# Output Force Density Saturation in COMSOL Simulations of Biomimetic Artificial Muscles
## 1 Introduction  
Traditional robotic actuators are electric motors and hydraulic systems. Electric step motors [1] provide precision of motion, utilize convenient power, and are scalable to a degree, making them the common choice in small robotic systems. As the actuation force is the Lorentz force, a significant magnetic field must be applied, usually by running large electric currents through copper solenoids. That generates significant Joule heating and leads to high power requirements. Electromagnetic motors have been found to be most effective for large displacements and less effective in small displacements [2]. The other major approach Is the use of hydraulic systems, e.g., in derricks, cranes, bulldozers, forklifts, industrial robots, and the US Army Mule walking robot [3, 4]. However, hydraulic actuators scale disadvantageously with system size, because shrinking the system decreases the piston area, so the same pressure produces less force. Hence, hydraulic systems are unsuitable for miniaturization. In addition, pneumatic actuators output less force than alternative solutions at larger displacements [2]. Hydraulic actuators perform comparably to pneumatic artificial muscles [5]. Finally, biomimetic applications, e.g., exoskeletons and prosthetics, require fluidity and fine motion control, which are difficult to achieve through hydraulics [6]. In view of the above limitations of traditional actuators, many modern applications, such as undersea drones, exoskeletal suits, all-terrain walker drones, prosthetics, and medical assists and augments, look for alternative solutions, such as artificial muscles [7,8]. For example, Dielectric Elastomer Actuators (DEAs) [9, 10, 11, 12] utilize the deformation of a polymer slab due to the electrostatic force between the charges built-up on electrodes under externally applied voltage. This force scales as the inverse square of the distance between the electrodes. Therefore, it is small for macroscopic distances but increases disproportionally at microscopic scales. (Incidentally, the same principle can be used to build a capacitive high-sensitivity gauge of mechanical deformation [13,14], as well as a pressure sensor [15].) Such devices can be arrayed longitudinally, to increase elongation distance, and laterally, to increase force output. However, the unit cost would make the simple arraying and stacking of individually fabricated devices prohibitively expensive if traditional manufacturing was used. As a result, more recent attempts tend to be limited to small devices, such as single-chamber microrobots [16, 17, 18] and HASEL tentacle actuators [19,20]. Multi-material 3D printing is promising [21, 22, 23, 24], but still in its infancy. We pioneered an alternative solution [25]: artificial muscles built by 3D printing microfluidic channels, wherein the channels become embedded wires and electrodes for micro capacitors arranged to form muscle fibers [26]. Applying voltage to an array of micro capacitors makes them contract in unison, producing an electrostatic actuation force. That force is transmitted to the outside world via the surrounding polymer material, functioning as tendons. The resulting electrostatic actuators would be scalable, energy-efficient, and offer a high force-to-weight ratio [25]. The COMSOL simulations [25] indicated up to 33 MPa force density at current advertised limits of fabrication capability. Those simulations were limited to 1 \(\times\) 1 \(\times\) 1 and 2 \(\times\) 2 \(\times\) 1 devices [25]--devices containing one or two micro capacitors in each of the lateral directions and just one micro capacitor in the longitudinal direction of the artificial muscle. The output force density scales are the inverse square of the separation distance between the plates in the micro capacitor, so best results are achieved when the individual capacitors are microscopic.