## 1 Introduction  
Traditional robotic actuators primarily consist of electric motors and hydraulic systems. Electric step motors [1] are widely used in small robotic systems due to their precision, convenient power usage, and scalability. These motors operate through the Lorentz force, requiring a substantial magnetic field generated by large electric currents through copper solenoids, resulting in significant Joule heating and high-power demands. Electromagnetic motors are most efficient for large displacements but less effective for small displacements [2]. Hydraulic systems represent another major approach and are used in applications such as derricks, cranes, bulldozers, forklifts, industrial robots, and the US Army Mule walking robot [3; 4]. However, hydraulic actuators do not scale well with system size, as reducing system size decreases piston area, leading to reduced force at the same pressure. This makes hydraulic systems unsuitable for miniaturization. Additionally, pneumatic actuators produce less force than other solutions at larger displacements [2], and their performance is similar to that of pneumatic artificial muscles [5]. Biomimetic applications, such as exoskeletons and prosthetics, require smooth and precise motion control, which is challenging to achieve with hydraulics [6]. Given these limitations, modern applications like undersea drones, exoskeletal suits, all-terrain walker drones, prosthetics, and medical assist devices are exploring alternative solutions, such as artificial muscles [7; 8]. For instance, dielectric elastomer actuators (DEAs) [9; 10; 11; 12] utilize the deformation of a polymer slab due to electrostatic forces between charges on electrodes under an external voltage. This force increases inversely with the square of the distance between electrodes, making it more effective 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]. DEAs can be arrayed longitudinally to enhance elongation distance and laterally to increase force output. However, the high cost of traditional manufacturing methods limits the scalability of these devices, typically confining them to small applications like single-chamber micro-robots [16; 17; 18] and HASEL tentacle actuators [19; 20]. Although multi-material 3D printing holds promise [21; 22; 23; 24], it remains in its early stages. We have developed an alternative solution [25]: artificial muscles created by 3D printing microfluidic channels, which serve as embedded wires and electrodes for microcapacistors arranged to form muscle fibers [26]. Applying voltage to an array of microcapacitors causes them to contract simultaneously, producing an electrostatic actuation force transmitted via the surrounding polymer material, acting as tendons. These electrostatic actuators are scalable, energy-efficient, and offer a high force-to-weight ratio [25]. COMSOL simulations [25] demonstrated a force density of up to 33 MPa at current fabrication limits, though these simulations were restricted to small devices [25]. The output force density is inversely proportional to the square of the separation distance between the capacitor plates, with optimal results achieved at microscopic scales. The total force of a muscle fiber bundle is the sum of the forces generated by individual fibers arranged in parallel [25; 26]. In our previous research, we investigated the scalability of artificial muscle fibers, recognizing that each fiber generates a relatively small force, necessitating tens of thousands of microcapacitors for a practical macro-scale muscle. Direct simulation of such large-scale muscles was computationally prohibitive due to the limitations in computational power. To circumvent this, we hypothesized that the output force density would reach saturation as the array size increased, enabling us to estimate the force of larger arrays without direct simulation.