# Characterization of Control-Dependent Variable Stiffness Behavior in Discrete Muscle-Like Actuators
## Muscle-Like Actuators Composed of Discrete Building Blocks
### Mathematica Formulation  
Changing between the classes of the actuation units, which depends on the overall length of the actuator, is a nonlinear switching condition. The different classes of the actuator, discussed earlier in Section 2.3, give definition to the nonlinear phase portraits, a plot of the position versus velocity for the system, that will be shown and discussed in detail in this section. The setup in Figure 7 shows the simple actuator-mass-return spring configuration that was used in the simulations to be discussed shortly. The equations describing the variable stiffness are not easily described in closed form due to the piecewise-linear behavior described in Section 2. Equation (4) gives the sum of the forces about the common node for two actuation units in a series chain connection, \(i\) and \(i+1\), as shown in Figure 7.  
Figure 7: Horizontal oscillatory motion. Notice the conditional statement introduced that allows this general equation to be used for Class Zero-Class III (definitions of which can be found in Section 2.3). Equation (8) represents the force-length relationship for actuation units in Class III. However, in order to adjust this to be used with a unit in Class II the term \(k_{i}^{in}\) must be set equal to 0. This is the term quantifying the spring constant once the units have reached critical length and extend beyond the mechanical stops. Setting this equal to zero and substituting into Equation (8) will give a correct representation of the sum of forces for this Class. \(a_{i}\) will change to: \(a_{i}=\frac{(k_{i+1}^{in}+k_{i+1}^{in})}{k_{i}^{in}}\). If, however, the unit is in Class I, then the motor units on board the actuation unit are all contributing to the stiffness and are all inactive and the length has been extended beyond the inactive resting length, \(\ell_{i}>\ell^{o}\). If this is the case, then substituting \(k_{i}^{a}=0\) into the equation for \(a_{i}\) and Equation (8) will correctly categorize the sum of the forces for this state of the actuator. The last Class is Class Zero for which the force across the actuator is zero and neither the inactive nor the active motor units contribute to carrying the load. In this case, the load is only across the return spring seen in Figure 7, which corresponds to Equation (7) being undefined for Class Zero. These equations and conditions can then be written for any number of actuators in series and put in matrix form as seen in Equations (9)-(12). Whereas the \(a_{i}\) matrix is defined in Equation (10), \(c_{i}\) is defined in Equation (12), and the individual actuation unit lengths are given by Equation (11). \[A\cdot\ell=C \tag{9}\]  
\[A=\begin{pmatrix}1&-\frac{k_{i+1}^  
setup of Figure 7. In these simulations, a setup of two actuation units was chosen each with six motor units. Choosing two actuation units allows the actuator to display the discretization effects seen in the phase portraits and control input, however, the theoretical formulation developed in the previous section is valid for \(n\) units in a chain and for any number of motor units per actuation unit, and it can also extend easily to bundles of chains. Adding more actuation units decreases the discretization effects making it appear more and more linear until at some point the quantization is so linear it can be more or less considered continuously variable. However, most practical applications will have a small enough number that quantization is necessary. There are four cases shown in Figure 8: (1) a simple harmonic motion between a linear spring and our actuator; (2) a switching method based on time; (3) a switching method based upon actuator position, right or left of the origin; and (4) a switching method based on the sign of the velocity, so above or below the origin. Thus, it can be said that the third and fourth methods are a type of quadrant based switching, but with a limited number of activation patterns and activation levels dictated by the number of actuation units and their arrangement within the actuator.