## 2 Mathematical Modeling Strategy
### Development of Mathematical Model 1
#### 2.1.2 Actuator Dynamics  
An electromechanical actuator (EMA) is made of an electric DC motor (that corresponds to the electrical section) and an associated mechanical gearbox (that defines the mechanical section). EMAs are categorized in both rotative and linear stages. A linear actuator is as follows: geared and direct-driven linearly operated EMA. The geared linear electromechanical actuator generally consists of a DC motor with a lead-screw (or ball-screw or roller-screw) and a nut assembly. The motor shaft connects to the lead screw using a shaft coupling or a transformation gearbox. The DC motor transforms the electrical energy to the respective rotational motion of the shaft. Hence, the lead screw converts the rotary motion into proportional linear motion. The motor shaft rotates either anti-clockwise or clockwise, depending on the voltage input polarity. Once the rotational motion is transferred to the lead screw, the nut assembles the needed motion (XN), either forward or backward, depending on the forced rotational direction of the shaft. In the direct-driven linear actuator (considering both electrical and mechanical sections), the nut is the rotational element, and the lead screw represents the translational element. The lead screw structure is accommodated inside the actuator itself, considering the technical composition of the DC motor. The interiorly threaded nut is placed inside the rotor. The rotor and the nut rotate when the engine receives electrical energy. This rotational motion of the nut forces the linear lead-screw motion. The shaft attached to the ending section of the lead screw has slotted grooves. This configuration restricts the rotational motion of the associated lead screw simultaneously with the nut. In terms of their functionalities, the two mentioned actuators are not significantly different. Nonetheless, the direct-driven EMA has a compact size compared to the geared EMA. This configuration makes the direct-driven EMA a suitable and appropriate technical option for aerospace, robotics, and other industries. Modeling a single EMA is crucial for developing a multi-element set of actuators operating as HRA. This section presents the mathematical model of a direct-driven linear EMA. The electric circuit implies that the supplied electrical power using a given voltage \((V_{a})\) forces the current \(i_{a}\) flow whose conduction is opposed by the conducting path resistance \((R_{a})\), inductance \((L_{a})\) and a back electromechanical force or e.m.f. for short \((V_{b})\), which is equivalent to \(K_{e}\dfrac{d\theta_{m}}{dt}\). The voltage balance is expressed as:  
\[K_{e}\dfrac{d}{dt}\theta_{m}+L_{a}\dfrac{d}{dt}i_{a}+R_{a}i_{a}=V_{a} \tag{27}\]  
Here \(K_{e}\) represents the back e.m.f parameter. The parameter \(\theta_{m}\) characterizes the angular variation of the DC motor. In the mechanical part, inertia, damping, and frictional settings are concentrated parameters. When the electromagnetic torque performed by the electrical section is supplied to the mechanical section of the actuator (lead-screw and nut) rotational element, it moves at a given rate of \(\dfrac{d}{dt}\theta_{m}\), which corresponds to the given moment of inertia \(J_{m}\). The angular speed of both sections, the rotor and the nut are the same because they are attached to each other [13]. The variables labeled as \(x_{s}\) and \(x_{l}\) represent linear variations of the lead screw and the load correspondingly. The matrix \(K_{s}\) defines the relative stiffness existing between the lead screw with respect to the nut. The damping between the bearing and rotor is defined by the variable \(C_{s}\). The produced torque \(\tau_{m}\) is operating in opposition by damping torque \(\tau_{d}\), load torque \(\tau_{l}\) and the inertial torque \(\tau_{l}\).