## 2 Spacecraft Attitude Dynamics, Determination and Control  
The research work considers a spacecraft orbiting around a planetary body, namely the Earth, whose generic configuration is shown in Figure 1. The spacecraft attitude motion is described as the rotation of the rigid body frame, \(B\), with respect to an inertial reference frame, \(I\). Without loss of general applicability, the inertial reference frame can be selected to be an Earth-centered inertial (ECI) coordinate frame. In this paper, the \(I\) frame coincides with the J2000 ECI reference frame, which is defined with the Earth's mean equator and equinox at 12:00 terrestrial time on 1 January 2000 [27]. The rigid body frame \(B\) is centered in the spacecraft's center of mass, \(O_{B}\), and it is aligned with the spacecraft's body axes, \(\mathbf{\hat{b}}_{1}\), \(\mathbf{\hat{b}}_{2}\) and \(\mathbf{\hat{b}}_{3}\). Without loss of generality, the solar arrays are aligned with \(\mathbf{\hat{b}}_{2}\), and their normal direction, \(\mathbf{\hat{n}}_{S}\), has components in \(\mathbf{\hat{b}}_{1}\) and \(\mathbf{\hat{b}}_{3}\). Note that \(\mathbf{\hat{n}}_{S}\) is fixed in \(B\), since the research assumes no SADA to rotate the plane of the solar arrays. The spacecraft attitude dynamics is described according to the Euler equation of rigid body motion as:  
\[\mathbb{I}\dot{\boldsymbol{\omega}}+\boldsymbol{\omega}\times\mathbb{I} \boldsymbol{\omega}=\mathbf{t}_{d}+\mathbf{t}_{c}, \tag{1}\]  
where \(\boldsymbol{\omega}=[\omega_{1},\omega_{2},\omega_{3}]^{\mathrm{T}}\) is the vector of spacecraft angular velocities, expressed in body frame and representing the rotation of \(B\) with respect to \(I\). \(\mathbb{I}\) is the inertia tensor of the spacecraft body, which is, in general, a full \(3\times 3\) matrix, because the axes of \(B\) are not always coincident with the principal axes of inertia. Note that the inertia tensor is assumed to be constant in time (e.g., \(\mathbb{I}=0\)), since mass variations, flexibility effects, or component movements are not considered. Finally, \(\mathbf{t}_{d}\) is the vector of disturbance torques, and \(\mathbf{t}_{c}\) is the vector of control torques. The disturbance torques considered in this research work are due to both external and internal effects. They are modeled according to the international standards for system simulation and algorithms verification [28; 29]. In particular, gravity, magnetic, atmospheric drag, radiation pressure torques are considered, as well as internal micro-vibrations induced torques and electro-magnetic interaction disturbances. The control torque is expressed in Equation (1) according to the actuation device that is applyingthe control action to the system. The paper considers both momentum exchange devices and magnetic torquers as possible alternative actuators. The momentum exchange devices are assumed to be reaction wheels, whose control action is executed by varying the spin rate of a rotating mass contained inside the spacecraft body. This assumption is motivated by analyzing existing space systems, being the reaction wheels the most common typology of momentum exchange actuators for spacecraft, either small or large classes. An \(i\)-th reaction wheel applies a relative torque to a spacecraft thanks to the exchange of its angular momentum, \(\mathbf{h}_{w_{i}}\), with the main body. The angular momentum of the reaction wheel is expressed as:  
\[\mathbf{h}_{w_{i}}=I_{w_{i}}\mathbf{\omega}_{w_{i}}, \tag{2}\]  
where \(I_{w_{i}}\) is the moment of inertia of the \(i\)-th reaction wheel around its spin axis, and \(\mathbf{\omega}_{w_{i}}\) is the relative angular velocity of the \(i\)-th wheel with respect to \(B\). With this formulation, the inertia of the non-spinning wheel is already included in the inertia tensor of the spacecraft, \(\mathbb{I}\), and Equation (2) represents the additional angular momentum of the wheel due to its spinning state.