## 1 Introduction  
The development of BRDs has attained a significant degree of maturity over the last three decades. Novel advances in mechanical design, electrical instrumentation, and automatic control methods have forced BRDs to exert more complex activities such as autonomous walking, climbing stairs, and jumping, among others [1]. Nowadays, there is a scientific and technical trend in developing more efficient BRD configuration, which includes using lighter and more resistant materials, faster digital processors, and more advanced electronic sensors and actuators [2]. This last aspect has attracted attention because actuators' characteristics define most BRD mobilization abilities. There are diverse actuator options for BRDs, including permanent magnet and brushless motors, steppers, pneumatic muscles [3], multi-motor drive systems [4], and more recently, linear actuators [5]. Linear actuators are devices that are conformed by an electrical motor (usually a DC with a permanent magnet) connected to a ball-screw driving structure (Figure 1). This device is used increasingly in robotics due to its working characteristics, including the device scalability, setup versatility, stack setting, larger torque, the electrical equivalent operation similar to DC motors actuators, and motion robustness concerning external perturbations, as well as long periods of activity. Using linear actuators for mobilizing BRD has several advantages (compared to other actuation strategies). A heavy actuation device can be placed near the hip joint, limiting the leg inertia, and some others [6]. Despite the mentioned benefits, some drawbacks must be overcome if the linear actuators are going to be considered a feasible variant for forcing the BRDs motion. Among others, linear actuator dynamics are usually not considered in the device modeling description. The restricted range of motion in the actuator (which implies limited articulation angular displacement) are the main disadvantages that should be corrected. All the benefits of using linear actuators as driving elements for BRDs have been emphasized by various studies. The Technical University of Munich constructed the humanoid robot LOLA [7]. They proposed the design of a linear actuator based on a ball screw drive attached to the main structure by Cardan joints. It is used to maintain the moment of inertia of the thigh. Furthermore, the WL family of robots, designed by Waseda University, uses linear motion actuators for its Stewart platform-like mechanism that is oriented as a novel biped walking type wheelchair [8; 9]. The study introduced in [10] describes a double linear ball-screw drive structure for the ankle joint. In comparison, ref. [11] provides a design where both the knee and ankle possess single and double linear actuator structures, respectively. Most existing studies where the linear actuator has been the active driving element of biped robots have not considered the electromechanical dynamics that characterize it. Such dynamics include the motor's temporal evolution and connection to the ball-screw configuration. Moreover, the reported studies consider that the linear actuator has an  
Figure 1: Models considered for the BRD assuming the two configurations for the linear actuators forming a structure in the device. (**a**) Comparison of trajectory tracking performance between the proposed and PID controllers at the hip joint in the biped robot; (**b**) comparison of trajectory tracking performance between the proposed and PID controllers at the hip joint in the biped robot. unbounded range of linear displacement. Both characteristics are going to be analyzed in this study and used as limitations for the automatic control design that may enforce the exerting of a gait cycle (even if the BRD is suspended). The problem of introducing linear actuator dynamics in the BRD augments the complexity not only of its mathematical model but also of the automatic control design that forces the regulated movement of the BRD [12].