Sims–Flanagan Transcription

• Sims–Flanagan Transcription is a direct optimization method for low-thrust trajectories that approximates continuous acceleration with impulsive delta-V increments and conic arcs.
• It divides missions into legs between planetary events and frames each leg as a discretized nonlinear programming problem aimed at maximizing final spacecraft mass.
• High-fidelity extensions improve the method by replacing impulsive approximations with continuous thrust integration and adaptive mesh timing via the Sundman transformation.

Sims–Flanagan transcription is a direct optimization method for low-thrust trajectories in which a low-thrust leg is approximated by a sequence of impulsive velocity increments connected by conic, that is, two-body, ballistic arcs. A mission is divided into legs between planetary events, and each leg is discretized into segments whose controls enter a nonlinear programming problem (NLP) with objective of maximizing final spacecraft mass. In the formulation discussed in "Towards a High Fidelity Direct Transcription Method for Optimisation of Low-Thrust Trajectories," the method is also the baseline from which a higher-fidelity extension is constructed by replacing impulsive segments with continuously thrusting dynamics and replacing a uniform time mesh with an adaptive mesh induced by a Sundman transformation (Yam et al., 2010).

1. Origins, scope, and role in low-thrust design

The contribution is positioned as a direct extension of Sims and Flanagan’s 1999 method and of ideas developed within ESA’s Advanced Concepts Team, including work presented at CELMEC V (Yam et al., 2010). Within that lineage, the classical Sims–Flanagan transcription is treated as a preliminary-design method whose main virtues are speed and robustness, while its principal weakness is low dynamical fidelity.

In this framework, a mission is decomposed into legs between planetary events. On each leg, the continuous low-thrust acceleration is replaced by $N$ impulses, one per segment, and the segments have equal duration in time. The decision-space dimension remains comparatively small because the control on each segment is represented by impulse magnitude and direction rather than by a dense control history.

The 2010 extension is framed not as an abandonment of the Sims–Flanagan philosophy, but as an evolution of it toward a transcription that can also generate operationally meaningful trajectories while keeping the decision-space dimension essentially unchanged (Yam et al., 2010). This suggests a bridging role between preliminary design and later-phase design: the broad segment-based, forward/backward, match-point architecture is preserved, while the local dynamical model and mesh definition are changed.

2. Classical formulation

For a leg starting at time $T_0$ and ending at time $T_f$, the classical transcription assigns equal time duration to each segment,

$\Delta t = \frac{T_f - T_0}{N}.$

The maximum admissible impulse magnitude per segment is derived from the maximum engine thrust and the segment duration:

$$\Delta V_{max} = ( F_{max}/m) (T_f - T_0) / N. \tag{1}$$

Mass is propagated segment by segment using the rocket equation:

$$m_{i+1} = m_i \exp\!\left(-\frac{\Delta V_i}{g_0 I_{sp}}\right). \tag{2}$$

The transcription is based on a multiple-shooting-like construction. For each leg, the state is propagated forward from the departure boundary to a match point, usually at mid-leg, and backward from the arrival boundary to the same match point. Between impulses, propagation uses a two-body model. The spacecraft state is

$S = \{r_x,r_y,r_z,v_x,v_y,v_z,m\}.$

With forward and backward match-point states denoted by $S_{mf}$ and $S_{mb}$, the mismatch vector is

$$S_{mf} - S_{mb} = \left\{\Delta r_x,\Delta r_y,\Delta r_z,\Delta v_x,\Delta v_y,\Delta v_z, \Delta m \right\}. \tag{3}$$

A feasible trajectory requires this mismatch to be below a tolerance. In practical NLP form, these are the nonlinear constraints enforcing that the forward and backward half-legs meet (Yam et al., 2010).

The decision variables listed for the original Sims–Flanagan NLP are: departure epoch $T_0$0; departure hyperbolic excess velocity $T_0$1 relative to Earth; for each leg and each segment, impulse magnitude and direction; for each swingby, incoming and outgoing hyperbolic excess velocities relative to the planet; for each swingby $T_0$2, swingby epoch $T_0$3; and arrival epoch $T_0$4. For a rendezvous mission, arrival relative velocity is not a decision variable because it is zero by construction. The constraints are impulse magnitude bounds, match-point constraints, and boundary conditions. The objective is to maximize final spacecraft mass, equivalently a minimum-propellant problem under fixed or partially free timing and boundary conditions. The NLP is solved with SNOPT using SQP.

3. Identified limitations of the impulsive transcription

Two principal deficiencies are identified in the classical method (Yam et al., 2010). First, the use of impulsive $T_0$5s to represent low-thrust propulsion can be too crude, because a real low-thrust spacecraft applies continuous acceleration over finite time rather than an instantaneous change in velocity. The conic-arc plus impulse approximation may therefore fail to represent the true trajectory dynamics accurately unless the number of impulses is made large.

Second, the use of an insufficient number of segments reduces fidelity and can also reduce optimality. If the number of impulses is increased to better approximate the continuous-thrust dynamics, the NLP becomes larger and slower. The original method therefore trades realism for speed and robustness.

The paper explicitly states that the Sims–Flanagan method has been mainly limited to preliminary design because the impulsive transcription is not expected to be accurate at the operational level. The practical consequences listed are that the optimized trajectory may not satisfy the real continuous-thrust dynamics, the resulting control history may be non-operational, more impulses are needed to approximate fast-varying trajectory regions particularly near the central body where speed is higher, and increasing the number of segments slows optimization.

The Earth–Mercury example makes this limitation visually explicit: with only 30 segments, the impulsive solution exhibits large gaps between impulses on a fast-rotating trajectory, making it an unfair representation of the actual low-thrust path (Yam et al., 2010). A common misconception is that a better numerical objective value in the impulsive transcription necessarily indicates a better trajectory. The numerical results discussed later contradict that interpretation, because the optimistic impulsive approximation can yield a higher final mass while remaining infeasible with respect to the real continuous-thrust dynamics.

4. High-fidelity extensions

The higher-fidelity transcription introduces exactly two improvements (Yam et al., 2010).

The first is replacement of impulsive segment models by continuously thrusting segment dynamics. Instead of applying an impulsive $T_0$6 at each segment and propagating with Keplerian conic arcs between impulses, the method uses a continuous thrust vector that is constant over the segment and integrates the spacecraft equations of motion over each segment. The thrust magnitude and direction are optimization variables. This replaces the impulse approximation by finite-duration thrust arcs and allows the propagation to include perturbations if desired.

The second is online mesh optimization using the Sundman transformation,

$T_0$7

where $T_0$8 is the distance from the central body. The leg is then divided into equal $T_0$9 segments rather than equal $T_f$0 segments. Since

$T_f$1

segments correspond to shorter times when $T_f$2 is small and longer times when $T_f$3 is large. The mesh therefore becomes denser near the central body, where spacecraft speed is usually larger and more control resolution is useful.

This online mesh adaptation differs from the standard setup in that segment durations in time are no longer fixed or equal a priori. Instead, the optimizer determines the trajectory in $T_f$4-space, and the implied time spacing adapts automatically with the radial distance along the current iterate. The paper emphasizes that this costs only one additional variable and one additional constraint per leg (Yam et al., 2010).

Taylor integration is introduced as enabling numerical machinery for the continuous-thrust segment model, not as a third conceptual transcription change. That distinction is important: the conceptual modifications concern the segment dynamics and the mesh, whereas Taylor integration addresses the computational burden of repeated finite-burn propagation inside the optimizer.

5. Mathematical structure and NLP realization

For algorithmic comparison of integrators, the paper considers a nondimensional fixed-thrust problem:

$T_f$5

where $T_f$6 are constant thrust accelerations over the segment. Under the Sundman transformation, these equations become

$T_f$7

where derivatives are with respect to $T_f$8.

For the higher-fidelity trajectory model in physical units, the continuous-thrust equations in time-space are

$T_f$9

with constant $\Delta t = \frac{T_f - T_0}{N}.$0 over each segment. The segment-wise controls are the Cartesian thrust components $\Delta t = \frac{T_f - T_0}{N}.$1. The natural path constraint implied by the original maximum-thrust model is

$\Delta t = \frac{T_f - T_0}{N}.$2

Applying the Sundman transformation yields the $\Delta t = \frac{T_f - T_0}{N}.$3-space equations

$\Delta t = \frac{T_f - T_0}{N}.$4

where all derivatives are with respect to $\Delta t = \frac{T_f - T_0}{N}.$5. The relation between $\Delta t = \frac{T_f - T_0}{N}.$6-span and time of flight is enforced through

$\Delta t = \frac{T_f - T_0}{N}.$7

The implementation assumes $\Delta t = \frac{T_f - T_0}{N}.$8 and solves for $\Delta t = \frac{T_f - T_0}{N}.$9 so that Eq. (8) is satisfied. Each leg therefore acquires one additional variable $$\Delta V_{max} = ( F_{max}/m) (T_f - T_0) / N. \tag{1}$$0 and one additional constraint enforcing time-of-flight consistency (Yam et al., 2010).

The forward/backward match-point defect concept remains unchanged in structure. One integrates from each end to the midpoint and enforces coincidence of the state vectors, so the defect constraints remain nonlinear equality constraints, but they are now generated by finite-burn propagation instead of impulse-plus-conic propagation. The NLP is clearly of the form

$$\Delta V_{max} = ( F_{max}/m) (T_f - T_0) / N. \tag{1}$$1

Here $$\Delta V_{max} = ( F_{max}/m) (T_f - T_0) / N. \tag{1}$$2 includes epochs, swingby variables where present, and segment controls. The paper does not introduce switching functions, normalized controls, or alternative control parameterizations beyond piecewise-constant Cartesian thrust vectors.

6. Computation, validation, and interpretation

Because the continuous-thrust model requires numerical integration repeatedly inside the optimizer, computational efficiency is critical. The paper compares Runge–Kutta–Fehlberg integrators with a Taylor integration scheme on 10,000 randomly generated fixed-thrust Cauchy problems (Yam et al., 2010). Accuracy is assessed by forward propagation for a random final time $$\Delta V_{max} = ( F_{max}/m) (T_f - T_0) / N. \tag{1}$$3, then backward propagation for the same duration, with real propagation error defined as

$$\Delta V_{max} = ( F_{max}/m) (T_f - T_0) / N. \tag{1}$$4

The reported result is that Taylor integration provides roughly an order-of-magnitude speed gain while maintaining or improving accuracy. At $$\Delta V_{max} = ( F_{max}/m) (T_f - T_0) / N. \tag{1}$$5, the paper reports: RKF 5(6), speed $$\Delta V_{max} = ( F_{max}/m) (T_f - T_0) / N. \tag{1}$$6 s and max error $$\Delta V_{max} = ( F_{max}/m) (T_f - T_0) / N. \tag{1}$$7; RKF 7(8), speed $$\Delta V_{max} = ( F_{max}/m) (T_f - T_0) / N. \tag{1}$$8 s and max error $$\Delta V_{max} = ( F_{max}/m) (T_f - T_0) / N. \tag{1}$$9; and Taylor, speed $$m_{i+1} = m_i \exp\!\left(-\frac{\Delta V_i}{g_0 I_{sp}}\right). \tag{2}$$0 s and max error $$m_{i+1} = m_i \exp\!\left(-\frac{\Delta V_i}{g_0 I_{sp}}\right). \tag{2}$$1. The empirical recommendation is $$m_{i+1} = m_i \exp\!\left(-\frac{\Delta V_i}{g_0 I_{sp}}\right). \tag{2}$$2 as a good speed/accuracy compromise for the Taylor integrator.

Validation is carried out on an Earth–Mercury rendezvous mission with initial mass $$m_{i+1} = m_i \exp\!\left(-\frac{\Delta V_i}{g_0 I_{sp}}\right). \tag{2}$$3 kg, maximum thrust $$m_{i+1} = m_i \exp\!\left(-\frac{\Delta V_i}{g_0 I_{sp}}\right). \tag{2}$$4 mN, specific impulse $$m_{i+1} = m_i \exp\!\left(-\frac{\Delta V_i}{g_0 I_{sp}}\right). \tag{2}$$5 s, launch date Apr. 9, 2007, arrival date Aug. 22, 2013, launch $$m_{i+1} = m_i \exp\!\left(-\frac{\Delta V_i}{g_0 I_{sp}}\right). \tag{2}$$6 km/s, and time of flight $$m_{i+1} = m_i \exp\!\left(-\frac{\Delta V_i}{g_0 I_{sp}}\right). \tag{2}$$7 years. For this example, thrust and specific impulse are assumed constant rather than using a full solar electric propulsion model (Yam et al., 2010).

Method	No. of Variables	Optimal Final Mass (kg)
Impulsive	97	392.9
Continuous $$m_{i+1} = m_i \exp\!\left(-\frac{\Delta V_i}{g_0 I_{sp}}\right). \tag{2}$$8-space	97	382.4
Continuous $$m_{i+1} = m_i \exp\!\left(-\frac{\Delta V_i}{g_0 I_{sp}}\right). \tag{2}$$9-space	98	387.0

Several interpretive points are emphasized. The impulsive method produces the highest final mass numerically, but only the continuous trajectories are feasible with respect to the real dynamical model and therefore usable into later design phases. The continuous time-space solution has lower final mass because it respects the actual continuous-thrust dynamics, so the optimistic impulsive approximation is removed. The continuous $S = \{r_x,r_y,r_z,v_x,v_y,v_z,m\}.$0-space solution improves final mass relative to continuous time-space,

$S = \{r_x,r_y,r_z,v_x,v_y,v_z,m\}.$1

showing that adaptive segment timing improves optimality. The variable count remains almost unchanged—97 variables for impulsive and continuous time-space, 98 for continuous $S = \{r_x,r_y,r_z,v_x,v_y,v_z,m\}.$2-space—confirming the claim that fidelity is improved without materially increasing problem dimension.

The figures discussed in the paper qualitatively validate the approach. The impulsive trajectory shows visible gaps between impulsive corrections, especially problematic for the fast-rotating Mercury transfer; the continuous time-space method fills the gap with actual low-thrust arcs; and the continuous $S = \{r_x,r_y,r_z,v_x,v_y,v_z,m\}.$3-space method uses smaller segment durations near Mercury than near Earth, demonstrating online mesh adaptation (Yam et al., 2010).

In implementation terms, the method preserves the familiar segment-based forward/backward match-point structure while changing the control parameterization from impulse magnitude and direction to constant thrust vector components $S = \{r_x,r_y,r_z,v_x,v_y,v_z,m\}.$4, replacing analytical Kepler propagation with numerical integration of finite-burn dynamics, replacing discrete rocket-equation updates with continuous mass depletion, and, in $S = \{r_x,r_y,r_z,v_x,v_y,v_z,m\}.$5-space, adding $S = \{r_x,r_y,r_z,v_x,v_y,v_z,m\}.$6 as a propagated state with $S = \{r_x,r_y,r_z,v_x,v_y,v_z,m\}.$7 together with the integral consistency constraint of Eq. (8). The tradeoffs are explicit: impulsive transcription is fast, robust, and low-dimensional but low-fidelity and optimistic; continuous thrust in time-space is physically realistic but more expensive; continuous thrust in $S = \{r_x,r_y,r_z,v_x,v_y,v_z,m\}.$8-space adds adaptive segment timing and improved optimality at the cost of one extra variable and one extra constraint per leg.