Closed-Form Solution of the l1-Optimal Bi-Impulsive Coplanar Orbital Transfer Problem

Abstract

S ATELLITE orbital transfer has been studied based on two main approaches: impulsive maneuvers and continuous-thrust maneuvers. Impulsive models, in spite of being approximate, are important because they enable mission designers to obtain a preliminary fuel budget as well as maneuver scheduling. Coplanar maneuvers usually consume less propellant than more general transfers wherein the orbital planes are modified. In this context, bi-impulsive maneuvers, such as the Hohmann transfer, are useful because they require a minimum number of thruster firings for each transfer between nonintersecting orbits. Under certain conditions, Hohmann transfers are optimal in the sense of minimizing the total velocity change. However, Hohmann transfers are designed to transfer a satellite between two collinear position vectors. When the initial and final position vectors are not collinear, one should consider other strategies. Several authors have studied the problem of optimal impulsive transfers. Munick et al. [1] provided a solution for optimal transfers from elliptic orbits to coplanar nonintersecting circular orbits using two velocity impulses. Lawden [2] derived a set of algebraic equations whose solution yields the optimal bi-impulsive transfer between two coplanar elliptic orbits. The obtained set of equations must be numerically solved. Battin [3] wrote an ‘2 cost function for a bi-impulsive transfer between twopositionvectors, assuming that the initial and final position and velocity vectors are all coplanar. The minimization of this cost function relies on numerical methods, since an eight-order algebraic equation must be solved. Battin’s development leads to necessary conditions, but no closed-form expressions for the optimal solution are provided. Broucke and Prado [4] studied the optimal coplanar transfer problem with two, three, and four impulses. They also obtained sets of algebraic equations that need to be numerically solved. Won [5] derived analytical conditions for the bi-impulsive transfer between elliptic coplanar coaxial orbits that minimize either the required velocity change or the transfer time. Ma et al. [6] developed an approach using the eccentricity vector, and derived equations to obtain the optimal bi-impulsive transfer between coplanar orbits. As in the previouslymentioned cases, the equationsmust be numerically solved.Conway and Pontani [7] derived optimal four-impulse orbital rendezvous maneuvers when both the chaser and target are in the same circular orbit with an initial phase difference of rad. They obtained general expressions for the change of velocity required by each impulse, and numerically optimized the total velocity change. An implicit assumption in most of the aforementioned works is that the propellant consumption is measured by the Euclidean norm of the velocity change vectors. This is appropriate if the satellite uses a single thruster to perform themaneuver, i.e., it regulates the attitude to align the nozzle with the required vector thrust. However, as also noted by others [8], in satellites that use multiple body-fixed orthogonal thrusters the mass consumption is measured by the ‘1-norm of the thrust vector rather than Euclidean norm. Thus, if Euclidean norms were used to find the fuel-optimal transfers for satellites equipped with body-fixed orthogonal thrusters (e.g., the Mango satellite in the PRISMA mission [9]), a nonfuel-optimal maneuver would be determined. Therefore, for satellites using orthogonal thrusters the optimization approach should be modified. With this consideration in mind, this work formulates the ‘1 cost function for a general coplanar transfer between two noncollinear position vectors with prescribed terminal velocities, and analytically determines the set of points that satisfy the necessary conditions for minima. Since this set includes only 14 points, it is easy to evaluate all of theminima candidates and obtain the globally optimal bi-impulsive maneuver for the mentioned scenario, avoiding numerical optimization processes. The derived analytical solution enables real-time onboard implementation.