# Analytical Attitude Guidance Planner for Multiple Ground Targets Acquisitions
## 1 Introduction
Therefore, in general, \(AGP\) cannot be evaluated on-board the satellite, but have to be computed offline and uploaded by ground control. This solution could limit the mission flexibility and efficiency, impacting the capability to plan the acquisition of new targets and reducing the number of targets that may be acquired. Based on the recent advent of small satellites [26] and the exploitation of machine learning for remote sensing tasks [27; 28; 29; 30], some authors have already explored the opportunities provided by on-board processing of remote sensing images [31; 32; 33; 34; 35]. However, all these applications would quite benefit from on-board autonomous planning capabilities, so that the satellites can entirely focus on observing and monitoring hazards and disasters on the planet. The aim of this work is to design an \(AGP\) in order to obtain an algorithm that requires a very limited computational effort to make it applicable in real-time on-board applications. Specifically, the work focuses on the design of an autonomous satellite maneuver planner for multiple ground target acquisition, having as input the ground targets positions with respect to the Earth-centered Earth-fixed (\(ECEF\)) reference frame (expressed in terms of latitude (\(\mu\)), longitude (\(\lambda\)), and altitude/elevation (\(h\)) above mean sea level) and the time in which each acquisition should be performed (\(t_{start}\)). The \(AGP\) outputs are represented by reference profiles that must be tracked by any external controller (linear PID controller, LQR controller, SDRE controller, etc.). Indeed, guidance and control can be split so that the former provides the reference kinematics, i.e., the quaternion and angular velocity profiles, while the latter tries to track them as well as possible. Thus, the goal of the paper is to develop a reference guidance policy for the \(GNC\) module in terms of attitude kinematics profiles (quaternions, angular velocities, and angular accelerations) and control policies. An analytical optimal closed-form solution was developed to perform an \(AGP\) and has been carried out by dividing the problem into two parts: in the first phase, the satellite attitude must be brought from the current state (at initial time \(t_{0}\)) to the desired final state in order to start ground target acquisition (\(t_{start}\)). Therefore, the first phase can be seen as a boundary value problem that can be solved by a point-to-point maneuver in a fixed time (\(\Delta T_{APPG}=t_{start}-t_{0}\)) and is named \(APPG\) (attitude point-to-point generator). In this phase, the guidance is approximated by means of polynomials, which always return a mathematical solution, but not all of them are exploitable. In fact, physical constraints for the maximum allowed control torque and angular velocity are considered in this work. The second phase aims to keep the payload line of sight pointed toward the ground targets for the time required for acquisition (\(t_{ac}\)). This phase is named the attitude target pointing generator (\(ATPG\)). The solution of this phase is achieved by geometric resolution, finding the attitude at each instant through two Euler rotations (around a principal axis). Although from a chronological point of view the two steps take place as indicated above, the algorithm requires them to be performed in reverse, i.e., the \(ATPG\) is evaluated before the \(APPG\). This is because the \(APPG\) must take into account the final desired state at time \(t_{start}\) before executing the point-to-point maneuver. For this reason, the \(ATPG\) is presented first, followed by the \(APPG\). The paper is organized as follows. In Section 2, the orbital and attitude dynamics of the satellite are reported along with the solution setup. Section 3 describes the attitude target pointing generator phase, i.e., the solution strategy to track the ground targets, while Section 4 deals with the attitude point-to-point generator, i.e., the first part of the proposed solution. In Section 5, numerical simulations and results are carried out.