## 1 Introduction
In addition, the linear actuator attached to the BRD also introduces a mechanical reconfiguration in the BRD because of the actuator's mechanical structure, which is now giving up a new mass distribution [13], forcing new damping and restricting the angular displacement at each articulation of the BRD (Figure 1). Including linear actuators in BRD electromechanical configurations introduces novel challenges for the automatic control design. Despite the development of automatic controllers for BRD that has been studied for a long time, the dynamical nature of linear actuators implies some new design paradigms, where the actuator now plays a mechanical role as well as exerting the driving force to the BRD [14; 15]. The characteristics of linear actuators force an increase in the number of states in the actuated BRD and bound the control action considering the limited velocity of the linear actuator and the restricted screw movement range. These particular settings must be considered in the control design requirements for BRD, which are adjoined to the presence of modeling imprecision and external uncertainties. All these challenging aspects can be overcome if the proposed control design is robust but, at the same time, can satisfy the successful tracking of the designed reference trajectories that define the needed articulation movements for the BRD. Most of the reported control designs for BRD have not considered such restrictions. Furthermore, many studies have not considered BRD designs' movement restrictions at each joint. Hence, an imprecise gain tuning process for the controllers aimed to mobilize each of the joints, and the robot may manifest unwanted effects in the curse of the transient period. These reasons have motivated the design of automatic controllers that may consider all the referred states or/and control limitations. Penalty state-dependent function, convex set projections, composite state convex sets, and adaptive gain barrier Lyapunov function have been considered the most preferable options to assess the effect of state restrictions on the control design. Barrier Lyapunov function (BLF) is considered a preferable option that could be related to restricted articulation movement. The number of applications of BLF has grown in recent years due to the benefits of restricted systems' transient and steady operation. Diverse classes of BLF have been around since the beginning of the previous century, considering the logarithmic, integral, and tangent variants, just to list some of them. Robotic devices with biped configurations controlled by adaptive gains defined by the application of BLF have shown significant benefits over systems using traditional control formulations such as state feedback, extended state approaches, sliding modes, and many others. However, there is a growing number of applications using adaptive state-dependent gains for the proposed controllers, which are calculated using controlled BLF. However, no reported study proposes the application of adaptive controllers based on BLF for linear actuators for BRDs. This study introduces a novel design for a state-dependent adaptive controller for a BRD, including the presence of linear actuators in the robot configuration. The control design considers state restrictions in the linear actuator and all the BRD articulations while tracking reference trajectories corresponding to developing a regulated gait cycle for a BRD. The study also includes the dynamics of the DC motor that drives the linear actuator motion. The central objective of this study is to present the development of an adaptive automatic control method to resolve the trajectory tracking control design related to a BRD that considers that each articulation is driven by a regular electrical linear actuator.