# An Iterative Guidance and Navigation Algorithm for Orbit Rendezvous of Cooperating CubeSats
## 1 Introduction  
The use of small satellites in modern space missions has become an important factor in the aerospace industry, thanks to the versatility, reliability, and low cost of these platforms. As the scenarios involving small satellites are growing in numbers and complexity, more advanced technological solutions are needed to address the new tasks of these missions. Autonomous guidance, navigation, and control (GNC) systems for small satellites, for example, are crucial for implementing challenging tasks, such as rendezvous and docking [1], formation flying [2], and space debris removal [3]. Guidance algorithms for rendezvous are usually based on the classical Hill-Clohessy-Wiltshire (HCW) equations, either under the impulsive thrust approximation or assuming finite thrust. Seminal publications on impulsive rendezvous have been written by [PERSON] [4; 5], who investigated minimum-fuel rendezvous trajectories including two, three, or four velocity changes. By using the HCW equations, [PERSON] [6] focused on finite thrust rendezvous with an upper bound on thrust magnitude, and proved that at most seven thrust/coast intervals can occur. Later, [PERSON] and [PERSON] [7] investigated fuel-optimal rendezvous relative to a point in a general Keplerian orbit, and demonstrated that nosingular arc exists if the orbit is noncircular. Most recently, [PERSON] and [PERSON] [8] addressed the problem of finding a variety of minimum-fuel rendezvous trajectories by using finite thrust and proved a remarkable symmetry property of optimal rendezvous paths. Although all the preceding contributions employ the HCW equations, [PERSON] and [PERSON] used the nonlinear equations of Keplerian motion, for the purpose of minimizing fuel consumption [9] under the impulsive thrust approximation. [PERSON] et al. [10] found both impulsive and finite thrust, fuel-optimal rendezvous trajectories by using a particle swarm algorithm. Other works use alternative deterministic and heuristic methods for rendezvous optimization and guidance, such as the glideslope multipulse technique [11], optimally timed trajectory correction maneuvers [12], \(\mathcal{H}_{\infty}\) and \(\mu\)-synthesis techniques [13]. However, in actual operational scenarios, rendezvous trajectories are affected by orbit perturbations, which make both the HCW linear equations and the Keplerian nonlinear equations relatively inaccurate for precise orbit rendezvous. Moreover, the assumption of perfect information on the relative spacecraft dynamics is not realistic, and an accurate rendezvous is likely to be infeasible without an effective navigation system, due to measurement noise or unavailable variables needed to compute the guidance command. This circumstance points out the need of an integrated, iterative guidance and navigation architecture, capable of performing correction maneuvers on the basis of the estimated state provided by the navigation system. This paper analyzes the design of a relative guidance and navigation architecture for the rendezvous of two CubeSats, a target and a chaser, which are assumed to travel orbits in close proximity. The proposed guidance algorithm is based on the iterative application of the HCW linear equations of motion [14], written in terms of spherical coordinate displacements, under the assumption that both vehicles travel nearby orbits with small eccentricities. This simple approach aims at finding the correction maneuvers, modeled as impulsive velocity changes, that are necessary to maintain the chaser on track to the target. The orbital motion of both spacecraft is described by integrating the nonlinear equations of motion, which can include all the orbit perturbations relevant to the dynamical context. The iterative nature of the guidance strategy allows accounting for the update of the target estimated orbit, provided by the navigation state, which is obtained from the measurements of the on-board sensors.