Modeling of TCP Muscles for Understanding ...

Proceedings of the ASME 2017 International Mechanical Engineering Congress and Exposition IMECE2017 November 3-9, 2017, Tampa, Florida, USA

IMECE2017-71982

Modeling of TCP Muscles for Understanding Actuation Behavior Farzad Karami Humanoid, Biorobotics and Smart Systems (HBS) Laboratory, Department of Mechanical Engineering The University of Texas at Dallas Richardson, Texas - 75080, USA [email protected]

Yonas Tadesse Humanoid, Biorobotics and Smart Systems (HBS) Laboratory, Department of Mechanical Engineering The University of Texas at Dallas Richardson, Texas - 75080, USA [email protected]

ABSTRACT I. INTRODUCTION

Twisted and Coiled Polymers (TCP) muscles are actuators that generate force and linear displacement in response to thermal stimuli. Their length changes significantly by heating due to a high negative coefficient of thermal expansion (CTE). A mathematical model for predicting the behavior of TCP muscles is essential for exploiting maximum advantage from these actuators and also controlling them. In this work, a simple, practical, and accurate model for predicting the displacement of TCP muscles, as a function of input electrical actuation and load, is derived. The problem is broken down into two, i.e. modeling of the thermal and thermo-elastic part. For the first part, a differential equation with changing electrical resistance term is derived. In the next step, by using a temperaturedependent modulus of elasticity and CTE as well as taking the geometry of muscles into account, an expression for displacement as a function of temperature and load is proposed. Experimental actuation data of a TCP muscle is used for verifying the model and investigating its accuracy. The thermal part shows a good agreement between the simulation and experimental result. The displacement part also has a good accuracy for medium and high actuation currents but there is a mismatch in very high current magnitudes. The cause of the discrepancy is explained and recommendations are made for the best performance of TCP muscles.

Use of thermal actuation for creating desired motion has a long history. It is the most simple and straightforward method for making displacement in structures. But, the commonly low coefficient of thermal expansion limits this method to applications with small traveling distance. This value ranges from near zero for some alloys, such as Sitall, to large values for specific types of polymers. Even for the material with a large CTE, the expected linear displacement cannot exceed few millimeters in an allowable temperature range of the chosen material. The convenience of using thermal actuation has convinced many researchers to work on finding the materials with a higher response to a change in temperature, with higher CTE, or using innovative modifications in the geometry to magnify the limited displacement into a large motion. Several material types with high CTE, mostly polymers, and composites, have been developed and tested. The alternative method, using an amplifying geometry, is proven to be less intuitive but less expensive and easier to achieve. Using the alternative geometries, instead of single string straight actuator, would add to the possible stroke. The most well-known of such configurations is helical springs which show significant stroke with an equal load, in compare with a single rod under axial load. This method has also been shown effective with Shape Memory Alloys (SMA). Using them in a linear helical form can significantly increase the linear displacement comparing to an actuator of a stretched SMA wire with the same length [1].

Keyword: Phenomenological Modeling, Smart Materials, Actuators, TCP muscle, Thermomechanical Modeling, Artificial Muscles

Using polymers as the precursor fiber for the helical spring, instead of metal wire, and exploiting a large negative CTE of polymers, which is magnified by the helical geometry, is the cornerstone of a newly proposed actuator called TCP muscles

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[2-4]. Due to an increase in the actual length of the fiber and also reducing the effective stiffness of a TCP muscle it can exhibit a considerably larger displacement in response to heating. Usually, muscles are heated up by electrical current, sent through conductive fibers implemented inside the muscle. This actuator can be used in many application where low cost, ease of use, compact size, and low weight are necessary. These qualities, make TCP muscles a very good choice for light weight and economic robots [5].

The dynamic behavior of the TCP muscles is investigated briefly so far [8, 9]. A simple mass-spring linear model is the most common method to express the behavior of TCP muscles. There are some other works focusing on the mechanical behavior of these actuators. In [10], a top-down approach is exploited to express a relationship between microscopic properties of the material and macroscopic behavior of TCP muscles. The authors focused on the static and steady-state behavior, ignoring the transient response of the muscles to the actuation. In the other work, a statistical thermodynamics approach is used to explain the change in the structure of polymers with respect to temperature [11]. They propose that mechanical characteristics are a function of the volume ratio of different phases inside the polymer structure. In their work, they adopt the normal distribution to predict the phase conversion with taking the temperature as a random variable and for mechanical part, a linear model of coiled springs is used. In [12], a kinematic relationship between the elastic constants of the material and a twisted member is proposed and the effect of geometry, such as the helix angle of muscles, on the elastic behavior of the muscle is investigated.

To use these artificial muscles in applications requiring length control, having a mathematical model to link the input signal to the output is necessary. This model should describe the thermomechanical behavior of a TCP muscle to give a prediction of the length based on geometry, temperature, and load. The modeling problem can be broken down into two disciplines: thermal and mechanical. The thermal aspect of the modeling concerns with finding a relationship between the actuation and the rise and fall in the temperature of a TCP muscle. There are different physics taking roles in this phenomenon, such as the electro-physical behavior of the conductive part as well as heat transfer between members inside muscles and the surrounding environment. The study of conductive part, which has been less investigated so far, mainly focuses on the change of electrical resistance and its effect on the overall performance. On the other hand, the heat transfer mechanism consists of conductive heat flow through the muscle, convective heat transfer with the surrounding environment, and energy exchange via radiation. Among the possible heat transfer mechanisms, the convection heat transfer is a matter of interest as the others have fewer effects on the actual thermal behavior of the muscle. Due to stretching the conductor member along the length of muscles as well as a negligible radius, the conductive heat transfer contribution would be small. On the other hand, radiation dominates the heat transferring in high temperature, e.g. higher than 1000 C, and it can be ignored in room condition at which muscles are used.

There are some works focusing on heat transfer coefficients, mainly h as the convection heat transfer coefficient, of TCP muscles and similar actuators. Research for finding the coefficients for the cylinders, wires, and rods, which are generally the shape of TCP muscles, is a well-stablished topic [13]. The problem with using their results is that the shape of the TCP muscles is not an exact cylindrical wire and the closely packed helical shape makes a considerable deviation from smooth wires characteristics. The purpose of this work is to develop a model of displacement of TCP muscles with a mathematical phenomenological approach. In this way, a thermodynamic analysis of the TCP muscle, as the control volume, is applied. The effect of temperature on the electric resistance of the conductor is modeled as a variable heat source and a modified differential equation for predicting the temperature rise is obtained. Neglecting this phenomenon results in a significant deviation of the model from the real temperature. As the convective heat transfer parameters show little sensitivity to the different conditions, they are estimated by using a curve fitting on a part of test result and then applied for the rest of simulations. The model output and the test results are in a good agreement and accuracy of the electrical input-temperature model is assured. The authors expressed the displacement of the muscle is as a function of temperature and load to express the constitutive equation like other smart materials. Two factors have been identified that shape the response of the muscle to an actuation: CTE and modulus of elasticity which both depend on temperature. Using experimental results from previous tests, expressions for both as functions of temperature are obtained and used in the model. For verifying the model, experimental data are used for different actuation levels. This data consist

The investigated actuation method is based on Joule heating of the muscle via passing electrical current through a conductive fiber woven with the polymer fibers. The resistance of the conductive, commonly made by metal, causes an energy conversion from electrical to heat from. This heat would raise the temperature of the muscle with respect to boundary conditions. Heating up of metals causes an increase in the electrical resistance and, consequently, a growth in the amount of produced heat energy. Polymers are sensitive to temperature, therefore, heating changes the mechanical properties significantly. Given that TCP muscles are made of polymers, it is necessary to investigate the change in characteristics, during the actuation process, to make any modeling reliable. There is much research for every type of polymers and their characteristics in different temperature levels. There are also some works dedicated to developing a quantified model for temperature dependent characteristics of polymers [2, 6, 7].

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time history of displacement and temperature as well as actuating current.

fibers inside TCP muscles, the change is modeled with a linear relation as follows:  T   0 1   T  T0   (3) Where α is called temperature coefficient of resistivity, ρ0 is resistivity in a reference temperature and T0 is the reference temperature. Assuming a constant geometry, the equation can be rewritten in term of resistive: R T   R0 1   T  T0   (4) Substituting equation (4) in equation (1), one can write:

II. Modeling The geometry of TCP muscles is a long string in longitude direction with negligible dimensions in the two others. Given that heat flux starts from the conductive fibers which are implemented along the length of the muscle, and with an assumption of uniform resistance, one can assume that the heat flux flows mainly from the conductive fibers toward the surface of the muscle, with a negligible flow in the longitudinal direction. On the other hand, the thickness of the muscle, in which the flux travels, is very small compared to its length. These lead to a small Biot number and validity of lumped parameter assumption.

mc p

dT   hA T  T  i 2 R dt

(1)

Where m is mass, cp is the specific heat capacity, h is the coefficient of convection heat transfer, A is area exposed to the ambient air, i is the actuation current and R is the electric resistance of the TCP muscle. The resistance of conductors is a function of material and geometry. It is defined as: R

l A

(5)

It is convenient to assume same resistivity reference and ambient temperature and rearrange equation (5) to the following form: d T  T  mc p   hA  i 2 R0  T  T   i 2 R0 (6) dt The interesting result of the above equation is that if the magnitude of i2R0α is more than hA, then the differential equation will be unstable and the temperature soar. This rise would change the structure and, consequently, the governing equation of heat transfer. The solution of the above equation for an initial temperature equal to ambient temperature is:  hA R0i 2   t R0i 2 1  e mc p  (7) T  T   2  hA  R0i     The elastic coefficients of Nylon6/6, which is commonly used as the precursor for TCP muscles, considerably depends on temperature. In high temperature, as experimental finding indicates, the elastic modulus of Nylon 6/6 drops to a 20% of its value in room temperature. Granted that the displacement of a TCP muscle is related to the elastic modulus of its material, high temperature can change the performance of TCP muscles. Figure 1 shows the elastic modulus versus temperature. The figure is replicated from an experiment date [14].

Taking the length of the TCP muscle as control volume, the boundary conditions would be free convection heat transfer with the surrounding. There is also a heat generation by Joule heating inside the control volume. Using the first law of thermodynamics for the defined control volume, we have: mc p

dT   hA T  T  i 2 R0 1   T  T0   dt

(2)

In the above equation, ρ is resistivity, a fundamental property of a material, l is the length and A is section area of the conductive part. It is well-known that the resistivity of the metals varies with temperature. For metals, such as conductive

Figure 1: Elastic modulus of Nylon 6/6 vs temperature and strain rate obtained from [14].

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  c1T  c2

According to the above data, the strain rate has no significant effect on the elastic modulus in low rates and high temperature. Regarding the performance of TCP muscles, the strain rate of actuation falls in low values and its effect can be neglected. By assuming a strain rate of 0.001 1/s, a curve fitting is used to determining a relationship between elastic modulus and temperature. The best-fitted function is obtained as the following form:

(14) This value can be used for calculation of the contraction in a straight member, but, the change in the length of coiled muscles is different from the change of the precursor fibers defined by the linear thermal expansion equation. To find the change in height of TCP muscles, the geometry of the muscle should be taken into account. The height of a helix have the following relation with diameter and length:

E T   aT b

(8) Using the classic linear model of the compression/extension springs for the overall stiffness of TCP muscles, there is a linear dependency between the elastic coefficients of the material and the stiffness of a TCP muscle. The following equation depicts this relation: k

D 4G 8d 3 N

H  L2   D 2

(15) Where, D and H are diameter and height of the helix, respectively, and L is the length of the precursor. Assuming small displacement due to heating, linear temperaturedisplacement relation, and an equal CTE in all directions, by differentiating the equation (15) and using thermal linear expansion model, the following equation is obtained for the change in the height of a TCP muscle due to thermal expansion:

(9)

Where D is the coil diameter, G is the modulus of shear, d is the precursor diameter and N is the number of coils. Using the relation between modulus of elasticity, modulus of shear and Poisson ratio for isotropic linear materials, the equation (9) can be rewritten in the following form.

k

 D4  E   3 8d N  2 1    

th   H 

D 4  aT b    8d 3 N  2 1    

(10)

(11)



(16)

(17) H  H 0   th   el (17) Which, Δth is thermal displacement and Δel is elastic displacement due to tension. Substituting the thermal displacement from the equation (16) and using the linear spring model along with the stiffness from the equation (11) into the equation (17) yields:

(12) k (T )   T b Where E is the elastic modulus and µ is the Poisson ratio of the material. The γ coefficient, which represents all temperature independent parameters, has been obtained from a loaddisplacement test in room-reference temperature. The elongation of the muscle due to a tension is obtained as follows: F k

 D D L2   D 2

The whole change in the length of the actuator can be considered as having two separate contrasting components. One is the elongation due to tension and the other is contraction due to heating. These two components, with an assumption of unchanged geometry during actuation, can be added independently to give the total displacement of the muscle, as it follows:

With the assumption of unchanged geometry during the actuation, one can present the stiffness as a function of only temperature and convert equation (11) to the following form:

 el 



 L20  D02      T  T0  H0   H0

Substituting the E from equation (8) into the equation (10) we have: k

L L L2   D 2

 L2  D02  H T   H 0   0    c1T  c2 T  T0  H0   H0 D 4  aT b   3  F 8d N  2 1    

(18)

This is an algebraic equation giving the displacement as a function of temperature and load. Along with the differential equation in the equation (6), this equation can give a prediction of displacement as a function of input actuation (electric current) and load, which can also be presented as a disturbance.

(13)

Where F is the load on the muscle. The result of the test and parameter values are presented in the simulation and experiment section. The main characteristic that made some materials suitable for use in a twisted and coiled configuration, as an actuator, is their high negative coefficient of thermal expansion. According to a research [3], there is a temperature dependent relation between CTE and temperature. We adopt a linear function which is presented as:

III. Simulation and experimental results In this section, the result of simulation along with experimental result, for verifying the mathematical model, are presented. The test stand used for the experiments consisted of a TCP muscle actuated by a controlled current. Temperature and displacement of the muscle were measured by a thermostat and an electronic

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ruler, respectively. The muscle was fixed from one end and a weight was hung at the other side. To obtain the parameters of the TCP muscle, we carried out an experiment. The experimental setup is shown in Figure 2. It consists of a laser displacement sensor (Keyence LK-G152) for displacement measurement, a load cell (Futek LSB200) for force measurement, a power supply (BK Precision 1687B) for applying power and a National Instrument DAQ for data collection. The experiments were performed at three different step input (0.5A, 0.6A, and 0.65A) and each test has 4 repeating cycles. For all experiments, the actuation time is 30 s and the cooling time is 45 s, a period of total 75 s. The experimental data is stored for further analysis. A type-E thermal couple is attached to the muscle for measuring the temperature.

Figure 3: Displacement vs load in an isothermal experiment with the TCP muscle Geometric parameters of the muscle, used in the test and simulation, are depicted in the following table. Table 1: Parameters of the TCP muscle used in the experiment Initial Length Initial Length Initial Diameter Specific heat[15]

1100 mm 250 mm 0.75 mm 1700 J/KgK

To quantify and validate the thermal model of TCP muscles, the test result of an actuation including heating and cooling periods is used. To assure the validity of the model, the coefficient of heat transfer is extracted from the test results. Due to high complexity in the geometry of the muscle, finding the exposed area would be a difficult task. Therefore, the value of convection heat transfer times exposed area, instead of only convection coefficient, would be identified. Curve fittings are used for finding this value. Curve fitting result is shown in Figure 4

Figure 2: Experimental setup for TCP muscle characterization The CTE is adopted from [3], in the form of a linear function of temperature and given by:

 T   3.50 106 T  4.20 104 (1/ K )

L0 H0 D0 cp

(19)

Where T is in Kelvin. A curve fitting to find elastic modulus versus temperature, based on data of Figure 1, is done. It leads to the following equation:

The hA value extracted from Figure 4 curve fittings, using the equation of heat transfer for cooling, is 0.1 with an error of 5%. This value is used for simulation of heating part. Figure 5 shows the simulation results and the test result for both heating and cooling cycle. As it is shown, the thermal model has a good accuracy and could follow the experimental data well. It is also observed that as the temperature rises, a slight diverging between the test result and simulation output shows up which is investigated by the author

E T   70.14T 1.079 GPa

(20) It is required to have the stiffness of the actuator in room temperature as a reference for calculating it in other temperatures. Generally, there are two methods for obtaining the stiffness. The first is calculating with geometric parameters together with appropriate assumptions on mechanics of the actuator. This method required a precise knowledge of the parameters, which are not easy to measure in general. The second is to conduct an isothermal load-displacement test in room temperature, extracting the stiffness coefficient from the test result, and using it as a reference value for higher temperatures. The following figure shows this test result. Using Figure 3, a stiffness of 225.3 N/m is obtained for the TCP muscle.

To investigate this different, electric resistance of the TCP muscle, as the most effective factor in heating phase is plotted as a function of temperature in Figure 6.

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Figure 4: Fitting result of experimental data of cooling phase .

Figure 5: Simulation result and experimental data of temperature in TCP muscles during an actuation and cooling Process for three actuation current levels.

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According to the results presented in Figure 6, the linear resistance-temperature relation is held to 100C. But, it doesn’t follow the linear pattern after that point by reaching a maximum point and then fall, which is clear in the third experiment for 0.65A actuation. This matter shows its effect in the temperature rise of the TCP muscle for 0.6A and 0.65A actuation. They fall short of simulation, which is done by linear resistancetemperature assumption. This phenomenon can be explained by a significant change in the material properties or structure of the TCP muscle due to high temperature and is an interesting topic for a future study. The final analysis is dedicated to studying the displacement of the TCP muscle. The aim is to find the validity and accuracy of the model proposed for displacement of the muscle. For the first step, the performance of the muscle is studied in a constant load of 50 grams, applied as a dead weight.

Figure 6: Electric resistance vs temperature for three actuation currents of the experiment

Figure 7: Simulation result and experimental data of displacement of TCP muscles during four cycles of actuation and cooling for three actuation current levels. phenomenon shows up in the cooling phase, it has a little practical importance for most of the applications. The other deviation between experiment and simulation is at the maximum displacement with the 0.65A actuation. The displacement is the test is stopped and then flatted after a certain time but the simulation result shows a rise as far as the actuation current is applied. This behavior is due to reaching the coils to contact point preventing further contraction.

As it is shown in the first image of Figure 7, the accuracy of the model for actuating current of 0.5A is good and the simulation track the experiment result very well. For the second experiment with a 0.6A current, there is a slight difference between simulation and experimental result. As the simulation result for temperature agrees with the corresponding experimental result, this mismatch would be related to the mechanical behavior of TCP muscles. There is a difference in mechanical behavior of TCP muscle in loading and unloading as well as heating and cooling [16]. It is called hysteresis and some models are proposed to model it. Given that this

IV. Discussion and Conclusion In this paper, a practical electro-mechanical model is proposed for predicting the behavior of a TCP muscle actuated by Joule

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heating. At the first step, a differential equation of heat transfer is developed as a model of actuation-temperature relation. The effect of a change in electrical resistance is also incorporated into the model and the simulations show a considerable contribution of this factor in the overall behavior. In the second step, the effect of temperature on the structure of TCP muscles is modeled and quantified by modeling its effect on the length, through thermal expansion, and on the stiffness by changing the modulus of elasticity. The outcome of this analysis is a nonlinear algebraic equation yielding the magnitude of displacement as a function of temperature and load. For verifying the obtained model, a set of simulation is done and the results are compared with the experimental data. The model shows a good accuracy in predicting temperature based on the current actuation. The performance of the displacement model is also investigated. The model exhibits a good accuracy in medium current actuation levels, with a peak temperature under 1000C, for both heating and cooling phases. As the actuation current, and consequently the temperature, increases, the simulation cooling curves diverges from the experimental data. It is explained as a result of a drop in heat generation and also slight hysteresis inside the material. The effect of reaching the coils to each other is also observed in high current actuation as a stop in contraction of the muscle. The author’s recommendation is to avoid the high current action because reaching to the proximity of contact point results in resistance drop and also higher hysteresis. The high temperature also brings about an irreversible change in the structure of muscles that degrades their performance or damage them totally. According to the test and simulation results, the recommended maximum working temperature would be 110⁰C. The model can be improved by taking the effect of viscoelasticity and hysteresis into account, which are dominant physical phenomena in high-performance applications.

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