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

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 $$\Delta v$$ -allocation problems analytically. A distinct feature of these iso-impulse 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 $$\Delta v$$ optimality of general three-dimensional time-fixed rendezvous solutions. Simultaneous allocation of $$\Delta v$$ at two and three APs is formulated, which allows for generating $$\Delta v$$ -optimal solutions while satisfying a constraint on individual impulses such that $$\Delta v \le \Delta v_\text {max}$$ . A key result is that all iso-impulse solutions can be classified into 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- $$\Delta v$$ , iso-impulse, and three-dimensional trajectories under nonlinear two-body dynamics. To illustrate the utility of the method, geocentric examples are considered including the apogee-raising phase of the CAPSTONE mission subject to an operational constraint on the maximum magnitude of the individual impulses.