# Characterization of Control-Dependent Variable Stiffness Behavior in Discrete Muscle-Like Actuators
## 4 Experimental Verification of Length-Based Hardening  
To verify the variable stiffness characteristics discussed in Sections 2.1-2.3, an experiment was designed that would enable the actuator to be given an actuation pattern and then steadily lengthened while keeping the activation pattern constant throughout. The setup of the experiment can be seen in Figure 16 and is described as follows. A chain of four actuation units of the type described in [PERSON] et al. [2] (see Figure 16) makes up an actuator mounted between two fixtures where one of the fixtures is secured to an optical table while the other fixture is mounted to a micrometer stage. In series with the actuator is a Futek 1 lb force cell whose voltage output was measured using a LeCroy (WaveRunner 604 Zi, Teledyne LeCroy, Rockland County, NY, United States) oscilloscope. The fixture mounted on the micrometer stage gave the ability to lengthen the actuator in steps as fine as 0.025 mm, however, for this experiment steps of approximately 1.2 mm were sufficient. The drive circuitry supplying current to each motor unit is a custom design, described in [PERSON] et al. [2]. The digital signal supplied to the circuitry is from an National Instruments DAQ and manually activated through a National Instruments LabView Virtual Instrument. Taking data sets was performed by setting a constant activation pattern and then increasing the length of the actuator in steps of approximately 1.2 mm, the actual excursion recorded directly from the micrometer stage, and measuring the force at each step. This was repeated with a different activation pattern after resetting the stage. At the extremes of extension, the restoring force of the spring overcame the magnetic force holding the solenoid plunger in the solenoid housing, and data points could not be collected beyond this ultimate length, which varied with the activation pattern (in practice, this ultimate length constitutes failure and should be avoided). Table 2 shows the activation patterns that were used in the experiment and  
Figure 14: Demonstration of the uncontrollable linear springs. Figure 15: Antagonist control phase portraits: (**a**) antagonistic control with observer; and (**b**) antagonistic control with observer and disturbance Force. their corresponding activation levels and the figure where their data is graphed and compared to the theoretical model. The experiment was performed for different actuation patterns and levels. The experiment was also run for the same actuation level repeatedly, while varying which motor unit was used; to evaluate the effects of manufacturing variations, the motor units were activated in different combinations. For example, if the activation level is 4 for a certain actuation cell, then given a total number of motor units of six, the different combinations for activation four motor units is given \(\binom{n}{k}=\frac{n!}{k!(n-k)! }\) for a total of 15. Figure 17a shows the experimental data plotted with standard deviation error bars with an activation pattern of [4 0 1 3]. This plot includes four data sets for the same activation pattern just with different choices for which solenoid to activate from those available on each cell. Although there is some deviation in the slope of each segment due to manufacturing variation, the hardening trend described in Section 2 is evident. The data stop before the actuator reaches the last hardening point of the actuator, this happens because the restoring force of the springs became too great and overcame the magnetic field of one or more of the motor units. The plot shown in Figure 17b shows a different activation pattern but the same activation level, eight motor units. Note that this activation pattern has different hardening behavior than the previous one and the data follows the trend predicted by the model. Again, the data do not go to excursions as far as the model predicts because the spring force overcame the magnetic field force.