\text{M}} \left( {T - M_{\text{s}} } \right) < \sigma < \sigma_{\text{f}}^{\text{cr}} + C_{\text{M}} \left( {T - M_{\text{s}} } \right)} \hfill \\ { - \frac{{1 - \xi_{{{\text{s}}0}} }}{2}{ \sin }\left( {\frac{\pi }{{\sigma_{\text{s}}^{\text{cr}} - \sigma_{\text{f}}^{\text{cr}} }}\left( {\sigma - \sigma_{\text{f}}^{\text{cr}} } \right)} \right)\frac{\pi }{{\sigma_{\text{s}}^{\text{cr}} - \sigma_{\text{f}}^{\text{cr}} }},\;{\text{if}}\; T < M_{\text{s}} , \sigma_{\text{s}}^{\text{cr}} < \sigma < \sigma_{\text{f}}^{\text{cr}} } \hfill \\ {\frac{{\xi_{{{\text{s}}0}} }}{{2C_{\text{A}} }}{ \sin }\left( {a_{\text{A}} \left( {T - A_{\text{s}} - \frac{\sigma }{{C_{\text{A}} }}} \right)} \right)a_{\text{A}} ,\;{\text{if}}\; T > A_{\text{s}} , C_{\text{A}} \left( {T - A_{\text{f}} } \right) < \sigma < C_{\text{A}} \left( {T - A_{\text{s}} } \right) } \hfill \\ {0, \;{\text{otherwise}}} \hfill \\ \end{array} } \right. \\ \end{aligned}$$

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It should be noted that the terms $g_{1} \left( {T,\sigma } \right)$ to $g_{6} \left( {T,\sigma } \right)$ are formed by some bounded constants, cosine and sine functions. Therefore, these terms are bounded.