Self-sensing for DEA

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The capacitance of a DEA, the ESR of its electrodes, and leakage current through the EPR of the dielectric membrane have been identified as key parameters that will provide valuable information regarding the mechanical state and health of a DEA. Acquiring this information is a fundamental step towards enabling greater control of the electromechanical behaviour of a DEA. In the prior art, no example of a system that can reliably estimate all three of these parameters in real-time whilst applying a driving voltage has been presented.
In this chapter the development of a novel self-sensing system for DEA specifically targeted for portable devices will be discussed. In a departure from the prior art, Pulse Width Modulation (PWM) was used to simultaneously generate a large scale DC offset to control the level of actuation and a small scale electrical oscillation to facilitate sensing the electrical parameters. This chapter specifically addresses the first objective of this thesis: an algorithm for estimating the key electrical parameters of DEA during both static and transient operating conditions that is suitable for a portable device is derived.

Pulse Width Modulation applied to DEA

Sensing the capacitance of a DEA requires the application of a periodic electrical disturbance. In the prior art this was achieved by superimposing a high frequency, low amplitude AC voltage signal onto the DEA actuation signal. This necessarily requires a tone generator, and as discussed in Chapter 2 is not well suited to the capabilities of low-power, high-gain DC-DC converters. As demonstrated in the examples presented in Chapter 2 it also couples the dynamics of the DEA to the dynamic behaviour of the power supply, and prevents the power supply from being used to control multiple DEA independently.
In a departure from the prior art, PWM has been chosen as the method for controlling the voltage across the DEA. A PWM signal is a square wave with a fixed frequency and an adjustable duty cycle, where the duty cycle is the ratio of the time the signal is high to the period of the signal (see Figure 3-1). By making the period, T, of the PWM signal sufficiently small relative to the electrical and mechanical time constants of the DEA, controlling the duty cycle of the signal (t/T) controls the average voltage across the DEA, which governs the degree of actuation. At the same time, the rapid switching of the PWM signal introduces small scale oscillations to this voltage, and it will be shown in this chapter that this is can be used to sense the electrical parameters of the DEA.
In normal operation the voltage across the DEA will always be less than or equal to the voltage of the power source^*. The power supply therefore is acting purely as a current source and is not required to sink current. This is particularly well suited to power supplies with a rectified output. Such a configuration however necessitates an alternative discharge path for the DEA. This can be achieved by passive or active means e.g. a resistor connected in parallel with the DEA or a second switch dedicated to discharging the DEA.

A circuit for acquiring raw feedback for DEA self-sensing

Sensing the electrical parameters of a DEA requires measuring the voltage across the DEA and the series current through the DEA. It is informative to consider the model of a DEA as part of a functionally complete circuit in order to better understand its behaviour (Figure 3-2, with the DEA shown enclosed in the blue rectangle). The power supply is based upon a compact, low power DC-DC converter connected to the DEA via a high speed, high voltage opto-coupler (OC100HG, Voltage Multipliers, Inc.)^†. The output current of the opto-coupler, i.e., the current that flows from the DC-DC converter to the DEA, is proportional to the input current through the low voltage LEDs of the opto-coupler. By using a PWM signal to turn on and off the current through the input LEDs of the opto-coupler, the DC-DC converter and opto-coupler combined act as a PWM current source (I_source).

Fundamental equations for self-sensing

Estimating the electrical parameters of a DEA requires acquiring and interpreting measureable electrical signals from a circuit containing a DEA. With reference to Figure 3-2, these measureable signals are the series current through the DEA (i_series) and the voltage difference between the terminals of the DEA (V_DEA). It is important however to consider what these signals represent.
Using Kirchhoff‟s Current Law, the series current is the sum of the current through C_DEA and the leakage current through R_EPR. Current is equivalent to the rate of change of charge with respect to time, and the charge on a capacitor is the product of its capacitance (C_DEA) and voltage (V_C). The current through C_DEA is therefore the first derivative with respect to time of C_DEAV_C, taking care to note both C_DEA and V_C potentially have non-zero derivatives. Thus the series current through the DEA is the sum of three components: the current that is charging the DEA (C_DEAdV_C/dt), the current induced due to the rate of change of the capacitance (V_CdC_DEA/dt) and leakage current through R_EPR (i_EPR) (Equation 3.3).
Using Kirchhoff‟s Voltage Law, the voltage across Circuit A (red box, Figure 3-2), is equivalent to V_C. The voltage difference between the terminals of the DEA (V_DEA) is therefore equal to the sum of the voltage across the electrodes and V_C. To go one step further, the voltage across the electrodes can be expressed as the product of the series current and the resistance of the electrodes (R_ESR), thus V_DEA can be expressed in the form of Equation 3.4.
Equations 3.3 and 3.4 form the foundation upon which the self-sensing algorithm is based. It is clear however that there are 2 equations but 6 unknowns (C_DEA, dC_DEA/dt, V_C, dV_C/dt, i_EPR, and R_ESR). Additional steps are required to reduce this system to one that is solvable.

Evolution of a DEA self-sensing algorithm

The details of the final design of the self-sensing algorithm for DEA that was developed for this thesis are presented in Section 3.5, however it is informative to briefly summarise previous versions of the algorithm to provide context for the final design.

Self-sensing for DEA – Iteration #1

The first iteration of the DEA self-sensing algorithm focused on estimating the capacitance of the DEA [105]. With reference to Figure 3-2, the DEA was simplified to be a variable capacitor, i.e., R_ESR=0, R_EPR=∞, and V_C=V_DEA. By selecting the frequency of the PWM signal to be much greater than that which can be reproduced mechanically by the DEA to be sensed, it could be assumed that the mean capacitance of the DEA was not affected by the oscillatory component of the actuation waveform.
If the capacitance is assumed to be constant for a single cycle of the PWM signal, i.e., a zero-order approximation, and leakage current is assumed to be negligible, the second and third terms on the right hand side of Equation 3.3 disappear. Thus the capacitance of the DEA could be determined from the slope of the voltage across the DEA during the “off” portion of the PWM input signal when the DEA was discharging (Equation 3.5). Where the period of the PWM signal was much shorter than the RC time constant of the DEA circuit^*, the DEA voltage waveform was approximately linear for this period, and could be estimated by applying a linear least squares fit to the acquired voltage data.
The first iteration of the self-sensing algorithm was limited in that it did not account for the resistance of the electrodes, leakage current, or the current induced due to the rate of change of the capacitance of the DEA. In particular, a sudden change in the capacitance of the DEA initially resulted in the estimated capacitance changing in the opposite direction to the actual capacitance (see Figure 3-3). Furthermore, leakage current and the voltage drop across the resistance of the electrodes affected i_series and dV_DEA/dt such that an estimation of C_DEA would be offset from its true value. Nevertheless, it was simple and was well suited to applications where the relative steady state capacitance was important and leakage current was negligible. O‟Brien successfully implemented Iteration #1 of the self-sensing algorithm in several biomimetic arrays, including a linear array of 4 mechano-sensitive bending DEA elements that were made to actuate when they detected a small perturbation from their equilibrium position [106], an oscillating ball-on-rails system that used diaphragm DEA to autonomously propel a ball around a circular track [107], and to demonstrate autonomous travelling waves in planar and inflated DEA [108].

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Self-sensing for DEA – Iteration #2

The second iteration of the self-sensing system sought to address some of the shortfalls of the first. Clearly current due to the rate of change of the capacitance would need to be accounted for to improve the dynamic response of the self-sensing. Furthermore, it was necessary to compensate for the effects of leakage current at high electric fields. The first step therefore was to revisit the simplifications that were made of Equation 3.3, and determine a method for incorporating the effects of the second and third terms from the right hand side into the self-sensing process.
To enable the second and third terms of Equation 3.3 to be grouped together, leakage current was expressed in terms of the voltage across the capacitance of the DEA and R_EPR. Thus Equation 3.3 became Equation 3.6.
Equation 3.6 could be further simplified by evaluating the parameters at a point in time where dV_C/dt = 0, thus eliminating the first term on the right hand side. At this point in time, it is therefore possible to combine the effects of the rate of change of capacitance and leakage current into a single error term K (Equation 3.7). By assuming K remains constant for a short
The point in time at which dV_C/dt=0 occurs is during the period when the PWM signal is transitioning from „on‟ to „off‟, or vice versa. For a conventional PWM signal with a very high slew rate, i.e., where the transition between „on‟ and „off‟ is effectively instantaneous, it is entirely impractical to evaluate the state of the circuit the point in time when dV_C/dt=0. To improve this situation the slew rate of the PWM signal used for self-sensing was capped, i.e., the rising and falling edges of the PWM waveform became ramps rather than step changes. This expanded the period of time over which the transition between the on and off states occurred and made it feasible to determine the point in time when dV_C/dt=0 (Figure 3-4).
period after it has been calculated, C_DEA can be estimated using Equation 3.8 for any point in time where dV_C/dt≠0. Furthermore, once C_DEA is known, finite differences can be used to calculate dC_DEA/dt. This can be substituted back into Equation 3.7 to enable the calculation of R_EPR, and thus leakage current.
Equations 3.7 and 3.8 enabled capacitance and leakage current to be determined provided the voltage across the capacitance of the DEA, V_C, was known. Recalling Equation 3.4 however, it was necessary to account for the resistance of the electrodes before capacitance and leakage current could be estimated. Figure 3-5 illustrates the influence of R_ESR on V_DEA (top) for a short period of time centred on the point in time that a PWM input current signal transitions from the „off‟ to the „on‟ state (bottom). Note for both V_DEA waveforms shown in the top plot of Figure 3-5, V_C is the same. However, for the dashed blue line R_ESR=0, thus V_DEA = V_C, while for the solid red line R_ESR=500 kΩ, thus V_DEA has a current dependent term. This is reflected by the difference between the two lines. Most notably however, when R_ESR is significant, „corners‟ appear in the V_DEA waveform as the PWM signal changes state (see V_DEA for R_ESR=500 kΩ at t=t_ref and t=t_f from Figure 3-5). This feature provides a mechanism that can be used to evaluate R_ESR. By rearranging Equation 3.4 to be in terms of V_C (Equation 3.9), the correct value for R_ESR will transform the solid red line from the top graph of Figure 3-5 to the dashed blue line, thereby eliminating the corners.
With reference to Figure 3-5, the necessary information to determine R_ESR can be obtained by using regression to fit straight lines to the voltage data and the series current data from Part 1, and the series current data from Part 2, and fitting a parabola to the voltage data from Part 2 (Equations 3.10–3.13). By substituting V_{DEA(Part 1)} and i_{series(Part 1)} into one expression for V_C, and V_{DEA(Part 2)} and i_{series(Part 2)} into second expression for V_C, the correct value for R_ESR will make the slope of each expression for V_C equal at t=t_ref. Thus R_ESR can be found using simple algebra (Equation 3.14).
The accuracy of this self-sensing algorithm was verified using a numerical simulation created in MATLAB (R2008a, The Mathworks, Inc., Appendix 1). Based on the results of the numerical simulation, the second iteration of the self-sensing algorithm addressed all of the shortfalls of the first iteration. The resistance of the electrodes was accounted for, and accurate estimations of capacitance and leakage current were possible for static and dynamic operating conditions. Attempts to implement this system in a practical system however were only moderately successful. Accurate estimations of the electrode resistance were possible. Estimations of capacitance and leakage current had the correct nominal value, however, they were very noisy. In particular, calculating K (Equation 3.7) was highly sensitive to noise in the feedback signal, which translated to noise in the estimated parameters of the DEA. Further work was required to reduce the sensitivity of the self-sensing algorithm to noise in the raw feedback.

Self-sensing for DEA – Iteration #3

A simple but powerful breakthrough in the development of the self-sensing algorithm was the realisation that the derivative terms of the Equation 3.3 could be approximated using terms based on finite differences. This meant that rather than attempting to quantify the derivative of an inherently noisy feedback signal for a specific instant in time, the derivatives could be approximated using multiple data points spanning a short period of time. By using multiple data points to estimate average values for the parameters of interest, the influence of noise can be greatly attenuated. The system of equations used for self-sensing therefore become Equations 3.15–3.17^*.

1 Introduction
1.1. Evolving robotics
1.2. Muscle: nature‟s smart actuator
1.4. Electroactive polymers
1.5. Dielectric Elastomer Actuators
1.6. Artificial muscles using DEA: Opportunities and Challenges
1.7. Research objectives and thesis outline
1.8. Contributions of this thesis
2 Literature review
2.1. The significance of the electrical parameters of DEA
2.2. Self-sensing DEA
2.3. Chapter Summary
3 Self-sensing for DEA
3.1. Pulse Width Modulation applied to DEA
3.2. A circuit for acquiring raw feedback for DEA self-sensing
3.3. Fundamental equations for self-sensing
3.4. Evolution of a DEA self-sensing algorithm
3.5. 3D DEA self-sensing
3.6. Visualisation of the 3D self-sensing process
3.7. Chapter Summary
4 Experimental validation of DEA self-sensing
4.1. Introduction
4.2. Self-sensing feedback circuit design
4.3. ESR and leakage current validation
4.4. Capacitive sensing verification
4.5. Integrated self-sensing of DEA
4.6. Conclusions
4.7. Chapter Summary
5 Position and stiffness control of DEA
5.1. Introduction
5.2. Design and tuning of a position controller for an expanding dot DEA
5.3. Position control experimental methods and results
5.4. Design and tuning of a stiffness controller for an expanding dot DEA
5.5. Stiffness control of an expanding dot DEA
5.6. Conclusions
5.7. Chapter summary
6 Cyber-pain for DEA
6.1. Introduction
6.2. Characterisation of leakage current
6.3. Real-time estimation of leakage current using DEA self-sensing
6.4. Conclusions
6.5. Chapter summary
7 Conclusions and future work
7.1. Thesis summary
7.2. Contributions of this thesis
7.3. Applications of self-sensing
7.4. Future work
7.5. Related Publications