Design and control of a novel compliant differential shape memory ...

Sensors and Actuators A 225 (2015) 71–80

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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna

Design and control of a novel compliant differential shape memory alloy actuator Zhao Guo a , Yongping Pan a , Liang Boon Wee b , Haoyong Yu a,∗ a b

Department of Biomedical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117575 DSO National Laboratories, 20 Science Park Drive, Singapore 118230

a r t i c l e

i n f o

Article history: Received 6 January 2014 Received in revised form 14 January 2015 Accepted 15 January 2015 Available online 11 February 2015 Keywords: Shape memory alloy Compliant SMA actuator Bio-inspired design Saturated PI control

a b s t r a c t This paper presents the design and control of a novel compliant differential shape memory alloy (SMA) actuator with significantly improved performance compared to traditional bias and differential type SMA actuators. This actuator is composed of two antagonistic SMA wires and a mechanical joint coupled with a torsion spring. The differential SMA wires are utilized to increase the response speed, and the torsion spring is employed to reduce the total stiffness of SMA actuator and improve the output range. Theoretical models that include the stiffness equations of the SMA wire as well as the dynamics of three different SMA actuation systems are introduced and compared. Simulation and experimental results have proved that this new actuator can provide larger output angle compared to conventional SMA actuators under the same conditions. Moreover, regulation and tracking control experiments have demonstrated that this compliant differential SMA actuator achieves higher response speed compared to the bias SMA actuator using compatible PI controller. The tracking performance is further improved by the saturated PI controller. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Shape Memory Alloy (SMA) is an intelligent material that can remember its original shape at low temperature and return to the pre-deformed shape by heating over a threshold temperature. Because of this unique property, it has been receiving considerable attention as mini-actuator in recent years [1]. Among the SMA materials, Nitinol (Ni–Ti) alloy has been widely employed for miniactuator design due to its high strain (up to 8%). Compared to conventional (electric, hydraulic, and pneumatic) actuators, SMA actuators have the advantages of high force to mass ratio, biocompatibility, small size, simple mechanical design, and silence operation. These advantages make them suitable for a wide variety of applications, like soft robotics [2,3], robotic surgical systems [4,5], and grippers [6,7]. Nevertheless, SMA actuators present several disadvantages, such as, low energy efficiency, slow response rate, nonlinearity, which encourage additional effort in special mechanical design and advanced control strategy. SMA wire can only achieve unidirectional actuation, thus it is necessary to provide a recovery force via a weight, a spring or another antagonistic SMA wire to realize bidirectional movement.

∗ Corresponding author. Tel.: +65 6601 1590. E-mail address: [email protected] (H. Yu). http://dx.doi.org/10.1016/j.sna.2015.01.016 0924-4247/© 2015 Elsevier B.V. All rights reserved.

Typically, there are two classes of SMA actuators: bias type SMA actuator and differential type SMA actuator [8,9]. The former one, composed of a SMA element and a bias spring, has slow response as the speed is determined by the cooling process and the passive spring. The latter one, which consists of two antagonistic SMA elements, has faster speed of response than the bias actuator, but it consumes more power and the angle is restricted by the stiff antagonistic SMA elements. SMA actuators with large working range, contraction strain and quick response are desirable for different purposes in many fields [10]. To achieve this goal, a series of new SMA-based actuators or mechanisms have been developed. For instance, Grant and Hayward reported a differential SMA actuator which is comprised of 12 SMA wires in a helical arrangement to produce larger strains [11]. Zhang and Yin presented a SMA-based artificial muscle composed of 16 parallel SMA wires and a simple linear spring to improve the driving force [12,13]. Paik and Wood introduced a bidirectional SMA folding actuator to produce two opposing 180◦ motions [14]. Park et al. [15] proposed a differential spring-biased SMA actuator for a bio-mimetic artificial finger. In this design, the spring was directly connected to the SMA wire, which may absorb the contraction length of SMA wires. Compared to stiff actuators, compliant actuators which incorporate passive elastic element like spring or damper possess the ability of absorbing shocks, storing and releasing energy, and safe interactions with the environment [16]. These characteristics have

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attracted strong interest from the areas of rehabilitation and surgical robotics. Many types of compliant actuators, such as series elastic actuator [17,18], variable stiffness actuator [19], and pneumatic artificial muscle [20] have been developed. Several advanced materials such as ionic polymer metal composite (IPMC) [21], electro-active polymers (EAP) [22] are proposed as soft actuators for their large deformations, but they can generate only small forces. Unlike these soft polymers, SMA wire can generate large forces due to its large Young’s modulus [8]. However, the use of such SMA materials in compliant actuator design has not yet been well investigated. Inspired by the biological structure of human joint driven by antagonistic muscles with tendons, this paper introduces a compliant differential SMA actuator from the perspective of stiffness: a soft spring is integrated into an antagonistic SMA actuator, intending to reduce the total stiffness and increase the compliance of the differential SMA actuator, without compromising the advantages of conventional SMA actuators. The rest of this paper is organized as follows. Section 2 describes the design of the compliant differential SMA actuator, along with the implementation of two other different SMA actuators. Section 3 presents the mathematical models as well as a saturated PI controller developed for the compliant differential SMA actuator. The simulation and experimental results are shown in section 4. Section 5 discusses the potential application of this actuator and the limitation of this study, with concluding comments presented in Section 6.

rotate around the axis. An encoder is connected to the axis by a coupling. The rotational angle of the axis can be measured by the encoder. That means the position of the load and coupler #1 can be detected. The working process of this actuator is described as follows: when the upper SMA wire is activated by heating, the coupler #1 rotates due to the active contraction force of SMA wire, the load follows the rotation of coupler #1, thereby the torsion spring starts to twist and transmits the active force from coupler #1 to coupler #2, the lower SMA wire will be stretched to restrict the rotation, by contrast, when the lower SMA wire is heated to contract, the active force is transmitted in the opposite direction. Based on this principle, we can heat the antagonist SMA wire and cool the opposite one to increase the response speed of this actuator. 2.2. Comparison to other SMA actuators

In this section, we introduce the design of the compliant differential SMA actuator and the experimental testing bench. We implement three configurations using the same mechanical setup for the performance comparison of three kinds of SMA actuators.

To compare the performance of this actuator with the two traditional SMA actuators, we implement three configurations to represent three types of SMA actuators using the same mechanical setup. As shown in Fig. 3, they are (a) bias SMA actuator (coupler #1 is connected with an SMA wire, coupler #2 is fixed by a hard iron wire, recovery force comes from the torsion spring), (b) differential SMA actuator (two SMA wires are connected to coupler #1), and (c) compliant differential SMA actuator (two SMA wires are linked to two couplers respectively, the torsion spring is hidden inside). Fig. 3 includes the CAD model of each actuator and its equivalent mechanical model. Two SMA wires generate the tensile force with a variable stiffness ki and a damping factor bi (here and the following i = 1, 2). The torsion spring transmits the SMA force with a constant stiffness ks . The system damping between two couplers is represented by bs . mi represents the mass of each coupler and the mass of SMA wire can be neglected (mSMA  mi ). The load applied on coupler #1 is represented as a mass mload . Comparison of these three SMA actuators will be discussed in the experimental section.

2.1. Compliant differential SMA actuator

3. Actuator modeling and controller design

Human elbow joint is actuated by antagonistic skeletal muscles (biceps and triceps) which are connected to the bone through the tendon (Fig. 1(a)). Inspired by this biological structure, we propose a new SMA-based compliant differential actuator to mimic the extension/flexion motion of human joint. As shown in Fig. 1(b), this SMA actuator is composed of two antagonistic SMA wires, a torsion spring and two cylindrical couplers. A load is applied on coupler #1 by threaded connection. The antagonistic SMA wires, behaving like artificial human muscles, are directly connected to the couplers. SMA wires provide the active force for bi-directional motion of the actuator. The torsion spring is used to mimic the human tendon and is packaged inside the couplers. As illustrated in Fig. 1(b), two legs of the torsion spring (0◦ deflection, left hand wind) are restricted by the slots in the couplers. The active contraction force produced by the SMA wire can be transmitted from one coupler to the other by the torsion spring. In this design, we select a soft torsion spring. Its stiffness is lower than that of the SMA wires. The torsion spring has two functions: one is to store the energy and provide the recovery force for the SMA wire; the other is to reduce the total stiffness of the actuator. Fig. 1(c) and (d) shows the integrated Computer Aided Design (CAD) model and prototype of the compliant differential SMA actuator. The whole structure of this actuator is simple and easy to implement for different applications. An experimental testing bench designed to evaluate the performance of this compliant differential SMA actuator is presented in Fig. 2. The actuator is assembled into a base, supporting by two bearings on two sides. The coupler #1 is fixed on the axis by a small screw (not shown in this picture), while the coupler #2 can free

In this section, we present general theoretical models that include the equations of the SMA wire as well as the dynamics of the actuation system. These models are developed in several steps for clarity. First, the nonlinear stiffness of the SMA wire is given based on the constitutive model. And then, the dynamic equations are developed for the controller design.

2. Actuator design

3.1. SMA modeling 3.1.1. Constitutive equation According to Liang and Rogers [8,23], the constitutive equation of SMA wire can be described as the relation of stress , strain ε, temperature T and martensite fraction  (0 ≤  ≤ 1,  = 1 means SMA totally in martensite phase,  = 0 means SMA in austenite phase). The general form is written as ˙ = E ε˙ + ˝˙ + T˙

(1)

where E, ˝ and  represent the Young’s modulus, phase transformation constant and thermal expansion coefficient, respectively. For SMA material, ˝ is a constant and can be expressed as ˝ = −εL E, where εL is the maximum recoverable strain of SMA. Since the influence of  on the strain is much smaller than that of ˝, the thermal expansion part in Eq. (1) can be neglected. The constitutive equation is simplified as ˙ ˙ = E ε˙ − εL E .

(2)

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Fig. 1. (a) Structure of human elbow joint; (b) the proposed bio-inspired human joint mechanical model, two SMA wires, two couplers and a torsion spring play as the antagonistic muscles, bones and tendon respectively; (c) the integrated design in CAD model and (d) the prototype of the compliant differential SMA actuator.

Fig. 2. (a) CAD model and (b) prototype of the experimental test bench.

The Young’s modulus is the function of martensite fraction, expressed in following expression E = EM + (1 − )EA

(3)

where EM and EA are the Young’s modulus constants corresponding to martensite and austenite phases, respectively. Taking into account the initial condition of SMA wire (0 , ε0 , 0 ), the constitutive Eq. (2) is given by



t

˙ E(ε˙ − εL )dt + 0 .

=

(4)

0

where A and l are the cross-sectional area and the initial length of SMA wire, respectively. k is the stiffness of SMA wire, from Eq. (5), which is written as k=

A . εl

Considering two terminal conditions, where the SMA wire in twinned martensite phase without load (0 = 0, ε0 = 0,  = 1), and full austenite without load (0 = 0, ε0 = 0,  = 0) [24], since the phase transformation rate is equal to zero, thereby from Eq. (4), the SMA stiffness in full martensite phase and austenite phase are given by

The contraction force of SMA wire satisfies F = A = kεl

(5)

(6)

kM =

EM A EA A , kA = . l l

(7)

Fig. 3. CAD models and equivalent mechanical models for three SMA actuators: (a) bias SMA actuator; (b) differential SMA actuator; (c) compliant differential SMA actuator.

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Table 1 Parameters of the SMA wire and the compliant actuator. Parameter

Value

Parameter

Value

EM EA As Af Ms Mf mw R l0 m1 , m2 mload J1 , J2 bs ks

28 GPa 75 GPa 88 ◦ C 98 ◦ C 72 ◦ C 62 ◦ C 6.8 × 10−4 kg/m 20 /m 0.37 m 8g 100 g 0.38 kg (mm)2 0.5 0.0018 N m/1◦

CA CM Tamb A Aw Cp εL εA h0 h2 rl r1 ,r2 b1 ,b2 rs

10MPa/◦ K 10MPa/◦ K 20 ◦ C 4.9 × 10–8 m2 290.45 × 10–6 m2 320 J/Kg ◦ C 2.3% 5% 20 0.001 20 mm 15 mm 2 7.5 mm

3.1.2. Heat transformation model SMA has inherent hysteresis during the transformation between martensite phase and austenite phase. The martensite fraction  is modeled by SMA temperature and stress [8,23]. During the heating transformation (martensite phase to austenite phase)  is given by =

M 2





cos[aA (T − As ) + bA ] + 1

for As +

  ≤ T ≤ Af + CA CA

(8)

1 − A 1 + A cos[aM (T − Mf ) + bM ] + 2 2   ≤ T ≤ Mf + for Ms + CM CM

(9)

3.1.3. Heat dynamics The heat for phase transformation is generated by applying a voltage to the SMA wire. A SMA heat transfer model is formulated to describe the rate of temperature change due to a change in voltage of the wire and the convective heat loss to the environment. This model can be defined by the first-order dynamic equation [25]. V2 − hAw (T − Tamb ) R

J1 ¨ 1 = F1 r1 − b1 ˙ 1 − ks (1 − 2 ) − bs (˙ 1 − ˙ 2 ) − mload grl cos 1 J2 ¨ 2 = ks (1 − 2 ) + bs (˙ 1 − ˙ 2 ) − F2 r2 − b2 ˙ 2

.

(11)

where Ms , Mf , As , Af are the start and finish transition temperatures associated with martensite and austenite phase transformations. aA , bA , aM and bM are constants derived from four transition temperatures, aA = /(Af − As ), bA = −(aA /CA ), and aM = /(Ms − Mf ), bM = −(aM /CM ). CA , CM are material coefficients.  M , A are the initial martensite fractions for each transformation.

mw cp T˙ =

to the y axis, is defined as  i . Fi is the SMA contraction force and ri is its moment arm, respectively. s is the torque of torsion spring. The moment arm of the load is r1 . Ji represents the inertia constant of each coupler. The main parameters of the compliant actuator are shown in Table 1. Taking into account the damping and stiffness in the actuation system, the dynamic equation of each coupler is given by



and during the cooling transformation (austenite phase to martensite phase)  is given by =

Fig. 4. (a) Simplified schematic diagram, drawing with applied forces and (b) the equivalent mechanical model of the compliant differential SMA actuator.

(10)

where mw is the mass per unit length, cp is the specific heat constant, R is the SMA’s resistance per unit length, V is the applied voltage, Aw is the wire surface area, Tamb is the ambient temperature and h is the heat convection factor, which is a function of the temperature with the parameters h0 and h2 , h = h0 + h2 T 2 . Two NiTi SMA wires (0.25 mm diameter, the initial length is 0.37 m (Twinned martensite phase), Flexinol, Dynalloy, Inc.) are selected for our actuator design. Their properties are summarized in Table 1.

For comparison, we build the dynamics of the two other SMA actuators based on the mechanical model in Fig. 3, the equation of the bias SMA actuator is written as J1 ¨ 1 = F1 r1 − b1 ˙ 1 − ks 1 − bs ˙ 1 − mload grl cos 1 .

(12)

The differential SMA actuator’s dynamic model is given by J1 ¨ 1 = F1 r1 − b1 ˙ 1 − F2 r1 − b2 ˙ 1 − mload grl cos 1 .

(13)

Hence, the contraction force is related to the SMA stiffness and strain. The strain rate of SMA wire ε˙ i is derived from kinematics as a function of its joint velocity ˙ i ε˙ i = ri ˙ i /li .

(14)

Based on the modeling process, a schematic diagram of the equations involved in the overall mathematical models of the compliant actuation system is shown in Fig. 5. This model clearly illustrates a complex multiple-input multiple-output (MIMO) system with cross-coupling effects. Two independent output variables

3.2. Actuator modeling A simplified schematic diagram and its equivalent mechanical model of the compliant differential SMA actuator are presented in Fig. 4. The couplers are actuated under the function of the upper and lower SMA wires. The rotational angle of each coupler, relative

Fig. 5. Block diagram of the mathematical model of the actuation system.

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Specifically, the tracking performance can be improved by adjusting the thickness of the boundary layer. If si is over the boundary layer i , SMAs are activated by the high voltage ViH to get quick response. If si enters the boundary layer, within proper tuning of the control gains, the voltage is reduced to avoid overshoot for smoother and more stable response. If the value si is lower than zero, the controller switches to zero for cooling the SMA wire. The magnitude of the boundary layer is determined experimentally to protect the SMA wire from overheating. 4. Simulation and experimental results In this section, we provide a description of the experimental setup. Three types of experiments have been performed to test and compare the performance of the compliant actuator with the other two conventional actuators. 4.1. Experimental setup

Fig. 6. (a) Flow diagram of the position control system for the compliant differential SMA actuator, and (b) structure of the SPI control strategy.

 1 and  2 can be controlled by applying voltages on two SMA wires separately. Since the load is applied on coupler #1,  =  1 represents the output angle of the actuator.  r is the reference angular position. 3.3. Saturated PI Controller design From the actuation system modeling derived in Section 3.2, it is clear that the SMA actuator position can be controlled by the voltage signals. It is worth noting that the actuation model is important for the simulation and controller parameters selection although it is not directly applied to controller design. In this section, a saturated Proportional Integral (PI) controller is designed to demonstrate the performance of the proposed compliant differential SMA actuator. The flow diagram of the position control system is presented in Fig. 6(a). Two control voltages produced by the controller are applied to the upper and lower SMA wires to get the desired position. The structure of the saturated PI controller is shown in Fig. 6(b). A basic control law si for each SMA wire is defined as follows:



si = ci + cPi  + cIi

4.2. Comparison for the three kinds of SMA actuators

dt,

(15)

where  =  − r is the position error. cpi , cIi are the proportional and integral gains, respectively. These parameters are chosen for desired tracking performance. It has been observed that the passive SMA wire in the different SMA actuator can produce a few millimeters of slack when it cools [10]. This will cause the wire to contract enough to take up the slack before it begins to pull the coupler. Slack in the wire will slow the response speed of the control system. To solve this problem, ci is set in a small voltage to keep SMA wires in tension when the position error is zero, for the purpose of reducing the slack of SMA wires. Each si is restricted by a boundary layer between a high voltage ViH and zero. Thus, the control voltage Vi for each SMA wire is defined as

Vi =

⎧ V if si ≥ i ⎪ ⎨ iH ⎪ ⎩

The experimental setup has been developed to evaluate the performance of this compliant differential SMA actuator. Before to set the experimental setup, each actuated SMA wire (twinned martensite state) is passively extended to the fully de-twinned martensite state. As shown in Fig. 7(b), an iron stick linked with a standard weight (100 g) is connected to the coupler #1 of the actuator as the testing load. The pre-strain of each SMA wire is extended to 0.03. Including the load force, the initial strain of the upper wire is equal to 0.0336. Fig. 7(a) and (b) shows the schematic diagram of the control system and a picture of the experimental setup, respectively. Lab VIEW software (National Instruments Corp., TX) on a computer controls the position of the actuator using a power supply (Dual-tracking DC 6306D, Topward Electric Instrument Co., TW), an NI-PCI-6221 Data Acquisition (DAQ) card (National Instruments Corp., TX) and a voltage amplifier. The angular position of the actuator, measured by a shaft encoder (Omron E6B2-CWZ1X, Omron Inc., Japan) with a resolution of 2500 pulses per turn, was collected by the DAQ card. This DAQ card also provided two 16-bit resolution analog voltage outputs with a full-scale range of ± 10 V to the amplifier. Two analog voltage channels were augmented by the amplifier with a gain of 3.125 V/V and applied on two SMA wires separately. For comparison, saturated PI and PI (without boundary layer and initial constant value) control strategies were programmed in the Lab VIEW software and experimental tests have been conducted in the following.

si

if 0 ≤ si < i .

0

if si < 0

(16)

A comparison between this new compliant differential SMA actuator and the other two SMA actuators was performed in two aspects: stiffness and output angle. A Simulink model was developed in MATLAB/Simulink to simulate the joint motion of the three different SMA actuators with the same load and same input voltage in open loop. Based on the mathematical models of the actuation system and the SMA stiffness definition, we can get the stiffness of the upper SMA wire. Fig. 8 presents the simulated SMA stiffness for the three different SMA actuators in a function of recovery strain (0 → 0.015) and martensite fraction (1 → 0). It displays that the stiffness increases from martensite phase to austenite phase during heating process. The minimal value is equal to 3.7 kN/m when the SMA wire is in fully martensite phase. In this compliant actuator design, the stiffness of the torsion spring ks is chosen as 0.0018N m/1◦ ; its equivalent stiffness of an extension spring is 1.03 kN/m. This value is determined by the parameters of the spring (including the mean coil diameter, diameter of spring wire, number of turns, elastic modulus of the material, and so on), depending

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Fig. 7. (a) Block diagram configuration of the position control system and (b) experimental setup for testing.

Fig. 8. Stiffness of upper SMA wire during heating process, in which blue line, black line and red line, represent the results of the bias, differential, and compliant differential, SMA actuators, respectively. Abbreviation in the figure: BS: bias SMA actuator; DI: differential SMA actuators; CD: compliant differential SMA actuators.

on the limited space between the two couplers. Thus, this spring stiffness is lower than the SMA stiffness (ks < kM ). According to the equivalent mechanical models in Fig. 3, the passive stiffness of the compliant actuator includes the stiffness of the unheated SMA wire and the bias spring, which is equal to (k2 · ks )/(k2 + ks ) (because the torsion spring and lower SMA wire are connected in series). Thus, the values of passive stiffness of the three SMA actuators follow the order (k2 · ks )/(k2 + ks ) < ks < k2 . This compliant actuator has the smallest passive stiffness. I.e. the actuator presented in this paper has the smallest passive stiffness and is thereby the most compliant SMA actuator. The simulation angles for the three actuators during heating and cooling phases are shown in Fig. 9(a). Since it has the smallest passive stiffness, the compliant differential SMA actuator can provide larger output angle than the other two actuators. To prove this hypothesis and the simulation results, a constant voltage (1 V) with the same load was applied to the upper SMA wire of the three actuators in open loop, raising the temperature of the SMA wire by heating. The lower SMA wire was not activated and remained as a passive spring. The experimental results for the three actuators, including heating and cooling phases of the upper SMA wire, are shown in Fig. 9. The results confirm that the compliant differential SMA actuator provides the largest angle. The maximum angular motion of the compliant differential SMA actuator is close to 30◦ , while the bias SMA actuator is limited to 20◦ and the differential SMA actuator provides only 13◦ . This limitation in angular motion is due mainly to the high stiffness of the antagonist SMA wire. These experimental results provide strong evidence for the theoretical models and simulation results in Fig. 9(a). The dynamic response

Fig. 9. (a) Simulation and (b) experimental results of the three SMA actuators under the same applied voltage, showing the load angular position over the time.

of the three actuators shows a slight position overshoot before reaching steady-state, see Fig. 9(b). This behavior is explained with the aid of the heat dynamics, Eq. (10). During the heating phase, the SMA wire undergoes a phase change that persists until an energy balance between the power input and the power loss is reached. This energy transfer also corresponds to the temperature response of the SMA wire. Interesting, during the cooling phase, we observed that the three actuators have almost the same cooling time. It means that the rate at which the compliant differential SMA actuator returned to the relaxed state is faster than the other actuators. This faster rate of the compliant differential SMA actuator is provided by stored energy in the torsion spring. The above results clearly indicate that the compliant differential SMA actuator provides larger working range under the same voltage stimulating condition and a faster system response. 4.2.1. Remark The maximum speed of response of this actuator mainly depends on the input current/voltage through the SMA wire and the SMA wire’s stiffness. Larger current can increase the response speed, while the current should be limited to protect the SMA wire. In addition, the maximum working range of this compliant mechanism depends on the mechanical range of two couplers, the de-twinned strain of the SMA wire and the stiffness of the SMA wire. When all wires are subjected to provide the same initial conditions

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Table 2 Performance comparison of step response for different control methods. Step response angle (◦ )

0 → 10 10 → 20 20 → 10 10 → 0

Time to the stable state (s) PI BS

PI CD

SPI CD

4 4.5 6 6

4.2 3.5 2 2.5

0.6 0.5 1 1

Total reduction percentage 85% 89% 83% 83%

Note: Total reduction percentage: from SPI control on CD actuator to PI control on BS actuator. PI BS: PI control for bias SMA actuator, PI CD: PI control for compliant differential SMA actuator, SPI CD: saturated PI control for compliant differential SMA actuator.

Fig. 10. Step response of two actuators with different controllers, blue thick dashed line, red thin solid line, dark green thick solid line, and light green thick dash-dot line represent the reference (abbreviated as Ref), real value, current on upper SMA wire, current on lower SMA wire, respectively: (a) bias SMA actuator using PI control, (b) compliant differential SMA actuator using PI control, and (c) compliant differential SMA actuator using saturated PI control.

(pre-strain, current, temperature, length, etc.), the actuator with the softer joint will have a larger working range motion because the resistance force will be smaller and the wires can experience larger contraction. This is clearly shown in the simulation and experimental results. 4.3. Regulation results In the following sections, we compare the tracking performance between the bias SMA actuator and the compliant differential SMA actuator. The differential SMA actuator will not be compared in this test due to its limited angle range. Firstly, the experimental responses to a series of step inputs were examined. For comparison, the same PI controller for the two actuators and different controllers (PI and saturated PI) for the same compliant differential SMA actuator have been implemented. The gains of both controllers were optimized to minimize the position response time and improve the output accuracy. The voltage boundary value is limited under 4 V. Fig. 10 shows position responses of both actuators with different controllers to desired steps of 10◦ and 20◦ , and the currents applied on both SMA wires, respectively. Fig. 10(a) illustrates the step response of the bias SMA actuator with PI controller. We observe that this actuator can follow the reference within a relatively long time just under one channel current driving. During the heating phase, the rise time to the 10◦ step input is approximately 4 s. For the response to 20◦ , it is slightly longer and requires approximately 4.5 s. This is expected as the set point is further away from the zero position, so it takes a longer time for the actuator to reach the desired 20◦ set point. During the cooling phase, the average fall time from 20◦ to the 10◦ and from 10◦ to 0◦ is approximately

6 s. The joint angle reduces slowly to reach the set point under the recovery force from the torsion spring. Fig. 10(b) shows step response of the compliant differential SMA actuator using the same PI controller. It is observed that the response speed is increased under the function of two channels currents. Compared to the bias SMA actuator, the compliant actuator reaches the set point faster, while it costs in average 4 s to reach the stable value during the heating phase, and also an overshoot takes place in this tracking. The rise time is almost the same as the bias SMA actuator. Specifically, to increase the response speed of this actuator, the lower SMA wire was heated with a second channel current during the cooling phase. This reduces the fall time to 2.5 s. To further improve the performance, the saturated PI controller was employed to control the compliant actuator. As shown in Fig. 10(c), with properly adjusting the parameters, (set the original current on the SMA wire and increase the current during the cooling process), this controller eliminates the overshoot and further reduces the response time. Compared to PI controller, during the heating phase, the saturated PI controller reduces the rise time on the 10◦ , 20◦ step input from 4.2s to 0.6 s, and 3.5 s to 0.5 s, respectively. Note that, during the cooling phase, the falling time of step response from 20◦ to 10◦ , 10◦ to 0◦ reduces from 2.5 s to lower than 1 s. Step response comparisons between bias SMA actuator with PI controller, compliant differential SMA actuator with PI controller and compliant differential SMA actuator with saturated PI controller are shown in Table 2. The results clearly indicate that the rise and fall times of step response are decreased significantly, as well as the response speeds both in the heating and cooling phases are increased. The consumed time during cooling phase is reduced over 50% from bias actuator to compliant differential actuator with the same PI controller. Therefore, the compliant SMA actuator has better performance than the bias SMA actuator. Furthermore, our proposed saturated control method can further improve the response results over conventional PI controller. So we achieve a step response time reduction of more than 80%. 4.4. Tracking results This section presents the comparison of tracking responses to a series of sine inputs with varying frequencies. Fig. 11 shows the tracking results with 0.05 Hz sinusoidal reference trajectory. Fig. 11(a), is the result of PI control for bias SMA actuator, showing a little disturbance in the rising period of tracking and 2◦ error in the peak value. Some fluctuations appear in the recorded current of SMA wire, which cause the disturbance in tracking. The actuator can track the reference in the falling process with the help of the torsion spring. Fig. 11(b) illustrates the tracking results of the compliant differential SMA actuator under PI control. The tracking performance is improved obviously by controlling two current channels on the SMA wires. The real position smoothly follows the reference trajectory, while a time lag is observed at the beginning of rising and falling time of the sine wave. As shown in Fig. 11(c), the tracking

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Fig. 11. Experimental tracking results for 0.05 Hz sinusoidal reference trajectory showing (a) PI control for bias SMA actuator, (b) PI control for compliant differential SMA actuator, (c) saturated PI control for compliant differential SMA actuator, and (d) error comparison.

result using the saturated PI controller on compliant SMA actuator is presented. It illustrates that the saturated PI control method obtains more accurate and smooth tracking results than the PI control on the same actuator. Fig. 11(d) displays the error comparison among the three different tracking. With the same PI controller for the two different actuators, the error range is reduced from −1◦ to 2◦ to −1◦ to 1.5◦ , and the error is further decreased to −0.8◦ to 1◦ with the saturated PI control method on compliant differential SMA actuator. Fig. 12 shows the results with 0.1 Hz sinusoidal reference trajectory tracking with the three previous ways for 0.05 Hz tracking. Fig. 12(a) illustrates the tracking of bias SMA actuator under PI control. By adjusting the current on the upper SMA wire, the actuator can track the reference with a disturbance during the rising time, while it follows the reference in the falling time with a larger error. A time lag comes out because of the increased frequency. To improve the tracking performance during the falling time, the lower SMA wire in the compliant differential SMA actuator is activated. The real position can follow the reference with some fluctuations, presented in Fig. 12(b). However, it is hard to catch the peak value with this simple PI control. We introduce the saturated PI controller for the compliant differential SMA actuator, and a smooth, stable and accurate tracking is obtained in Fig. 12(c). Fig. 12(d) shows the

Fig. 12. Experimental tracking results for 0.1 Hz sinusoidal reference trajectory showing (a) PI control for bias SMA actuator, (b) PI control for compliant differential SMA actuator, (c) saturated PI control for compliant differential SMA actuator, and (d) error comparison.

error comparison among the different control ways. It is shown that the saturated PI method for compliant differential SMA actuator has the lowest error, limiting its range among −1◦ to1◦ . The output error of the compliant differential SMA actuator is smaller than the error of the bias SMA actuator, using for both the same PI controller. The error is reduced from −2.5◦ to1.5◦ to −1.5◦ to1.5◦ . The error has been further minimized with the saturated PI controller. In addition, it is obviously observed that the tracking performance is smooth and the actuator is able to reach the peak of the sinusoidal reference trajectory. Comparing the results of the same actuator with the same controller and different reference frequencies (from 0.05 Hz to 0.1 Hz), it is observed that each tracking error rises gradually as the frequency increases. It takes a finite amount of time for the actuators to track for the limited cycle time. The actuators begin to lag behind the desired trajectory due to the slow response speed of the SMA wire. Nonetheless, these figures demonstrate better tracking performance for the compliant differential SMA actuator over the bias SMA actuator with the same PI controller, and the tracking performance has been further enhanced with the substitution of the conventional PI controller for the saturated PI controller.

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5. Discussion From above experimental investigation, the compliant differential SMA actuator possesses the merits of both conventional SMA actuators. Although the differential SMA actuator has faster response than bias SMA actuator, its working range is limited for the increasing stiffness in the inverse phase transformation. In the current study, a soft spring is employed in the actuator design to reduce the stiffness as well as to enlarge the working range of differential SMA actuator. A saturated PI law has been developed that demonstrated better tracking performance by introducing two switching surfaces into the control structure. Experimental results also reveal that saturated PI controller has quicker response speed and higher accuracy than the PI controller in the position tracking of the same compliant differential SMA actuator. With this saturated PI controller, 0.1 Hz sinusoidal tracking on 0.25 mm diameter SMA wire has been implemented. The tracking frequency can be further improved with smaller diameter SMA wires, since these have lower heating and cooling times. Nevertheless, the smaller diameter SMA wires also provide lower contraction force. Therefore, in practical application, a tradeoff between the response speed and output force should be considered. In this study, we introduced a mechanical solution to improve the performance of conventional SMA actuators with a simple prototype. Such a design has the potential applications in robotic systems, such as for flexible multi-section surgical tools or rehabilitation finger design. According to real application, mechanical modification could be considered to meet the specific requirements: the size of coupling elements can be reduced to couple a smaller spring, multi-section SMA actuated joints can be connected together to increase the degree of freedom of the robot. Due to the limited strain of SMA wire, long wire was usually used to obtain large rotational angle in the traditional SMA actuators. Since our method increased the angle of the SMA actuator, it would be valuable to reduce the length of SMA wire for the desired motion. This study will be investigated in the future. It is also noteworthy mentioning that de-twinned strain of SMA wire is an important factor [26]. It can affect the working range of the SMA based actuator. In the current work, to minimize any possible influence on our results, the same loading and current condition to all three actuators were adopted to provide the same initial conditions. However, considering the importance of this factor, a further study is under consideration. In addition, the hysteresis phenomenon can be found in the open loop experiment for the SMA actuator during heating and cooling process. Since the primary focuses of the present study was the comparison of working range, response rate and control performance, we will pay more attention to the hysteresis testing and compensation of this SMA actuator using advanced control strategy in the future work.

6. Conclusion In this work, we introduced a novel compliant differential SMA actuator incorporating two antagonistic SMA wires and a torsion spring. This design allows a wider range of angular motion compared to conventional SMA actuators. Experimental comparison between the new compliant SMA actuator vis-à-vis conventional SMA actuators has been conducted to evaluate the performance enhancement in terms of tracking accuracy and settling time. Future work will focus on the design of advanced control strategies [27–29] and the applications to robotic systems for this compliant SMA actuator.

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Biographies Zhao Guo received his Ph.D. degree in mechanical Engineering from the Institute of Robotics, Shanghai Jiao Tong University (SJTU), Shanghai, China, in 2012. He is currently a Research Fellow at the Department of Biomedical Engineering, National University of Singapore (NUS). His research interests include compliant actuator design, SMA actuator for bio-inspired robots, exoskeleton, and neuromuscularmodel based robotic control.

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Yongping Pan received the Ph.D. degree in Control Theory and Control Engineering from the South China University of Technology, Guangzhou, China, in 2011. From 2011 to 2013, he was a Research Fellow of the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He is currently a Research Fellow of the Department of Biomedical Engineering, National University of Singapore, Singapore. He serves as the Reviewer for several flagship international journals. His research interests include adaptive control, computational intelligence, rehabilitation robots and embedded systems. Liang-Boon Wee received his PhD in Space Robotics at The University of Michigan, Ann Arbor, in 1993. Liang-Boon started his career as a project engineer in 1987, working on design of computer simulation and the development of flight control systems. Throughout his career in DSO, National Laboratories he has worked on various acquisition and development projects. His area of expertise has also expanded

to include guidance research, systems architecture research as well as computer vision and image processing research. His current appointment is Chief Engineer in the Guided Systems Division. Haoyong Yu received his PhD in Mechanical Engineering from the Massachusetts Institute of Technology (MIT), Cambridge, in 2002. He is an Assistant Professor of Department of Biomedical Engineering at National University of Singapore (NUS). Before joining NUS in September 2010, he worked in Defense Science Organization (DSO), National Laboratories of Singapore as a Principal Member of Technical Staff, where he worked on exoskeleton and humanoid robots as well as intelligent ground and aerial robots. His current research at NUS focuses on robotics for robotics in surgery, neuro rehabilitation, assistive technologies, and bio-inspired robots. Dr. Yu is a member of IEEE. He received the Best poster award in the 2013 IEEE Life Sciences Grand Challenges Conference.