--- title: 'PyKEP: Open-Source Toolkit for Interplanetary Design' url: https://www.emergentmind.com/topics/pykep type: topic --- # PyKEP: Open-Source Toolkit for Interplanetary Design Searching arXiv for the specified paper to ground the response. PyKEP is an open-source library focused on preliminary interplanetary mission design. In the treatment centered on "Designing Complex Interplanetary Trajectories for the Global Trajectory Optimization Competitions" [1511.00821], it is presented as a software environment that wraps fast, well-tested astrodynamics primitives and transcriptions to quickly assemble trajectory building blocks, evaluate feasibility, and feed optimization or tree-search algorithms. The chapter uses PyKEP both to introduce fundamental problems of space flight mechanics and to formulate a multiple asteroid rendezvous strategy related to the 7th edition of the Global Trajectory Optimization Competition, with particular emphasis on Lambert transfers, low-thrust transcriptions, phasing indicators, clustering, and beam-search-based global exploration [1511.00821]. ## 1. Position within preliminary interplanetary design PyKEP is used for a typical preliminary-design workflow in which bodies, ephemerides, and epochs are first defined; transfer legs are then computed via Lambert methods under impulsive assumptions or via transcribed low-thrust arcs; $\Delta v$, mass, and time budgets are sized; coarse grids such as porkchops are generated to identify windows; and a global search is performed using beam search, tree search, or heuristics guided by surrogate metrics, after which promising sequences are refined [1511.00821]. Within that workflow, the chapter employs PyKEP to query ephemerides and simple planet models, including JPL low-precision models for Earth and Mars; solve the Kepler initial value problem with Lagrange coefficients through `propagate_lagrangian`; solve Lambert’s boundary value problem through `lambert_problem`, including multi-revolution branches when desired; numerically propagate constant-thrust arcs with a Taylor-series integrator through `propagate_taylor`; transcribe and evaluate low-thrust optimal control problems with the Sims–Flanagan method through `sims_flanagan` spacecraft and leg objects; compute and use phasing indicators and clustering; and orchestrate tree searches, including beam search and MOBS, to generate, rank, and prune trajectory candidates [1511.00821]. The chapter also notes that PyKEP ships tools for gravity-assist sequence exploration, such as MGA and MGA-1DS, and that porkchop-style grid generation can be built by sweeping Lambert solves across departure and arrival grids. These are described as typical preliminary-design tools even though they are not the main focus of the chapter. A plausible implication is that the library is intended less as a monolithic mission optimizer than as a composable toolkit for rapid assembly of trajectory-analysis pipelines. ## 2. Core astrodynamics primitives and transcriptions A central building block is Lambert’s problem, formulated as a two-body boundary-value problem, $$ \ddot{\mathbf r} = -\mu \frac{\mathbf r}{r^3},\quad \mathbf r(t_s) = \mathbf r_s,\quad \mathbf r(t_f) = \mathbf r_f. $$ The inputs are $\mu$, $\mathbf r_s$, $\mathbf r_f$, $T=t_f-t_s$, the transfer geometry, and the multi-revolution index; the outputs are the departure and arrival velocities $v_s^\ast=v_1$ and $v_f^\ast=v_2$, with possibly multiple branches, including $0$-rev and higher-revolution solutions, and prograde or retrograde cases. PyKEP exposes this through the pattern ```python l = lambert_problem(rs, rf, tof, mu, is_retrograde, mr_max) v1 = l.get_v1()[i] v2 = l.get_v2()[i] ``` and uses Izzo’s solver. For zero-revolution transfers, the chapter reports near-constant runtime across geometries; in the chapter’s measurements, a $0$-rev Lambert problem took approximately $40\,\mu s$ on a single thread of an Intel i7-4600U, corresponding to roughly $250$k Lambert problems per second [1511.00821]. For unperturbed two-body propagation, PyKEP wraps the Lagrange $f$–$g$ formulation, $$ \mathbf r_2 = f(\Delta t)\,\mathbf r_1 + g(\Delta t)\,\mathbf v_1,\qquad \mathbf v_2 = \dot f(\Delta t)\,\mathbf r_1 + \dot g(\Delta t)\,\mathbf v_1, $$ via `propagate_lagrangian(rs, vs, dt, mu)`. The coefficients depend on $\mu$, the universal variable $\chi$, the Stumpff functions $c_2(\chi)$ and $c_3(\chi)$, and the geometry, allowing propagation without explicit numerical integration for the unperturbed two-body motion. For constant-thrust propagation, the chapter uses `propagate_taylor`, described as a Taylor-series integrator for perturbed Keplerian motion with constant thrust and mass flow. For low-thrust optimal control problems, the principal transcription is the Sims–Flanagan method. The chapter’s representative pattern constructs a `sims_flanagan.spacecraft`, defines initial and final spacecraft states through `sims_flanagan.sc_state`, and builds a `sims_flanagan.leg`; mismatch constraints and throttle constraints are then evaluated through `leg.mismatch_constraints()` and `leg.throttles_constraints()`. This makes the transcription directly usable as a feasibility filter or as a component in broader search strategies [1511.00821]. These components are used as interoperable primitives rather than as isolated solvers. Lambert arcs support impulsive screening and seeding, while Sims–Flanagan legs support low-thrust feasibility analysis and resource accounting. This suggests that PyKEP’s design emphasizes high-throughput evaluation of candidate transfers under preliminary-design assumptions. ## 3. Mechanical relations and trajectory evaluation The chapter places PyKEP’s software abstractions within standard space-flight mechanics. In addition to Lambert’s boundary-value problem and the Kepler initial value problem, $$ \ddot{\mathbf r} = - \frac{\mu}{r^3}\mathbf r,\quad \mathbf r(t_s) = \mathbf r_s,\quad \mathbf v(t_s) = \mathbf v_s, $$ it recalls elliptic Kepler’s equation, $$ M = E - e \sin E, $$ and the vis-viva relation, $$ v^2 = \mu \left( \frac{2}{r} - \frac{1}{a} \right). $$ These are described as relations used internally by `lambert_problem` and ephemerides routines and as underpinning analytics in two-body propagation and orbital energy checks [1511.00821]. Mass and velocity budgeting are organized around the Tsiolkovsky rocket equation, $$ \Delta v = I_{sp} g_0 \ln\!\left(\frac{m_0}{m_f}\right), $$ and, for constant thrust, $$ \dot m = -\frac{T}{I_{sp} g_0}. $$ For impulsive Lambert legs, $\Delta v$ accounting is performed by matching encounter relative velocities: $$ \Delta V = \Delta V_1 + \Delta V_2,\quad \Delta V_1 = \|v_1 - v_{dep}\|,\quad \Delta V_2 = \|v_{arr} - v_2\|. $$ For low-thrust legs, mass flow enters the Sims–Flanagan and Taylor propagators directly. The chapter also describes porkchop data generation as a grid over departure times $t_s$ and arrival times $t_f$, with metrics such as $C_3$, $\Delta v$, and time of flight. PyKEP facilitates these grids by looping ephemerides queries and Lambert solves. The concrete examples use `planet.jpl_lp('earth')`, `planet.jpl_lp('mars')`, and epochs in `mjd2000`, with explicit conversion of time of flight to seconds before calling `lambert_problem`. The text explicitly warns that misaligned units, for example mixing days and seconds, will corrupt time of flight and $\Delta v$; it also notes that the chapter’s examples often use non-dimensional units inside PyKEP and therefore require consistent unit handling when mixing m/s, AU/day, and related conventions [1511.00821]. A common misconception in preliminary mission design is that coarse dynamical models are intrinsically inappropriate for algorithmic search. The chapter instead adopts heliocentric two-body dynamics, Keplerian asteroid ephemerides, and constant-thrust arcs for the preliminary phase. This does not assert sufficiency for final flight design; rather, it identifies a deliberate trade between dynamical fidelity and tractable global exploration. ## 4. GTOC7-inspired multiple-asteroid rendezvous formulation The chapter’s principal case study is a multiple asteroid rendezvous problem related to GTOC7. The target set contains $N=16256$ main-belt asteroids with Keplerian ephemerides. The spacecraft parameters are $m_0=2000\,\mathrm{kg}$, propellant mass $m_p=1200\,\mathrm{kg}$, $T_{max}=0.3\,\mathrm{N}$, and $I_{sp}=3000\,\mathrm{s}$. The mission duration is limited to $\mathrm{tof}=6$ years, with a minimum stay time per asteroid of $t_w=30$ days. The start date lies in $t_0 \in [7500,12000]$ MJD2000, and the spacecraft can be placed on a chosen starting asteroid at a chosen date. Rendezvous requires perfect position and velocity match; the dynamics are heliocentric two-body; and low-thrust propulsion is used for transfers, evaluated through Sims–Flanagan and Taylor propagation. The stated objective is to maximize the number of asteroid rendezvous under the remaining time and remaining mass resources [1511.00821]. The global-search strategy is MOBS, a multi-objective beam search. A node is defined by the sequence of visited asteroids together with the non-dimensional remaining resources $$ r_t = 1 - \frac{t-t_0}{\mathrm{tof}},\qquad r_m = \frac{m-m_s}{m_p}. $$ At each node, the branch procedure first selects up to $BF$ candidate next asteroids through $k$-nearest neighbors in the orbital-indicator metric $d_o$; for each candidate, it then solves an earliest-arrival low-thrust optimal control problem to obtain the earliest feasible arrival time $t^\ast$; if such a solution exists, it solves a set of fixed-arrival minimum-mass optimal control problems with $$ t_f = t^\ast + i\cdot dt,\qquad i=1,\dots,\ell, $$ to capture feasible variants; and each feasible solution yields updated $r_t$ and $r_m$ and therefore an extended node [1511.00821]. Nodes at the same depth are ranked through a node value $r$. The options tested in the chapter are best mass $r_1=r_m$, best time $r_2=r_t$, the soft minimum $$ r_4 = \frac{r_m r_t}{r_m + r_t}, $$ and Pareto dominance $r_6$. The beam keeps the best $BS$ nodes for further expansion. The compact pseudocode in the chapter is: ```text Procedure MOBS(t0, A0): B = { [[A0], 1, 1] }; best = [[A0], 1, 1] while B not empty: L = ∅ for N in B: L = L ∪ Branch(N) # up to BF new nodes best = UpdateBest(L) B = Beam(L) # select up to BS nodes return best ``` The constraints handled in this formulation include departure and arrival windows $t_s, t_f \in [t_0, t_0+T_M]$ when computing phasing, mission-horizon tracking through $r_t$ across legs, and enforcement of the minimum stay time $t_w$ between rendezvous events. Maximum time of flight per leg is enforced implicitly by the earliest-arrival and fixed-arrival checks. The chapter also notes that launch $C_3$ limits are not central to this belt-mission setup because the spacecraft is delivered to the starting asteroid [1511.00821]. ## 5. Phasing value, surrogates, and clustering The chapter introduces the phasing value as a multi-objective measure of the transfer quality between asteroids. With the spacecraft $S$ at asteroid $A_s$ at time $t_0$, Lambert problems are considered to a target asteroid $A_f$ over $(t_s,t_f)\in[t_0,t_0+T_M]$, with thrust feasibility constrained by $$ \Delta V \le \alpha_M \Delta T,\qquad \alpha_M = \frac{\tau_M}{m_s},\qquad \Delta T=t_f-t_s. $$ The associated objectives are $$ f_1 = \Delta V,\qquad f_2 = t_f, $$ subject to $t_s