Sensorless resistive-based control of shape memory alloy actuators in locking mechanism

Abstract

This article presents a sensorless resistive-based control method to control shape memory alloy wire actuators to be used in the development of a laparoscopic surgical locking system. Shape memory alloy wire electrical resistance is measured to control the martensite phase fraction of shape memory alloy wire. The martensite phase fraction is calculated based on mathematical heating model, resistance models, and the measured resistance. Experiments are conducted to evaluate the validity and performance of the method in the control of the locking system angle. The results show that the sensorless resistive-based control method accurately controls the wire without using any position sensors, resulting in the lower cost, size, and weight of the system. The results show very small steady-state errors, verifying the practicability of this method. The sensorless resistive-based control method has the potential to be used in many applications in which cost and space are limited and the use of an external sensor is impossible or costly.

Keywords

Shape memory alloy sensorless control resistance model smart materials locking mechanism

Introduction

Shape memory alloy (SMA) actuators are metallic materials that can regain their predetermined shapes by changing temperature. These actuators have been used in many applications, such as shape control of aircraft wings, micro-robot manipulation, robotics, micro-system precision control, and medical applications (Fu et al., 2004; Hirose et al., 1989; Huang et al., 2013; Kode and Cavusoglu, 2007; Kuribayashi, 2000; Madden et al., 2004; Moallem and Lu, 2005; Payandeh et al., 2001; Pfeiffer et al., 1999; Shahinpoor and Wang, 1994; Tu et al., 1998).

Recently, we have proposed an automatic suturing-ligation laparoscopic (LigLAP) method as an alternative method to stapling for surgical occlusion process (Yousefian et al., 2013). In this method, a locking mechanism is required to complete the ligation procedure. In addition, the actuator size in laparoscopic surgery devices is critical due to the small space available in laparoscopic surgeries. With this in mind, our attention is directed to use an SMA-based locking mechanism and the control of the SMA actuator. A complete and recent review of biomedical applications using SMA actuators is provided by Huang et al. (2013). The required background for laparoscopic surgery and the ligation method are presented in section “Automatic ligation in laparoscopic surgery.”

The control of SMAs is important and challenging due to their highly nonlinear behavior and hysteresis. Many methods have been presented in the literature (Asua et al., 2010; Kha and Ahn, 2006; Ma et al., 2004; Moallem and Tabrizi, 2009; Song, 2002; Troisfontaine et al., 1998) to control SMA actuators using external sensors. However, a few studies have used the electrical resistance of SMA actuators to control the position of SMA wires. In these studies, due to the highly nonlinear behavior of SMAs, non-model-based control methods, for example, neural networks, have been employed to map the displacement and electrical resistance of SMA actuators (Asua et al., 2010; Ma et al., 2004). The details of these methods are presented in section “Control of SMA actuators.”

The SMA ability to regain its predetermined shape is due to the reversible crystalline phase transformation. The phase transformation occurs because of the temperature and stress changes in the material structure. SMAs have two certain stable crystalline phases: austenite is a high-temperature phase and martensite is a low-temperature phase. When a material that shows this effect is deformed at low temperature, it will regain its original shape before deformation when heat input is provided. The characteristics of SMA result in significant strain recovery during the phase transformation.

In this article, we present a sensorless control method (sensorless resistive-based control (SRBC)) of an SMA-based locking mechanism based on resistive and heating models of SMA wires. The main objective is to control the position of the SMA-based locking mechanism, utilizing actual electrical resistance. Due to the linear relation of the strain of SMA wires and the martensite phase fraction (ξ_M), ξ_M is used as feedback in the position control of the SMA actuators. ξ_M is 1 for pure martensite and 0 for pure austenite phase (increasing the temperature decreases the ξ_M from 1 to 0). In this method, ξ_M is calculated based on actual electrical resistance, temperature, and stress of the SMA wire. The position of the SMA wire is calculated using experimental data gathering of position-martensite phase fraction. The details of SRBC method are presented in section “SRBC.” Several experiments are conducted to identify the required parameters of the resistive model, including temperature dependence of martensite and austenite and stress dependence of martensite and austenite. In addition, the validity and performance of the method are evaluated through an experimental verification procedure using a prototyped SMA-based locking mechanism. The details of the experimental setup and procedure as well as results and discussions are illustrated in section “Experimental verification.” The conclusion and future work are presented in section “Conclusion and future work.”

Control of SMA actuators

Modeling and control methods of SMA actuators have been extensively investigated in the literature. However, complex thermal–electrical–mechanical dynamics of SMA actuators and their temperature dependency make their control challenging and, of course, difficult. For position control, there are two types of control algorithms (Ma et al., 2004): (a) employing a reverse dynamic model to compensate for the nonlinearity and hysteresis characteristics and (b) incorporating a feedback loop. In the feedback loop, feedback signal in the control algorithms may include temperature, position, and/or electrical resistance of SMA actuators. Troisfontaine et al. (1998) have studied the feedback control of SMA-based micro-actuators using temperature. Also, position feedback has been used in several control applications with various position sensors. For instances, a potentiometer (Kha and Ahn, 2006), a linear variable differential transformer (LVDT; Song, 2002), and an encoder (Moallem and Tabrizi, 2009) have been used as position sensors. However, using a position sensor increases the cost, weight, and overall size. We have developed the SRBC method to avoid using any external sensors to control the SMA-based locking mechanism.

The electrical resistance of SMA actuators is not constant during the process of changing the temperature and can be used in the control loop. The resistance is one of the 23 properties of SMAs that may be changed due to phase transformation (Harrison, 1990). Although the relationship between SMA actuator strain and the electrical resistance is highly nonlinear, this relationship can be used to control SMA actuators.

Ma et al. (2004) and Asua et al. (2010) have presented sensorless methods to control SMA actuators and obtain an accurate and stable position control. To avoid using position sensors, electrical resistance of the SMA actuator has been utilized as feedback. The relationship between strain and electrical resistance of the actuator has been modeled using a neural network to predict the position of the SMA actuator. Neither studies have been used a mathematical model of SMA actuators. Proportional–derivative (PD; Ma et al., 2004) and proportional–integral–derivative (PID; Asua et al., 2010) controllers have been employed in the closed-loop control. Both studies experimentally evaluated their methods and results show accurate control of SMA positions. Asua et al. (2010) have also used a first-order equation for the contraction curve and have reported better results for neural network (the position error about 70 µm). Although these methods can control the position of the SMA wire without any external sensor, they use a non-model-based method to model the relationship between resistance and displacement. In addition, changing stress on the SMA wire is not predicted. This may affect the design and development of control strategies. In the method presented in this article, there is no need to use a non-model-based method due to the incorporation of resistance and heating models. In addition, stress is involved in the SRBC method as an input variable. Thus, stress can be changed without significant changes in our control strategy.

Kim et al. (2013) have also addressed these issues in their study on the position estimation of an SMA coil spring actuator using inductance. Displacement has been estimated under various stresses using inductance of an SMA coil spring. The inductance–displacement and resistance–displacement curves are compared, and the results show that the resistance–displacement is highly affected with stress, while only a small variation is observed for the inductance. Their experimental results show reasonable accuracy for displacement estimation of the SMA coil spring. However, this method can only be used for the SMA coil springs and is not applicable to control of SMA wires.

Automatic ligation in laparoscopic surgery

Laparoscopic or minimally invasive surgeries (MIS) are well known and preferred in comparison to open surgeries. These surgeries allow surgeons to complete surgery operations using laparoscopic surgical devices through small incisions, avoiding undue loss of blood, and eschew manipulation of tissues. Hence, the body’s response to surgery is with a reduced immune activation and catabolism (Heald et al., 1982; Kapiteijn et al., 1998). Laparoscopic surgery and the immune response have been reviewed by Vittimberga et al. (1998), and it has been concluded that the body’s response to laparoscopic surgery is very low. There are several advantages for MIS, such as shorter hospital stays and recovery periods, less scarring, less injury to tissue, more postoperative comfort, and less postoperative pain (Fuchs, 2002; Johansson, 2003; Tanović and Mesihović, 2003; Yousefian et al., 2013). Therefore, the use of laparoscopic surgeries has increased and is preferred in possible operations. For instance, in surgeries such as cholecystectomy, kidney surgery, and resection of rectal cancer, in which it is necessary to occlude vessels and ducts, the laparoscopic versions of these surgeries are the procedures of choice. Laparoscopic cholecystectomy has replaced open cholecystectomy since it was first reported in 1987 (Gollan et al., 1993) and is one of the most frequently performed laparoscopic operations (Bencini et al., 2003; Champault et al., 1996).

Currently, laparoscopic stapling devices are frequently used as a method of duct closure and vessel occlusion in these surgeries (Golash, 2008; Yeh et al., 2004). Although stapling devices are commonly used for closure of ducts and/or vessels, several complications and problems are reported. Technical failures and dangers of stapling devices are possible when the staples are not closed securely (Belgaumkar et al., 2009). In addition, bile leak is a known postoperative problem that could be caused by dislodgment and migration of the clips, as well as bile duct necrosis (Golash, 2008; Medina, 2005; Ng et al., 1999; Saha, 2000). Moreover, a relatively large space behind the duct and/or vessel is required when using stapling devices. Other problems include vascular and duct injury, bleeding from the liver bed, and biliary leak (Bezzi et al., 1995; Jones and Soper, 1996; Shamiyeh and Wayand, 2004). Furthermore, in some cases—for example, in the presence of chronic inflammation—using stapling devices is difficult and dangerous.

Recently, we have proposed an automatic suturing-LigLAP method as an alternative for occlusion process, using a double-layer suture to occlude vessels and ducts. It is required to hold the suture in order to complete the LigLAP procedure. In addition, the actuator size in laparoscopic surgery devices is critical due to the small space available in laparoscopic surgeries. With this in mind, we have designed, prototyped, and experimentally evaluated an SMA-based locking mechanism to hold the suture in the LigLAP process (Yousefian et al., 2013).

SRBC

This section presents the details of the SRBC as a sensorless control method for the SMA-based locking mechanism. First, we generally describe the method, and then the models and blocks are presented in details.

In the SRBC method, the heating and resistive models of the SMA wires are employed to calculate the martensite phase fraction, using the actual electrical resistance, enabling us to control the wire without using any external sensors. One advantage is that neural network or other non-model-based methods are not being used to model the relationship between the martensite phase fraction and the position of the SMA wire due to the fact that martensite phase fraction has a linear relationship with strain. This will extend the scope of SMA wires sensorless control to employ model-based control strategies in addition to non-model-based algorithms. Another advantage of the SRBC method is that it can be used with various stresses on the SMA wire using the mathematical relationship between the martensite phase fraction and stress. Resistance of SMA wires is a function of martensite phase fraction, which itself depends on temperature and stress of SMA wires (Kim et al., 2013; Schiedeck and Mojrzisch, 2011). Since the SRBC uses martensite phase fraction, it can control the SMA wire under various stresses.

The block diagram of the control system is shown in Figure 1. Output of the θ−ξ_M transformation block in Figure 1 is the desired martensite phase fraction (ξ_M) based on the desired angle (θ). ξ_M is calculated using the measured electrical resistance (R_t) and computed resistance functions (R_M and R_A), which are detailed in section “Resistance model.” The control scheme is designed based on the heating and resistance models. A proportional–integral (PI) controller is used to minimize the martensite phase fraction error (e_ξ). Input current is sent to the SMA wire via the power supply. The current is calculated based on the input power command and the SMA wire voltage. In addition, the actual resistance of the SMA wire is measured using the SMA wire voltage and the applied current. The heating and resistance models of SMA wires are presented in the following sections.

Figure 1.

Overview of the control system block diagram. Electrical current is applied to the SMA actuator based on the input power, using a programmable power supply. In addition, temperature is computed using the temperature model. Resistance functions (R_M and R_A) are calculated with the temperature and stress inputs. Martensite phase fraction is the output of the system, obtained from resistance model using calculated R_M and R_A, and actual electrical resistance. A PI controller is employed to control martensite phase fraction relevant to the desired angle.

Resistance model

In this section, we present the background and the details of the SMA wire resistance model. The model includes the Resistance Functions Calculator and Martensite Phase Fraction Calculator blocks in Figure 1.

According to the literature, Ikuta et al. (1988) have studied the resistance behavior of SMAs and designed a position control system using resistance feedback. It has been reported that the SMA wire has higher electrical resistance in martensite and lower in austenite. However, Novák et al. (2008) found that the resistance behavior of SMA wires is more complicated than the model reported by Ikuta et al. (1988). Increasing wire stress results in higher electrical resistance as well as increasing the martensite start temperature, martensite finish temperature, austenite start temperature, and austenite finish temperature. In addition, the electrical resistance linearly changes with temperature in the pure martensite and austenite phases. Novák et al. (2008) presented a resistance model for SMA actuators, where the specific resistivity of SMA wire is a function of specific resistivity of austenite, martensite, and R-phase with weights of austenite phase fraction, martensite phase fraction, and R-phase fraction, respectively. Song et al. (2011) have modified the model presented by Novák et al. (2008). In the modified model, the overall electrical resistance is a function of the electrical resistances of martensite and austenite. The effect of R-phase is negligible and eliminated to simplify the model. The modified resistance model is given by

R_{A} = R_{0 A} + (T - T_{0 A}) \frac{\partial R_{A}}{\partial T} + σ \frac{\partial R_{A}}{\partial σ}

(1)

R_{M} = R_{0 M} + (T - T_{0 M}) \frac{\partial R_{M}}{\partial T} + σ \frac{\partial R_{M}}{\partial σ}

(2)

where R_A and R_M are austenite resistance function and martensite resistance function, respectively; R_0A and R_0M are the nominal wire resistances of austenite and martensite at T_0A and T_0M temperatures, respectively; T is the temperature of the SMA wire; σ is the stress; and $(\partial R_{A} / \partial T) (\partial R_{M} / \partial T)$ and $(\partial R_{A} / \partial σ) (\partial R_{M} / \partial σ)$ are temperature and stress dependence of the resistance function of austenite (martensite), respectively.

The SRBC method measures the total resistance ( $R_{t}$ ) and uses the resistances (R_A and R_M) calculated using equations (1) and (2) to determine the martensite phase fraction based on the following equation

ξ_{M} = \frac{R_{t} - R_{A}}{R_{M} - R_{A}}

(3)

The temperature and stress dependence of the resistance functions of austenite and martensite are linear in equations (2) and (3). $\partial R_{A} / \partial T$ , $\partial R_{M} / \partial T$ , $\partial R_{A} / \partial σ$ , and $\partial R_{M} / \partial σ$ parameters are dependent on operating conditions, including the length of the SMA wire. These parameters have to be experimentally identified to be used in the resistance model. An experiment was conducted to find these parameters by studying the effects of stress and temperature on the resistance of the SMA wire. The experiment was conducted with three runs that each run included 50 trials. We have used a 200-µm SMA wire with a length of 40.54 cm as an actuator in the locking mechanism. In each trial, a random input power is applied to the SMA wire, ranging from 0 to 5 W. For each run, a specified stress on the SMA wire was applied, including 156.0509 MPa in run 1, 187.2611 in run 2, and 234.0764 in run 3. The resistance (R) is captured, and the temperature (T) is calculated in each trial based on the heating model that will be presented in section “Heating model.” Each trial was performed with 30 s of applied power, followed by 60 s with no applied power, which allowed the SMA wire to cool and return to the martensite phase. Figure 2 shows the resistance and temperature of the SMA wire in the 30 s heating mode for a sample trial at a 234.076 MPa stress.

Figure 2.

Resistance and temperature of the SMA wire under 234.076 MPa stress. By raising the temperature, the electrical resistance increases in the beginning, then decreases, and finally increases again, but with a smaller slope than the beginning.

The temperature dependence of the resistance functions of austenite and martensite ( $\partial R_{A} / \partial T$ and $\partial R_{M} / \partial T$ ) is the rate of resistance increase per 1°C enhancement in the temperature. $\partial R_{A} / \partial T$ and $\partial R_{M} / \partial T$ are calculated based on the measured resistance in the linear austenite and martensite regions. This is done by calculating the resistance increase divided by temperature difference for various pairs of points in the linear austenite and martensite regions in Figure 3 (slope of the line between each two points). The mean value and standard deviation (SD) of the calculated $\partial R_{A} / \partial T$ and $\partial R_{M} / \partial T$ are presented in Table 1. The linear austenite and martensite regions in Figure 3 are the temperature zones in which the resistance changes linearly. These regions are the temperatures below the transformation zone (for martensite) and the temperatures above the transformation zone (for austenite). Increasing temperature in the both martensite and austenite phases results in a linear resistance increase. However, in the transformation phase, the resistance is sharply reduced with the rise in the SMA wire temperature.

Figure 3.

Resistance of the SMA wire at different temperatures. The resistance of the SMA wire increases almost linearly and reaches the maximum amount in the linear martensite region, and then decreases to the minimum amount. Furthermore, it slowly rises in the linear austenite region.

Table 1.

Linear temperature and stress dependence of resistance functions of austenite and martensite.

Parameters	Values	Standard deviation
$\partial R_{A} / \partial T$	3.4 mΩ/°C	±0.069
$\partial R_{M} / \partial T$	35 mΩ/°C	±2.289
$\partial R_{A} / \partial σ$	0.3 mΩ/MPa	±0.019
$\partial R_{M} / \partial σ$	1 mΩ/MPa	±0.022

To find the stress dependence of the resistance functions of austenite and martensite ( $\partial R_{A} / \partial T$ and $\partial R_{M} / \partial T$ ), the resistance behavior of the SMA wire under the three various stresses has been studied and compared, as shown in Figure 4. $\partial R_{A} / \partial σ$ is calculated based on the rate of resistance increase per stress enhancement for the austenite linear region, and $\partial R_{M} / \partial σ$ is calculated based on the rate of resistance increase per stress enhancement for the martensite linear region. This is done by calculating the resistance increase divided by the difference between stresses at the same temperature, for various pairs of points in Figure 4. The mean value and SD of the calculated $\partial R_{A} / \partial T$ and $\partial R_{M} / \partial T$ are presented in Table 1.

Figure 4.

Resistance of the SMA wire for three runs of the experiment under different stresses. At the same temperature, larger stress causes the larger resistance of the SMA wire. The stress dependence of resistance functions is calculated by comparison of the SMA wire resistance curves for various stresses.

As shown in Figure 4, increasing the stress causes the increase in the resistance of the SMA wire and shifts the resistance curve upward as well as changing the temperature at which the transformation temperatures occur.

Heating model

In order to heat the SMA and increase its temperature quickly, the electric current is used based on Joule heating (equation (4)), which is also known as resistive heating. According to the literature, Joule heating is a convenient and practical heating method for small-size SMA wires (Sun et al., 2012). The effect of size is also critical (An et al., 2008). Therefore, the exact size of the SMA wire is included in the heating model. In addition, the nonuniform distribution of temperature using joule heating may be a problem in several applications (Huang, 2005); however, this effect is negligible in our application

P (t) = V (t) I (t) = R (t) {(I (t))}^{2}

(4)

Heating model block in Figure 1 calculates the SMA wire temperature using input power. In the heating model, the temperature of the SMA wire is a function of heat transfer. The heat transfer equation that relates the input power, equation (4), to the wire temperature is presented in equation (5) (Madill and Wang, 1998)

ρ cV \frac{dT}{dt} = P (t) - hA (T (t) - T_{\infty})

(5)

where ρ is the material density, c is the specific heat of the wire, V is the wire volume, T is the wire temperature, P is the input power, h is the convection heat transfer coefficient, A is the wire surface area, T_∞ is the ambient temperature, and t is the time.

Conduction is ignored due to the assumption that the temperature is constant in the thin wire. In addition, the radiation is negligible in comparison to convection at the operating temperature. The cooling process is slow and can be implemented by natural air convection, or using appropriate cooling fluids or forced air convection to have a faster cooling. The total convection heat transfer coefficient (h) for a long horizontal cylinder, for example, SMA wires, is the sum of natural convection heat transfer coefficient (h_n) and forced convection heat transfer coefficient (h_f), as seen in equation (6). The forced convection (h_f) is due to the airflow in the environment around the SMA wire

h = h_{n} + h_{f}

(6)

The natural convection heat transfer coefficient is calculated based on equation (7) (Lavine et al., 2011)

h_{n} = \frac{k}{D} N u_{D}

(7)

where k is the conductivity of air, D is the diameter of the cylinder, and Nu_D is the Nusselt number.

Morgan (1975) has suggested a power form expression for the Nusselt number as follows

N u_{D} = C {Ra}_{D}^{n}

(8)

where C and n are constants for a broad range of Rayleigh number (Ra_D). The constant numbers are shown in Table 2. In addition, Ra_D is calculated by equation (9)

R a_{D} = \frac{g β (T_{s} - T_{\infty}) D^{3}}{ν α}

(9)

Table 2.

Constant numbers of equation (8) for various ranges of Ra_D (Morgan, 1975).

C	N	Ra_D
0.675	0.058	10⁻¹⁰ to 10⁻²
1.02	0.148	10⁻² to 10²
0.85	0.188	10² to 10⁴
0.48	0.25	10⁴ to 10⁷
0.125	0.333	10⁷ to 10¹²

where T_s and T_∞ are surface and ambient temperature (°K), respectively; ν is the viscosity of air at $(T_{s} + T_{\infty}) / 2$ ; α is thermal diffusivity of air at $(T_{s} + T_{\infty}) / 2$ ; g is the acceleration of gravity, and β is ${((T_{s} + T_{\infty}) / 2)}^{- 1}$ .

The forced convection heat transfer coefficient (h_f) is calculated using equations (10) and (11). However, due to the fact that there is no significant airflow around the SMA wire in the experimental environment (V ≈ 0), the forced convection heat transfer is almost 0

N u_{D} = h_{f} (\frac{D}{k}) = 0.68 (R e^{1 / 2}) (P r^{1 / 3})

(10)

Re = \frac{VD}{ν}

(11)

In order to use the heating model, the convection heat transfer coefficient (h = h_n) should be identified. Table 3 presents constant parameters and calculated quantities for the heating model. The convection heat transfer coefficient, h, is calculated using other parameters presented in Table 3, which is equal to 99.1082.

Table 3.

Setup parameters related to the heating model.

L	0.4064 m	Pr	0.7
ρ	6.45 × 10³ kg/m³	ν	21.1 × 10⁻⁶ m²/s
C	465.2 J/kg °C	A	31 × 10⁻⁶ m²/s
V	1.276 × 10⁻⁸	K	31 × 10⁻³ W/m °K
D	2 × 10⁻⁴ m	Ra_D	0.0426
A	2.553 × 10⁻⁴ m²	Nu_D	0.6394
g	9.8 m/s²	h	99.1082

Experimental verification

This section presents the details of the evaluation of the SRBC method using a locking mechanism prototype. The prototype is used in an experimental setup that enables us to experimentally verify the SRBC method, by controlling the angle of the locking mechanism. For this purpose, the same SMA wire, as the one for the parameters identification, is used in the locking prototype as an actuator.

Experimental setup and procedure

A large-scale aluminum locking apparatus is prototyped and placed on a wooden base, as shown in Figure 5. The SMA wire is looped through the peg of the locking mechanism, and the two ends of the wire are fixed using two aluminum plates. Therefore, the contraction of the SMA wire causes a change in the angle of the locking mechanism. To provide heating to the SMA wire, a programmable power supply (30 V, 3 A, direct current (DC) power supply) is used. The two ends of the SMA wire are directly connected to the power supply. The output current of the power supply is controlled by a variable voltage signal, which is sent to the power supply via a data acquisition (DAQ) card. In addition, the voltage of the SMA wire is measured using another DAQ card. Due to the high internal resistance of the DAQ card as well as its high accuracy, no external circuit is required. The SMA wire electrical resistance is calculated using the measured SMA voltage and the applied current. To provide a specified stress for the SMA wire, the peg is fastened to a high-strength string, which is connected via a pulley to a weight. A low-friction potentiometer is connected to the hinge of the locking mechanism for angle measurement. The potentiometer is not involved in the control loop and it is used only for the experimental verification of the control system.

Figure 5.

Locking mechanism experimental setup. Aluminum prototype of the locking mechanism with a 40.54-cm SMA wire as an actuator. The two ends of the SMA wire are fixed by two aluminum plates. A low-friction potentiometer is connected to the hinge of the locking mechanism for experimental verification.

The stress on the SMA wire is constant in this setup based on the following assumptions: (a) the length of the string and the SMA wire are very long compared to the movement of the peg. Therefore, the two angles (β and γ in Figure 6) are constant during the contraction within a small margin of error, (b) the two angles are also equal (β = γ) because the SMA wire and the string are leveled with each other, and (c) the weight of the peg is negligible. Figure 6 depicts the configuration of the experimental setup in which the force applied to the SMA wire is provided by the weight. The relationship of string tension and the total pulling force of the SMA wire is calculated using equation (12)

T_{string} cos (γ) - F cos (β) - mg sin (β) = 0

(12)

where T_string is the tension of the string, mg is the weight of the peg, F is the total pulling force of the SMA wire, β is the angle of the SMA wire with vertical axis (dashed line), and γ is the angle of the string with vertical axis (dashed line).

Figure 6.

Schematic diagram of forces in the experimental setup.

Since the SMA wire is threaded through the peg, force applied to the SMA wire on each side is the half of F. Equation (13) shows the constant stress on the SMA wire, in which F is equal to the amount of the weight connected to the string (W)

σ = \frac{F}{2 π r^{2}} = \frac{W}{2 π r^{2}}

(13)

where σ is the stress and r is the radius of the SMA wire. F is the total pulling force of the SMA wire.

The range of the locking mechanism angle (θ) is from the initial point of 60° (completely open) and the final point of 28° (completely closed). For the purpose of this experiment, the range is mapped to θ = 0 at 60° and θ = 32° at 28°. Evidently, the range for changes of the angle can be adjusted by the length of the SMA wire used in the locking mechanism, as well as the size and shape of the peg. For example, if the SMA wire is longer, the angle range between the initial and the final position is greater. The initial position may be adjusted by the initial distance between the hole in the peg and the two ends of the SMA wire.

In order to control the angle of the locking mechanism, a PI controller is used, as shown in Figure 1. The controller gains (K_p and K_i) must be chosen carefully due to the hysteresis characteristics and highly nonlinear behavior of the SMA wire. This is more important when the controller performs in the range of the phase transformation. In this range, if the desired position (or martensite phase fraction) is passed, returning to the desired position (or martensite phase fraction) is very slow. This is due to the slow cooling process and hysteresis characteristics. The controller gains selection has also a direct effect on the response time of the output. Therefore, optimized gains must be chosen to satisfy both the response time and the output accuracy. For the final positions close to 32° (martensite phase fraction close to 0), larger gains have to be chosen to achieve a precise and faster response. Figure 7 shows a sample output angle measured by the potentiometer for the desired angle of 30° (very close to the maximum possible angle).

Figure 7.

Angle of the peg of the locking mechanism for the desired angle of 30°.

The goal of the locking mechanism is to hold the suture for the LigLAP procedure. The possible diameter range of the sutures is between 0.4 and 1.2 mm. Therefore, the operation range of the system is about 4° based on the range of sutures (see Figure 8 and equation (14)). Equation (14) presents the range for the desired angles of the locking mechanism (Δθ), based on various suture diameters (D)

Δ θ = co s^{- 1} (\frac{h - D_{max}}{L}) - co s^{- 1} (\frac{h - D_{min}}{L})

(14)

where Δθ is the range of changes in θ for various diameters of the sutures, h is the length of the leg, L is the length of the peg, and D_max and D_min are the minimum and maximum diameters, respectively.

Figure 8.

Diagram of the suture under the locking mechanism peg.

To find optimized controller gains, a wide variety of values for K_p and K_i have been tested. As predicted, especially for the desired angles that are not close to 32°, having K_p and K_i larger than appropriate values causes a positive error (the output is larger than the reference). In the next section, the steady-state errors for each desired angle are presented.

Results and discussion

The martensite phase fraction versus the measured angle of the locking mechanism (ξ_M−θ) curve is shown in Figure 9. The angle (θ) is measured using the potentiometer, and martensite phase fraction (ξ_M) is calculated based on the heating and resistance model and actual electrical resistance.

Figure 9.

Angle of the locking mechanism versus martensite phase fraction.

The experimental results in Figure 9 show that the relationship between the martensite phase fraction and the angle of the locking mechanism is approximately linear, indicating the effectiveness of using ξ as a control variable. The best obtained controller gains for the desired angles between 22° and 26° (Δθ = 4) are in the range of 0.3 < K_p < 0.4 and 0.08 < K_i < 0.18. Figure 10 shows the martensite phase fractions and the angles of the locking mechanism for the desired angles of 22°, 23°, 25°, and 26° during 10 s of the system operation, with the controller gains set to K_p = 0.35 and K_i = 0.14. These controller gains are obtained for the desired angle of 23° and used for the entire operation range based on the different suture diameters. Moreover, the steady-state errors for the several angles of the locking mechanism are shown in Table 4.

Figure 10.

Results for the angle of the locking mechanism and martensite phase fraction for various desired positions based on various sizes of sutures: (a) Desired angle=22, (b) Desired angle=23, (c) Desired angle=25, and (d) Desired angle=26.

Table 4.

Steady-state errors for several angles of the locking mechanism based on various suture diameters.

Desired angle (°)	22	23	25	26
Steady-state error (°)	0.25	0	0.2	0.5

In comparison with the non-resistive-based control methods in the literature (Kha and Ahn, 2006; Madden et al., 2004; Moallem and Tabrizi, 2009; Song, 2002; Troisfontaine et al., 1998), our resistive-based method can precisely control the position of SMA wires with the incorporation of the resistance model. The results confirm the validity of the resistance model, similar to the conclusions rendered by Song et al. (2011) and Novák et al. (2008).

Conclusion and future work

Due to the small space available in laparoscopic surgical devices, using external sensors is impossible or not preferred. The electrical resistance of SMA wires is an alternative feedback, in order to avoid using external sensors. In this article, an SRBC method for SMA wires to be used in the SMA-based locking mechanism for laparoscopic surgical devices is presented. Wire electrical resistance is measured to calculate the martensite phase fraction (ξ_M), which has a linear relationship with the strain of the SMA wire. This method has the advantage of using only a thin SMA wire without using any external sensors. In addition, wire electrical resistance can easily and accurately be measured in the application of SMA wire actuators. A hardware-in-the-loop control setup is used to control the angle of the locking mechanism using the method. The locking mechanism reaches the desired angle in all experimental trials. Very small steady-state errors in the experimental results (0.5° at most) show the accuracy of this method. As a result, the SRBC method accurately controls the SMA-based locking mechanism using only electrical resistance. It is concluded that the SRBC method is an alternative to the position/temperature feedback control algorithms. Moreover, the SRBC method does not require a non-model-based mapping for the relationship between electrical resistance and strain of the SMA wire, on the contrary to the methods that have employed mappings, such as neural network. In addition, SMA wires under various stresses can be controlled using this method, as the effect of stress is included in the resistance model of SMA wires. Furthermore, the mathematical model used in this article would suggest using model-based control strategies instead of trial and error control methods. Future work will design and develop SMA resistive-based control strategies to employ model-based control methods. In addition, the SRBC method can be used for other applications in which cost and space are limited and use of external sensors is impossible or costly.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

References

Huang

. (2008) A note on size effect in actuating NiTi shape memory alloys by electrical current. Materials & Design 29(7): 1432–1437.

Asua

Feutchwanger

García-Arribas

. (2010) Sensorless control of SMA-based actuators using neural networks. Journal of Intelligent Material Systems and Structures 21(18): 1809–1818.

Belgaumkar

Carswell

Chang

. (2009) The dangers of using stapling devices for cystic duct closure in laparoscopic cholecystectomy. Surgical Laparoscopy Endoscopy & Percutaneous Techniques 19(5): e194–e197.

Bencini

Boffi

Farsi

. (2003) Laparoscopic cholecystectomy: retrospective comparative evaluation of titanium versus absorbable clips. Journal of Laparoendoscopic & Advanced Surgical Techniques 13(2): 93–98.

Bezzi

Silecchia

Orsi

. (1995) Complications after laparoscopic cholecystectomy. Surgical Endoscopy 9(1): 29–36.

Champault

Cazacu

Taffinder

(1996) Serious trocar accidents in laparoscopic surgery: a French survey of 103,852 operations. Surgical Laparoscopy Endoscopy & Percutaneous Techniques 6(5): 367–370.

Huang

. (2004) TiNi-based thin films in MEMS applications: a review. Sensors and Actuators A: Physical 112(2): 395–408.

Fuchs

(2002) Minimally invasive surgery. Endoscopy 34(2): 154–159.

Golash

(2008) An experience with 1000 consecutive cystic duct ligation in laparoscopic cholecystectomy. Surgical Laparoscopy Endoscopy & Percutaneous Techniques 18(2): 155–156.

10.

Gollan

Bulkley

Diehl

. (1993) Gallstones and laparoscopic cholecystectomy. Journal of the American Medical Association 269(8): 1018–1024.

11.

Harrison

(1990) Measurable change concomitant with SME transformation. In: Duerig

Melton

Stöckel

. (eds) Engineering Aspects of Shape Memory Alloys. Oxford: Butterworth-Heinemann, pp. 106–209.

12.

Heald

Husband

Ryall

RDH

(1982) The mesorectum in rectal cancer surgery—the clue to pelvic recurrence? British Journal of Surgery 69(10): 613–616.

13.

Hirose

Ikuta

Tsukamoto

(1989) Development of a shape memory alloy actuator. Measurement of material characteristics and development of active endoscopes. Advanced Robotics 4(1): 3–27.

14.

Huang

(2005) Transformation front in shape memory alloys. Materials Science and Engineering A: Structural Materials Properties Microstructure and Processing 392(1–2): 121–129.

15.

Huang

Song

. (2013) Shaping tissue with shape memory materials. Advanced Drug Delivery Reviews 65(4): 515–535.

16.

Ikuta

Tsukamoto

Hirose

(1988). Shape memory alloy servo actuator system with electric resistance feedback and application for active endoscope. In: Proceedings of 1988 IEEE international conference on robotics and automation, Philadelphia, PA, 24–29 April, pp. 427–430. New York: IEEE.

17.

Johansson

(2003) Management of acute cholecystitis in the laparoscopic era: results of a prospective, randomized clinical trial. Journal of Gastrointestinal Surgery 7(5): 642–645.

18.

Jones

Soper

(1996) Complications of laparoscopic cholecystectomy. Annual Review of Medicine 47(1): 31–44.

19.

Kapiteijn

Marijnen

CAM

Colenbrander

. (1998) Local recurrence in patients with rectal cancer diagnosed between 1988 and 1992: a population-based study in the west Netherlands. European Journal of Surgical Oncology 24(6): 528–535.

20.

Kha

Ahn

(2006) Position control of shape memory alloy actuators by using self tuning fuzzy PID controller. In: 1st IEEE conference on industrial electronics and applications, Singapore, 24–26 May, pp. 1–5. New York: IEEE.

21.

Kim

Han

Lee

. (2013) Sensorless displacement estimation of a shape memory alloy coil spring actuator using inductance. Smart Materials and Structures 22(2): 025001.

22.

Kode

VRC

Cavusoglu

(2007) Design and characterization of a novel hybrid actuator using shape memory alloy and DC micromotor for minimally invasive surgery applications. IEEE/ASME Transactions on Mechatronics 12(4): 455–464.

23.

Kuribayashi

(2000) Micro SMA actuator and motion control. In: Proceedings of 2000 international symposium on micromechatronics and human science 2000 (MHS 2000), Nagoya, Japan, 22–25 October, pp. 35–42. New York: IEEE.

24.

Lavine

Incropera

Dewitt

(2011) Introduction to Heat Transfer. John Wiley & Sons, Inc.

25.

Song

Lee

(2004) Position control of shape memory alloy actuators with internal electrical resistance feedback using neural networks. Smart Materials and Structures 13(4): 777–783.

26.

Madden

Vandesteeg

Anquetil

. (2004) Artificial muscle technology: physical principles and naval prospects. IEEE Journal of Oceanic Engineering 29(3): 706–728.

27.

Madill

Wang

(1998) Modeling and L₂-stability of a shape memory alloy position control system. IEEE Transactions on Control Systems Technology 6(4): 473–481.

28.

Medina

(2005) Analysis and physics of laparoscopic intracorporeal square-knot tying. JSLS: Journal of the Society of Laparoendoscopic Surgeons 9(1): 113–121.

29.

Moallem

(2005) Application of shape memory alloy actuators for flexure control: theory and experiments. IEEE/ASME Transactions on Mechatronics 10(5): 495–501.

30.

Moallem

Tabrizi

(2009) Tracking control of an antagonistic shape memory alloy actuator pair. IEEE Transactions on Control Systems Technology 17(1): 184–190.

31.

Morgan

(1975) The overall convective heat transfer from smooth circular cylinders. Advances in Heat Transfer 11: 199–264.

32.

Kong

Lee

(1999) Migration of three endoclips following laparoscopic cholecystectomy. Journal of the Royal College of Surgeons of Edinburgh 44(3): 200–202.

33.

Novák

Šittner

Dayananda

. (2008) Electric resistance variation of NiTi shape memory alloy wires in thermomechanical tests: experiments and simulation. Materials Science and Engineering A: Structural Materials Properties Microstructure and Processing 481: 127–133.

34.

Rothe

Parameswaran

(2001) Design and development of actuating system for diagnostics application in minimally invasive surgery (MIS). In: Proceedings of the 23rd annual international conference of the IEEE Engineering in Medicine and Biology Society, Istanbul, 25–28 October, vol. 4, pp. 3473–3476. New York: IEEE.

35.

Pfeiffer

DeLaurentis

Mavroidis

(1999) Shape memory alloy actuated robot prostheses: initial experiments. In: Proceedings of 1999 IEEE international conference on robotics and automation, Detroit, MI, 10–15 May, vol. 3, pp. 2385–2391. New York: IEEE.

36.

Saha

(2000) Ligating the cystic duct in laparoscopic cholecystectomy. American Journal of Surgery 179(6): 494–496.

37.

Schiedeck

Mojrzisch

(2011) Design of a robust control strategy for the heating power of shape memory alloy actuators at full contraction based on electric resistance feedback. Smart Materials and Structures 20(4): 045002.

38.

Shahinpoor

Wang

(1994) Design and modeling of a novel fibrous shape memory alloy (SMA) actuator. In: 1994 North American conference on smart structures and materials, May 6, 1994, pp. 730–738. International Society for Optics and Photonics.

39.

Shamiyeh

Wayand

(2004) Laparoscopic cholecystectomy: early and late complications and their treatment. Langenbeck’s Archives of Surgery 389(3): 164–171.

40.

Song

(2002) Robust position regulation of a shape memory alloy wire actuator. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 216(3): 301–308.

41.

Song

Kubica

Gorbet

(2011) Resistance modelling of SMA wire actuators. In: International workshop on smart materials, structures & NDT in aerospace conference, Montreal, QC, Canada, 2–4 November, 10 pp. Available at: http://212.8.206.21/article/ndtcanada2011/papers/66_Gorbet_Rev2.pdf

42.

Sun

Huang

Ding

. (2012) Stimulus-responsive shape memory materials: a review. Materials & Design 33: 577–640.

43.

Tanović

Mesihović

(2003) Differences in the postoperative course and treatment in patients after laparoscopic and standard cholecystectomy. Medicinski arhiv 57(4): 219–222.

44.

Troisfontaine

Bidaud

Dario

(1998) Control experiments on two SMA based micro-actuators. In: Casals

de Almeida

(eds) Experimental Robotics V. Berlin and Heidelberg: Springer, pp. 490–499.

45.

Lee

Wang

. (1998) Design of fuzzy walking pattern (FWP) for a shape memory alloy (SMA) biped robot. In: IEEE international conference on systems, man, and cybernetics, San Diego, CA, 11–14 October, vol. 4, pp. 3266–3271. New York: IEEE.

46.

Vittimberga

Jr Foley

Meyers

. (1998) Laparoscopic surgery and the systemic immune response. Annals of Surgery 227(3): 326–334.

47.

Yeh

Jan

Liu

. (2004) Endo-GIA for ligation of dilated cystic duct during laparoscopic cholecystectomy: an alternative, novel, and easy method. Journal of Laparoendoscopic & Advanced Surgical Techniques 14(3): 153–157.

48.

Yousefian

Kia

Zadeh

(2013) On the design of shape memory alloy locking mechanism: a novel solution for laparoscopic ligation process. World Academy of Science, Engineering and Technology 78: 1644–1647.