# Characterization of Control-Dependent Variable Stiffness Behavior in Discrete Muscle-Like Actuators
## Muscle-Like Actuators Composed of Discrete Building Blocks
### Activation Patterns and Activation Levels  
There are many ways that the active and inactive motor units can be distributed over the actuation units that will total the same number of motor units active for the entire actuator, see Figure 6. The _activation level_ is defined as the sum of active motor units on each actuation unit. Thus, if four actuation units are in series, two motor units could be activated on the first actuation unit, five motor units on the second, one motor unit on the third and zero on the fourth. This can be represented by the vector [2 5 1 0]. An example of a different _activation pattern_ while keeping the same activation level would be [2 2 2 2]. However, another activation pattern that could be realized would be [6 2 0 0], each of these patterns totalling eight units active. However, specifying an activation pattern does notdetermine _which_ of the motor units on a given actuation unit are active, simply that two of the \(N=6\) motor units are active. It should be evident that a large finite number of different stiffness values that depends on the number of actuation units and the number of motor units used to make up the actuator can be achieved. The following equations represent the activation level with M being the number of active motor units per actuation unit. In the example illustrated in this section, four actuation units \(P=4\), connected in series are used. Each actuation unit consists of one PCB with six motor units, \(N=6\), mounted on each PCB. The number of different stiffness levels that can be selected via the control input can be calculated using binomial coefficients and the equation for combinations seen in Equation (1). \[\binom{n}{k}=\frac{n!}{k!(n-k)!} \tag{1}\]  
When calculating the number of stiffness levels,substitutions for n and k can be made, as shown in Equation (2). In the substitution, \(N\) stands for the number of motor units used and \(P\) denotes the number of actuation units for the actuator following similar notation to [7]. This equation can be simplified to Equation (3). \[\binom{N-1+P}{N-1}=\frac{(N-1+P)!}{(N-1)!((N-1+P)-(N-1))!} \tag{2}\]  
\[\frac{(N-1+P)!}{(N-1)!(P)!} \tag{3}\]  
For the length based hardening simulation that will be discussed later, if \(0\) is included as a state showing that there are no active motor units for that actuation cell, then four actuation units in series results in a total of 210 different stiffness levels that can be achieved through different activation patterns. For longer chains, it can easily be shown that by substituting the number of units in the equation for \(P\) and evaluating gives a determination for the new number of total activation levels. As an individual actuation cell lengthens and moves from Class II into Class III, the overall stiffness of the actuator should increase. This is because the actuation cell moves from having less than the maximum number of motor units \((M<N)\) providing stiffness, to a point where the springs of the inactive motor units \((N-M)\) engage with the mechanical stops and begin providing stiffness to the overall spring constant. The exception is if all the motor units are active \((M=N)\) or all the motor units are inactive \((M=0)\), in these cases there will not be a Class III distinction for this actuator or a critical length. The stiffness of an actuator, made up of four actuation cells, increases in a nonlinear fashion as each of the cells moves from Class II into Class III. This happens in distinct steps because of the  
Figure 6: Different activation patters while maintaining same activation level: (**a**) eight motor units active (highlighted in green) over two actuation units—shorthand written as a vector [4 3 0 1]; and (**b**) eight motor units active (highlighted in green) over two actuation units—Shorthand written as a vector [6 2 0 0]. difference in activation level between the four cells. The least active cell will reach Class III first, followed by the next least active cell and so on until the cell with the most active motor units reaches Class III.