**2.3. Modeling and control**

4 Will-be-set-by-IN-TECH

0.001 (0.025) 36.2 (1425) 0.02 (8.9) 45 0.18 0.15 0.0015 (0.038) 22.6 (890) 0.04 (20) 55 0.24 0.2 0.002 (0.050) 12.7 (500) 0.08 (36) 85 0.4 0.3 0.003 (0.076) 5.9 (232) 0.18 (80) 150 0.8 0.7 0.004 (0.10) 3.2 (126) 0.31 (143) 200 1.1 0.9 0.005 (0.13) 1.9 (75) 0.49 (223) 320 1.6 1.4 0.006 (0.15) 1.4 (55) 0.71 (321) 410 2 1.7 0.008 (0.20) 0.74 (29) 1.26 (570) 660 3.2 2.7 0.010 (0.25) 0.47 (18.5) 1.96 (891) 1050 5.4 4.5 0.012 (0.31) 0.31 (12.2) 2.83 (1280) 1500 8.1 6.8 0.015 (0.38) 0.21 (8.3) 4.42 (2250) 2250 10.5 8.8 0.020 (0.51) 0.11 (4.3) 7.85 (3560) 4000 16.8 14

One of the main limitations in SMA actuation speed is due to high latency that the the cooling time of the wire implies. Despite increasing the input heating power can reduce the heating time, large cooling times limit the operation frequency of the actuator. On average, NiTi wires with a diameter of 127*μm* typically requires an electrical current input about 320*mA* to contract in about 1*s* (nominal heating time) and relax in approximately 1.4*s* (nominal cooling time). In this case both contraction and recovery times would set a nominal actuation frequency about

Research to overcome this limitation has been oriented towards developing cooling systems for SMAs, aimed at decreasing the nominal cooling time involved during the recovery process. In this direction, temperature control methods have been proposed in [41]. Cooling systems based on Peltier cells [18] or active cooling [67], have been commonly used. However, nowadays bio-inspired robotic systems tend to be small and light, therefore other methodologies for enhancing SMA actuation speed must be addressed. For several years different strategies have been proposed to implement rapid control in the SMA wires [12],[19],[70],[72],[11],[71],[75]. A system consisting of rapid heating of the SMA was proposed by [12] aimed at increasing the overall actuation frequency by means of overloading the operation of SMAs. The term overloading refers to increasing the amount of input heating power to be delivered to the SMA wires. In [74] experiments carried out using a two degree-of-freedom Pantograph robot actuated by an antagonistic pair of SMA wires acting

Approximate Current for 1 Second Contraction (*mA*)

Cooling Time 158◦*F*, 70◦*C* "LT" Wire (seconds)

Cooling Time 194◦*F*, 90◦*C* "HT" Wire (seconds)

Pull Force *pounds* (*grams*)

Diameter Size

*inches*(*mm*) Resistance

**Table 1.** Characteristics of NiTinol®SMA wires [6].

**2.2. Improving the performance of SMA actuators**

0.416*Hz*, quite slow for many applications requirements.

The physical behavior of SMAs is more complex than many common materials: the stress-strain relationship is nonlinear, hysteresis is presented, large reversible strains are exhibited, and it is temperature dependent. This thermo-mechanical relationship can be described by formulating phenomenological models. Tanaka in [69] was one of the pioneers to study a stress-induced martensite phase transformation, proposing an unified one-dimensional phenomenological model that makes use of three state variables to describe this process: temperature, strain, and martensite fraction. His main contribution was to demonstrate that the rate of stress is a function of strain, temperature and martensite fraction rates. Later, Elahinia [7], [8] proposed an enhanced phenomenological model compared to other works [40], [31], [69] and also addressed the nonlinear control problem. This model was able to better describe the behavior of SMAs in cases where the temperature and stress states changed simultaneously. Their model was verified against experimental data regarding a SMA-actuated robotic arm [10].

Phenomenological models may provide some insights of SMA thermo-mechanical behavior that facilitate the development of control procedures. To control purposes, parameters' tuning is highly dependent of a modeling stage, but definitively phenomenological models are not the best choice for control design, especially if the goal is related to improving actuation speed. In this direction system identification is a promising alternative. As noted by [40], [19], [11], [71], [20], identified linear models for SMA can be developed. It has been demonstrated that the AC response of NiTi SMA wires behave as a first order low-pass filter.

#### 6 Will-be-set-by-IN-TECH 58 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

Section 4 of this paper details two different approaches for modeling SMAs; one based on identifying how electrical resistance change as a function of the input current [59], and the other based on identifying how the output torque produced by an antagonistic pair of SMA actuators change as as a function of the applied power [5]. Furthermore, [5] details how to take advantage of phenomenological models for simulating overheating problems when a SMA wires are overloaded. Attempting to perform this analysis on the real SMA actuators might cause several damage to the structure. Phenomenological models are really useful for determining the upper limits of applied input heating currents.

The control methods presented in [59] and [5] have been conceived for controlling a pair of antagonist SMA actuators. The antagonistic configuration is useful for having SMA actuators where each direction of motion can be controlled. In [59], the antagonistic pair of actuators must bend the structure of the fish robot, whereas in [5] the antagonistic SMA actuators are connected to a joint for providing the rotational motion. Other approaches in [11], [30], [41], [28] have demonstrated the advantages of using an antagonistic arrangement in terms of controllability. When the active actuator is being heated while the passive (antagonistic) is cooling, hysteresis effects are reduced due to the external stress that the active actuator applies on the inactive one above the austenite finish temperature.
