Soft actuators are of growing research interest due to their high flexibility, adaptability and light weight. Compared to their conventional rigid counterparts, soft actuators can easily achieve multi-degree-of-freedom motion and have high loadto-weight ratio. These characteristics give them great potential in applications like soft robotics [1] , drug delivery [2] , artificial muscles [3, 4] and so on. Soft actuators deform and change shapes when exposed to external stimuli such as electricity, heat, magnetic field, light, moisture, chemical, etc. Electrothermal actuators are a kind of soft actuators that driven by electricity. They are usually fabricated in a bimorph structure with two layers that have significantly different coefficient of thermal expansion (CTE). When applying an actuation voltage, a conductive path of the electrothermal actuator provides Joule heat as the stimulus for actuation. At elevated temperature, the large mismatch of thermal expansion between the two layers drives the actuator to bend.
A special characteristic of soft electrothermal actuators is their non-linear relationship between the actuation voltage and achieved bending curvature. Open-loop operation of electrothermal actuators usually has a long transient time and is susceptible to external environment disturbance or variations. Therefore, a closed-loop control for electrothermal actuators is crucial for the implementation of real-world applications with high accuracy requirements. Closed-loop control of soft actuators or other micro-scale devices can be realized using either an external sensor or the so-called self-sensing technology, also known as sensorless control. Adding an external sensor to the actuator not only enhances the fabrication complexity and costs, but will also adversely affect the flexibility and weight of the actuator. Instead, self-sensing technology integrates the actuation and sensing capabilities together, which eliminates the need for external sensors and simplifies the overall control system. Two most commonly used self-sensing methods are resistive self-sensing and capacitive self-sensing. For instance, in [5] , Merced et al. used a self-sensing approach to accurately control the displacement
Soft actuators are of growing research interest due to their high flexibility, adaptability and light weight. Compared to their conventional rigid counterparts, soft actuators can easily achieve multi-degree-of-freedom motion and have high loadto-weight ratio. These characteristics give them great potential in applications like soft robotics [1] , drug delivery [2] , artificial muscles [3, 4] and so on. Soft actuators deform and change shapes when exposed to external stimuli such as electricity, heat, magnetic field, light, moisture, chemical, etc. Electrothermal actuators are a kind of soft actuators that driven by electricity. They are usually fabricated in a bimorph structure with two layers that have significantly different coefficient of thermal expansion (CTE). When applying an actuation voltage, a conductive path of the electrothermal actuator provides Joule heat as the stimulus for actuation. At elevated temperature, the large mismatch of thermal expansion between the two layers drives the actuator to bend.
A special characteristic of soft electrothermal actuators is their non-linear relationship between the actuation voltage and achieved bending curvature. Open-loop operation of electrothermal actuators usually has a long transient time and is susceptible to external environment disturbance or variations. Therefore, a closed-loop control for electrothermal actuators is crucial for the implementation of real-world applications with high accuracy requirements. Closed-loop control of soft actuators or other micro-scale devices can be realized using either an external sensor or the so-called self-sensing technology, also known as sensorless control. Adding an external sensor to the actuator not only enhances the fabrication complexity and costs, but will also adversely affect the flexibility and weight of the actuator. Instead, self-sensing technology integrates the actuation and sensing capabilities together, which eliminates the need for external sensors and simplifies the overall control system. Two most commonly used self-sensing methods are resistive self-sensing and capacitive self-sensing. For instance, in [5] , Merced et al. used a self-sensing approach to accurately control the displacement 48th SME North American Manufacturing Research Conference, NAMRC 48 (Cancelled due to of a microelectromechanical actuator by monitoring the resistance of the integrated heater. Tang et al. [6] applied resistive self-sensing technology for the closed-loop position control of a twisted and coiled actuator. Sonar et al. [7] developed a closed-loop haptic feedback control of a soft pneumatic actuator. The actuator's inflation state was detected by the resistance change of the embedded thin biphasic metal films. In [8] , Ouyang et al. utilized the piezoresistive nature of a microactuator to realize its positioning control. Capacitive self-sensing is another important method. For instance, Gisby et al. [9] , Zhang et al. [10] and Rosset et al. [11] applied capacitive self-sensing technology for the closed-loop control of expanding area, actuation force and strain of dielectric elastomer actuators (DEAs), respectively. Dong et al. [12] adopted a capacitive self-sensing approach to realize the simultaneous actuation and displacement sensing of electrostatic drives.
In this work, an electrothermal actuator with directly printed microfilament heater was designed and fabricated. Two kinds of polymers, polyimide (PI) and polydimethylsiloxane (PDMS) were used as the structural materials due to their large mismatch of CTE. A microfilament heater was printed using a low-cost solder alloy Bi58/Sn42, compared to expensive carbon nanotubes [13] [14] [15] , graphene [16, 17] , and nanowires [18, 19] used in other similar researches. The microfilament heater was directly printed by electrohydrodynamic (EHD) printing technology [20, 21] , and embedded in-between the PI and PDMS layers. Owing to the direct patterning capability of EHD printing, the heater filament can be freely designed to be efficient for actuation, which is usually only possible in masked processes like lithography and etching. A maximum bending curvature of 1.0 1 cm  was observed under a voltage as low as 3V. Moreover, resistive self-sensing was adopted for the closedloop control of the actuator, which is a rarely studied area. The actuation effect is achieved from the temperature change of the actuator, which can be detected by the resistance change of the heater filament as they are linearly related. A PID (proportional-integral-derivative) controller was designed and implemented for the closed-loop control operation. Compared with open-loop operation, the PID controlled actuator demonstrated more rapid and precise response. The fabricated actuator was successfully demonstrated in applications as a soft lifter and soft gripper.
The customized design of the microfilament heater in the electrothermal actuator was fabricated by the EHD printing technology. The EHD printing system used in this study is shown in Fig. 1 (a), which is consisted of four sub-systems: a three-axis precision motion stage, a thermal control system, a high voltage supply, and a pneumatic dispensing system. The stage can move in XYZ directions with a 100 nm motion accuracy. The heating system can control the ink temperature up to 480 C  with a resolution of 0.5 C  . A high voltage is applied between the printing nozzle and the ground electrode. It provides the electrostatic force to eject molten metal out from the nozzle. The pneumatic dispersing system is regulated by a precision pressure valve to provide needed pressure to ensure a stable flow of the molten metal to the nozzle tip. In this study, the microfilament heater was printed using a nozzle with an inner diameter of 160 m  , the heating temperature was maintained at 260 C  , and printing voltage was 1700 V . Fig.  1 (b) shows the side view of a printing nozzle. The ejected flow clearly has a much smaller diameter than the nozzle. That is because the electrostatic force is able to overcome the surface tension of the printing ink, thus forming a fine jet. Fig. 1(c) shows an EHD printed high-resolution customized pattern.
To improve the overall performance of the soft electrothermal actuator, the embedded heater needs to be carefully designed. Two factors should be considered: area coverage and stiffness. The heater pattern needs to cover the whole actuator efficiently, while on the other hand, does not add too much stiffness to the actuator. The stiffness of the embedded heater could adversely affect the actuator's deformation and actuation capabilities, especially considering the fact that the Young's modulus of Bi58/Sn42 is much higher than that of PI and PDMS. The direct patterning capability of EHD printing technology allows free design of the heater pattern. We designed and compared two different heater patterns: a reciprocating zigzag pattern and a half circle curved pattern. Figs. 2(a) and (b) show the two printed heater patterns. Apparently, to cover the same heating area, the zigzag pattern has four parallel filaments on the bending plane, while the half circle curved pattern has only two filaments, indicating a lower bending stiffness. As a result, a large actuation can be expected from the half circle curved pattern. This has also been proven in our test in which both actuators were heated up to 60 C  , given all other conditions were the same, the half circle patterned heater reached a curvature about 0.4 1 cm  , while the zigzag pattern only reached 0.1 1 cm  . Moreover, compared with zigzag lines patterned heater, the heater with the halfcircle pattern can more easily resist strains induced during bending actuation [20, 21] . Therefore, in this study, the half circle patterned heater was selected for the fabrication of the soft electrothermal actuator.
The other factor that significantly influences the actuator's bending performance is the thickness of each layer in the bimorph structure. Since the PI layer thickness is fixed, the bending curvature of the actuator varies with the PDMS layer thickness. After converting the actuator into a bimorph model [22] , this relationship can be theoretically explored using the equation provided by Timoshenko [23] . By varying the thickness of the coated PDMS layer, and for a given temperature change, we can estimate the bending curvature theoretically against the thickness of PDMS (as shown in Fig.  2 (c) when the temperature change is set as from 20 C  to 60 C  ). The maximum curvature was achieved when the thickness of PDMS was approximately 0.25 mm . Too thick or too thin PDMS layer will reduce the actuation capability of the soft actuator. When the PDMS layer is too thin, it cannot generate enough stretching force to bend the PI layer and the heater. On the other hand, if the PDMS layer is too thick, the overall actuation will mostly be thermal expansion rather than bending. To verify the theoretical model, we fabricated a set of actuators with different PDMS layer thickness. These actuators were supplied with DC current and heated to 60 C  . The resulting curvatures were recorded and plotted in Fig. 2(c) . As it can be observed, the experimental results were generally in accordance with the theoretical values.
The fabrication process of the PI/Bi58/PDMS electrothermal actuator is illustrated in Fig. 3(a) . A layer of PDMS was first coated onto a glass slide, to make a suitable substrate for EHD printing. The PDMS was prepared using Sylgard 184 (from The Dow Chemical Company), and the silicone elastomer base and curing agent were mixed with a suggested ratio of 10:1. The mixed solution was then agitated and degassed before coating. After the PDMS was fully cured, the microfilament heater was printed on the substrate by EHD printing. Then a PI tape (from DuPont) was pressed against the heater to transfer it from the printing substrate to the PI tape. Thereafter, copper wires were connected to the printed heater for electric connection. Finally, a layer of PDMS, with controlled thickness, was coated onto the microfilament heater and the PI layer. When the top PDMS layer was fully cured, the actuator was produced by cutting the bimorph film into a strip. Fig. 3(b) shows the sectional view of the fabricated electrothermal actuator. The bottom layer is PI tape, which has a 30 m  thickness pure PI and 38 m  thickness silicone adhesive. The heater filament has a roughly semicircle cross profile with a diameter about 100 m  . The top layer is PDMS, whose thickness was optimized to about 0.25 mm for the largest bending curvature.
The electrothermal actuator was fabricated with the optimized heater pattern and layer thickness. Fig. 4 shows its actuation process, from initial state, fully activated state, to recovered state. The actuator was first characterized on its heating performance with regard to the applied electrical current. The temperature of the actuator was measured by an infrared camera. From the obtained thermal images, the average temperature of the actuator can be estimated. The current was applied at 0 s and turned off at 60 s. Fig. 5(a) shows the change of the average temperature as a function of time at different applied actuation currents. In general, the temperature of the actuator increased rapidly when the current was first applied. Then the temperature increased slowly and eventually approached and maintained a stable temperature within 60 s. Upon turning off the current, the temperature dropped down sharply at the beginning, and then gradually returned to the ambient temperature within 1 minute. The bending deformation of the soft actuators at five different actuation currents was also characterized, as shown in Fig. 5(b) . The supply current was turned on at 0 s and turned off at 60 s. The bending curvature was measured each 5 seconds, by comparing the actuator's profile with a specially designed curvature gauge. Since the deformation of the actuator is directly controlled by the temperature, the trend of the actuator bending is very similar to that of the temperature. In Fig. 5(b) , it can be observed that at 0.3 A (with about 3 V actuation voltage), a maximum bending curvature of 1.0 1 cm  was achieved.
The self-sensing mechanism of the electrothermal actuator is supported by its embedded alloy heater. When supplied with voltage, the heater filament heats up the actuator. We assume that the temperature change of the heater, denoted as h T  , and the temperature change of the overall actuator, denoted as a T  , are linearly correlated as:
According to previous report [24] and our experimental observation, the electrical resistance of Bi58/Sn42 generally varies linearly with temperature change:
where R  is the resistance change, 0 R is the original resistance in room temperature,  is temperature coefficient of resistance. From our previous study [22] , the bending curvature of the actuator is proportional to its temperature change:
where f is a function determined by the CTE ( 1
Young's modulus ( 1 E , 2 E ) and thickness ( 1 t , 2 t ) of the two films.
Combining formulas (1-3) , it can be found that the resistance change of the heater reflects proportionally the bending curvature of the actuator by R
The resistance change can be used as a sensing signal to sense the actuator's deflection. This is verified by the test results in Fig. 6(a) which shows the measured curvature against resistance change in percentage. They generally follow a proportional trend, and based on the fitted line,
. By using this resistive based sensing signal, we are capable to control the bending curvature of the actuator.
To use the resistive sensing signal for closed-loop control, a schematic of the control circuit is shown in Fig. 7 . A NI I/O device (USB-6259) was employed to collect the signal from the sensor and send the control command to the actuator. The voltage signal from the I/O device was amplified through a voltage amplifier to power the main circuit. In the circuit, a shunt resistor was connected in series with the electrothermal actuator. The shunt resistor has a 1  resistance, 1% tolerance and 0.25 W rated power. Voltage signals across the shunt resistor and electrothermal actuator were acquired via two analog input terminals on the I/O device. 1 U denotes the realtime voltage across the shunt resistor, and 2 U denotes the realtime voltage across the electrothermal actuator. Since the shunt resistor worked under a power lower than 0.1 W, it was treated as a constant resistor. 1 U numerically equals the current through the circuit as the shunt resistor has a unit resistance. Based on Ohm's law, the real-time resistance of the electrothermal actuator was obtained by the actuation voltage based on a PID control algorithm using the resistive sensing signal. Fig. 8 illustrates the feedback control loop of the electrothermal actuator. In the figure, ( ) R s  represents the reference signal, ( ) Y s represents the measured system output, ( ) E s is the difference between the reference signal and measured system output, ( ) C s represents the controller transfer function, ( ) U s represents the control signal, and ( ) G s is the transfer function of the soft actuator.
The performance of the designed PID controller was evaluated by step response and wave tracking. Fig. 9 (a) shows the step response for the closed-loop controlled actuator. As a comparison, Fig. 9 (b) shows the step response of open-loop operation. As it can be seen, the closed-loop controller significantly reduced the settling time, from the open-loop operation's over 40 s to less than 10 s . Meanwhile, the peak overshot was regulated to be within 5%. Fig. 9 (c) shows the sinusoidal wave tracking performance of the PID controlled actuator. The output signal closely follows a sinusoidal reference, which is difficult to be realized for open-loop configurations.
Soft actuators have tremendous potential applications in biomimetic and soft robotics areas. In this work, we have demonstrated that the fabricated electrothermal actuator can be used as a soft lifter and soft gripper. When used as lifters, soft actuators can lift samples that are much heavier than themselves. To test its lifting capability, the fabricated soft actuator was placed horizontally with one end fixed. As shown in Fig. 10(a) , the actuator bent downwards after loading the sample on the free end. Fig. 10(b) shows the actuated state, in which the sample was lifted more than 15 mm. The sample used here is a paperclip weighted about 0.4 g , which is four times the actuator's own weight (0.1 g ). The developed soft actuators can also be applied as soft grippers. They can be used to pick-and-place delicate objects due to their gentle interaction with the objects. Fig. 10(c) shows a soft gripper assembled by arranging four electrothermal actuators together and connecting them in a series circuit. It can easily pick up a foam block from a Petri dish.
In this paper, we designed and fabricated a soft electrothermal actuator with directly printed microfilament heater. The metal alloy based microfilament heater was printed by the high-resolution EHD printing process, and embedded inbetween the PI and PDMS layers. The heating and bending performance of the actuator was characterized. The soft actuator was able to achieve a maximum bending curvature of 1.0 1 cm  under a voltage as low as 3V. To precisely control the operation of the actuator, a self-sensing closed-loop control system was designed. The temperature-incurred actuator bending can be detected by the real-time resistance change of the embedded heater. A PID controller was designed using the resistive sensing signal on LabVIEW. Compared with openloop operation, the PID controlled actuator significantly reduced the settling time in step response, and demonstrated excellent tracking performance for complex command signals. The soft actuator was successfully demonstrated as a soft lifter and a gripper, which was able to lift sample that was four times its own weight and to pick-and-place gentle objects. 