Classification and Feasibility Assessment of Infinitely Many Iso-Impulse Three-Dimensional Trajectories

TL;DR

This work develops a comprehensive framework to classify and generate infinitely many iso-impulse, minimum-$\Delta v$ three-dimensional spacecraft trajectories under nonlinear two-body dynamics. It extends previous two-impulse results by introducing three-impulse base solutions and a generalized $\Delta v$-allocation across multiple impulse anchor positions, governed by a time-feasibility criterion. Trajectory solutions are organized into four layers—base solutions, feasible solution spaces, solution families, and solution envelopes—allowing analytic characterization of the complete iso-impulse solution space and enabling practical decisions for time-free, time-free rendezvous, and time-fixed rendezvous problems. The methodology is demonstrated on interplanetary Earth-to-Dionysus and geocentric transfers, yielding practical certificates of $\Delta v$ optimality and providing decision rules for base-solution selection and thruster-constrained impulse distribution, with envelopes visualizing the range of phasing-orbit periods involved. The results offer a fast, analytic pathway to design and certify minimum-$\Delta v$ impulsive transfers in 3D, informing mission planning and propellant budgeting in both interplanetary and planet-centric contexts.

Abstract

In two-body dynamics, it is proven that for a sufficiently long flight time, generating infinitely many iso-impulse solutions is possible by solving a number of $Δv$-allocation problems analytically. A distinct feature of these solutions is the existence of two impulse anchor positions (APs) that correspond to the locations of the impulses on time-free, phase-free, base solutions. In this paper, the existence and utility of three-impulse base solutions are investigated and their complete solution spaces are characterized and analyzed. Since two- and three-impulse base solutions exist, a question arises: How many APs should base solutions have? A strategy is developed for choosing base solutions, which offers a certificate for $Δv$ optimality of general three-dimensional time-fixed rendezvous solutions. Simultaneous allocation of $Δv$ at two and three APs is formulated, which allows for generating $Δv$-optimal solutions while satisfying a constraint on individual impulses such that $Δv \leq Δv_\text{max}$. All iso-impulse solutions are classified in four layers: 1) base solutions, 2) feasible solution spaces, 3) solution families, and 4) solution envelopes. The method enables us to characterize the complete solution space of minimum-$Δv$, iso-impulse, three-dimensional trajectories under the nonlinear two-body dynamics. To illustrate the utility of the method, interplanetary and geocentric problems are considered.

Figures (22)

• Figure 1: Earth-to-Dionysus phase-free two-impulse base solution and potential impulse APs ($t_{c_1} \neq 0, t_{c_2} \neq 0$).
• Figure 2: Schematics for the addition of one phasing orbit at the first AP on the Earth orbit in Fig. \ref{['fig:twoimp_base']}.
• Figure 3: A representative geocentric phase-free base solution with three potential impulse APs ($t_{c_1} = t_{c_2} = 0$).
• Figure 4: Circle-to-circle maneuvers: $\Delta v$-optimality for different types of maneuvers and number of impulses.
• Figure 5: Flowchart of selection of base solutions depending on the maneuver types.