# Analytical Attitude Guidance Planner for Multiple Ground Targets Acquisitions
## 2 Problem Description and Proposed Solution
### Reference Frames Review  
Before writing the dynamical model employed for the analysis, the required reference frames are reviewed in order to provide the right background for the description of the problem [36]. **Earth-Centered Inertial (ECI)**: The \(ECI\) coordinate system has its origin in the Earth's center of mass and it is fixed with respect to the stars. The \(Z_{ECI}\) axis is aligned with the Earth's rotational axis and points to the north, the \(X_{ECI}\) axis points in the direction of the _pernal equinox_, and the \(Y_{ECI}\) axis completes the right-handed orthogonal reference system. **Earth-Centered Earth-Fixed (ECEF)**: This reference frame has the same origin and the same \(Z\) axis (\(Z_{ECEF}=Z_{ECI}\)) as the \(ECI\) frame but rotates with the Earth. The \(X_{ECEF}\) is in the equatorial plane and points to the Greenwich meridian, while the \(Y_{ECEF}\) is taken in order to have a right-handed orthogonal reference system. **Local-Vertical Local-Horizontal (LVLH)**: It has the origin in the satellite's center of mass, \(Z_{LVLH}\) points to the center of the Earth, and \(X_{LVLH}\) is parallel to the local horizontal plane (which is the plane normal to the radial position vector connecting the center of the Earth and the satellite), its orientation is such that the satellite's velocity vector is contained in the \((X_{LVLH}-Z_{LVLH})\) plane, and its positive direction is taken to be concordant with the satellite velocity vector direction. The \(Y_{LVLH}\) axis is taken to form a right-handed orthogonal reference system. **Body-Fixed (BF)**: This frame corresponds to the satellite's principal inertia axes reference frame centered on its center of mass. The \(BF\) frame is useful to describe the attitude of the satellite. Indeed, the nominal reference attitude of the satellite is often given when the BF frame is aligned with the LVLH frame, and the attitude is described as the displacement of the BF frame with respect to the \(LVLH\) frame. **Payload Reference Frame (PRF)**: In this work, the payload consists of a camera mounted with a displacement with respect to the \(BF\) origin, as shown in Figure 1. Point \(O_{3}\) identifies the center of the charge-coupled device (\(CCD\)), which is the camera detector. The \(PRF\) frame has \(Z_{PRF}\) axis aligned with the optical axis and the other two axes (\(X_{PRF},Y_{PRF}\)) are taken to be a right triad. This reference frame is fixed with respect to the BF reference frame, so its rotational matrix \(R_{BF}^{PRF}\) is constant and is the payload mounting matrix. The reference frames described above are related to our problem as depicted in Figure 1. The geometry behind this problem is not intuitive, since the payload mounting matrix is formed by a roto-translation with respect to the \(BF\), and generally the camera view axis (\(Z_{PRF}\)) is not centered in the \(BF\) origin. This point is represented in Figure 1a,b, in which the ground target is indicated with the letter \(T\) and each reference frame is represented with different colors: the \(ECI\) is black, the \(LVLH\) is red, and the \(BF\)-\(PRF\) is blue. The latter are chosen with the same color since they move jointly. In Figure 1a, the origins of \(PRF\) and \(BF\) coincide, and any rotation of the satellite does not change the magnitude of the vector \(\overrightarrow{O_{2}T}\) providing the position of the target \(T\) in \(PRF\) (but the \(\overrightarrow{O_{2}T}\) components in \(PRF\) and \(BF\) change). In the case shown in Figure 1b, instead, \(PRF\) is displaced and rotated with respect to \(BF\) and, when a rotation of the satellite is performed, the vector \(\overrightarrow{O_{3}T}\) changes both magnitude and direction in \(PRF\), thus modifying the pointing geometry. This configuration makes the problem strongly non-linear and is the one considered in this work. It is worth noting that, at the state of the art, while payload rotation must necessarily be considered, displacement may not be.