## 2 Mathematical Modeling Strategy  
Including linear actuators on the biped robot structure introduces two possible modeling strategies. The first one considers the passive mechanical structure as the core part to be modeled and the linear actuators as the active section that provides the mobilization of the mechanical parts. The second strategy includes using the linear actuator as part of the mechanical structure. The difference between these strategies can be visualized considering the mechanical and active sections shown in Figure 1.  
This section presents the general characteristics of the modeling strategies. For both models, this study considers the following characteristics:  
* The biped robot can mobilize the legs on the sagittal plane,
* The contact points between the actuator tip and the structural section of the biped are not dynamic. Then, there is no dynamic contact (stiffness-damping).
* The linear actuator for each articulation is a lead screw and nut structure, which is developing the linear action related to the linear actuator.
* The actuator is activated by a DC motor, which is also considered part of the control design.
* All articulations in the biped robot are static based on a rotational joint which is justified using the argument related to the motion in the sagittal plane.
* The biped robot is suspended, and it does not have any impact contact with the supporting surface. These kinds of robots are aimed to serve as a basis for exoskeleton devices, which can contribute to the rehabilitation processes.  
Considering that \(q\in\mathcal{Q}\subset\mathbb{R}^{n}\) is the vector of generalized coordinates and \(\dot{q}\in\mathcal{Q}\subset\mathbb{R}^{n}\), then the kinetic energy \(T:\mathcal{Q}\times\mathcal{Q}\rightarrow\mathbb{R}^{+}\), \(T=T(q,\dot{q})\) and the potential energy \(U:\mathcal{Q}\rightarrow\mathbb{R}^{+}\), \(U=U(q)\) for the complete system may be obtained from the aggregated analysis of subsystems forming the biped robot. The energetic components of the individual subsystems can be calculated independently. Hence, the equations of motion are obtained with the calculation of the Euler-Lagrange equations:  
\[\frac{d}{dt}\bigg{(}\frac{\partial}{\partial\dot{q}_{i}}\mathcal{L}(q,\dot{q}) \bigg{)}-\frac{\partial}{\partial q_{i}}\mathcal{L}(q,\dot{q})=\tau_{i}- \Xi_{i} \tag{1}\]  
Here, we present the generalized forcing-dissipation terms \(\tau_{i}\in\mathbb{R}\), \(\Xi_{i}\in\mathbb{R}\), and the Lagrangian: \(\mathcal{L}(q,\dot{q})=T(q,\dot{q})-U(q)\). The principles justifying this formalism can be consulted in classical mechanics textbooks. Except for the differentiation that may become cumbersome for systems with many articulations, the Euler-Lagrange equations formalism can be applied straightforwardly because it is based on energetic-based functions. Basedon the calculus for the Lagrangian and the application of the Euler-Lagrange schemes, the complete model for a single leg of the BRD satisfies the following differential equation:  
\[M(q(t))\frac{d^{2}}{dt^{2}}q(t)+C\left(\frac{d}{dt}q(t),q(t)\right)\frac{d}{dt}q(t )+G(q(t))=\tau(t)-\Xi(t) \tag{2}\]  
where \(q\in\mathbb{Q}\subset\mathbb{R}^{3}\) is the vector of angular positions that includes all joints of a single leg in the exoskeleton. The inertia matrix is represented by \(M:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3\times 3}\), \(C:\mathbb{R}^{3}\times\mathbb{R}^{3}\rightarrow\mathbb{R}^{3\times 3}\) is the matrix of Coriolis and centrifugal forces, \(G:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}\) includes the terms related to the gravitational forces. The time-dependent term \(\Xi\in\mathbb{R}^{3}\) is the vector that represents the effect of viscous and dried frictions, external perturbations and modeling imprecision.