# Characterization of Control-Dependent Variable Stiffness Behavior in Discrete Muscle-Like Actuators
## 1 Introduction  
Bio-inspired robotic actuator designs seek to emulate certain desirable traits seen in human skeletal muscle systems, and can sometimes exceed them in performance. The design presented in this article utilizes skeletal muscle characteristics, including: compliance, redundancy to failure, and a modular building block or cellular architecture, see Figure 1. One other concept taken from human skeletal muscle systems is that of motor unit recruitment [1]. Motor unit recruitment is a discretized actuation strategy that provides human skeletal muscle with a plan to generate the particular motion dictated by the nervous system. Each motor unit is responsible for recruiting certain muscle fibers, spread throughout the muscle, to achieve the contraction. In this way, if a contraction of higher strength is needed, more motor units are recruited. An actuation scheme based on this idea can simply change which motor units are activated if one motor unit suddenly is rendered broken or useless. Such an actuation device offers a high resistance to failure, which is a very attractive feature in high-reliability applications. The key advantage is that a component failure results not in a total loss of motion for that particular robotic limb, but only a decrease in performance, meaning a decrease in the maximum force that can be produced. A discrete actuator design that uses \"on\"-\"off\" motor units must utilize such a recruitment strategy for its actuation and control. There are challenges associated with quantizing an analog control signal into activating discrete units, which will be addressed by showing two methods of quantizing such a signal. The first method is a constant scaling of an approximation for the force, length and number of total motor units active. The second method is through the use of a pre-computed look-up table referenced during run time. Compliance from unit-to-unit must also be a design component in order not to violate physical laws, as shown by [PERSON] et al. [2]. By introducing this compliance two things happen, the first is that it makes modeling the actuator very complex as each discrete unit now has an elastic element associated with it, but it also allows for the opportunity to vary the compliance in such a way as to introduce damping-like behavior where there was not before, which will be explained in more detaillater on. The units making up discrete actuators offer a redundancy that resists failure, for if one of the units is suddenly broken another unit can be recruited to \"pick up the slack\". These three characteristics make discrete actuators very desirable for use in robotic applications of various types, but modeling the dynamics of these actuators becomes quite complex because each discrete unit must be treated as a mass-spring system, which results in very high order systems of differential equations. High order systems makes constructing a closed form control law difficult. This paper provides a method for overcoming these hurdles using a Series Elastic Actuator model and an observer system. When modeling the dynamics of the actuator presented in this work, prior research in Series Elastic Actuators (SEAs) provides guidance in developing a controller to work with the actuator. An SEA was chosen to aid in the model development because its compliant element that operates in series with the motor is similar to the unit-to-unit compliance in a discrete muscle-like actuator. In the design of discrete actuators that are aligned in parallel formations there must be compliance coupling them together in order to adhere to physical laws. See Figure 2 for reference to this concept and read [2,3] for more detail. The SEA was one of the earlier developments made in compliant actuators and provided a platform used as a basis for many of the later designs. [PERSON] and [PERSON] [4] developed an actuator that included a series elastic element that gives the actuator a compliant nature.