Shape-Based Method and Velocity Hodograph

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

A shape-based method is an approximate analytical design technique for low-thrust trajectories: an analytical function with a few free parameters (the shape function) describes the trajectory geometry (typically the polar radius $r(\theta)$ or the velocity vector $\boldsymbol{v}(t)$ ), and inverse dynamics recovers the thrust acceleration $\boldsymbol{u}(t)$ and time of flight from the shape. It does not solve the full optimal control problem; instead it reduces an infinite-dimensional trajectory optimization to a finite-dimensional parameter optimization. It is fast and well-suited to global search, but the achievable shapes are limited by the chosen function family, so the solution is generally not truly optimal (Petropoulos & Longuski 2004; Conway 2010; Vellutini & Avanzini 2014).

Engineering role: shape-based methods are initial-guess generators and global pruners: rapidly producing near-feasible transfers among many candidates, then handing them off to direct collocation, HDDP, or indirect methods for refinement.

The Petropoulos Exponential Sinusoid

Under a central gravity field with tangential thrust, Petropoulos (Petropoulos & Longuski 2004) found that the exponential sinusoid

r(\theta) = k_0\,\exp\!\big[k_1\sin(k_2\theta+\phi)\big]

analytically satisfies the planar equations of motion when $\alpha=\gamma$ (thrust aligned with velocity). Substituting gives closed-form angular rate and thrust acceleration

\dot\theta^2 = \frac{\mu/r^3}{\tan^2\gamma + k_1 k_2^2 s + 1},\qquad a = \frac{\tan\gamma}{2\cos\gamma}\!\left[\frac{1}{\tan^2\gamma + k_1 k_2^2 s + 1} - \frac{k_2^2(1-2k_1 s)}{(\tan^2\gamma + k_1 k_2^2 s + 1)^2}\right]

where $\tan\gamma=k_1 k_2\cos(k_2\theta+\phi)$ and $s=\sin(k_2\theta+\phi)$ .

Feasibility condition $|k_1 k_2^2|<1$ . When $|k_1 k_2^2|\ge 1$ , $\dot\theta^2$ becomes negative for some $\theta$ (imaginary angular rate) or $a$ diverges; the shape is unflyable. This is the most important constraint of shape-based methods.

Izzo's multi-revolution Lambert analogue (2006). Given $r_1, r_2, \Delta\theta$ and a transfer time $t_f$ , the multi-revolution Lambert problem generalizes to exponential sinusoids: with $k_2$ fixed, a one-parameter family (parameterized by initial flight-path angle $\gamma_1$ ) passes through both endpoints; feasibility reduces to a quadratic inequality in $\tan\gamma_1$ , analytically bounding the feasible interval. This is a low-thrust Lambert solver for global optimization.

Other Shape Function Families

Inverse polynomial (Wall & Conway). Wall and Conway (2010) proposed fifth- and sixth-order inverse polynomials $r(\theta)=1/\sum_{i} a_i\theta^i$ , with more parameters and the ability to fit more general trajectories (including non-tangential thrust components), suitable for fixed-time rendezvous and interception. The price is the curse of dimensionality: grid search cost grows with order.

Velocity hodograph method. Represent the velocity vector $\boldsymbol{v}(t)$ (rather than position) as a shape function of time or true anomaly; recover position via $\dot{\boldsymbol{r}}=\boldsymbol{v}$ and thrust via the equations of motion. The advantage is direct handling of velocity boundary conditions, useful for rendezvous with velocity matching; the method has been applied to Earth–Mars, Earth–Mercury, asteroid and comet missions.

Forbes spiral, Lawden spiral, logarithmic spiral. Classic analytical families with fewer parameters and limited expressiveness, mostly used for teaching and quick screening.

CR3BP-Modified Exponential Sinusoid (Vellutini & Avanzini 2014)

The classical exponential sinusoid assumes a single central body; for Earth–Moon $L_1$ transfers lunar gravity is non-negligible. Vellutini & Avanzini augment the sinusoid with a deformation along the primary line

r(\theta) = k_0\,\exp\!\big[k_1\sin(k_2\theta+\phi) + k_3\theta\cos(k_4\theta+\phi)\big],

where the amplitude $k_3\theta$ of the cosine term grows linearly with angular travel, deforming the spiral toward the Moon as it moves away from Earth. The thrust profile is re-derived under the CR3BP equations (still with tangential thrust), yielding shorter transfer times and lower $\Delta V$ than the classical form for the same boundary conditions. The method extends to Sun–Earth $L_1$ and $L_2$ by replacing $\mu$ with $1-\mu$ .

Amplitude-Phase Shape Functions for Libration-Point Periodic Orbits

For low-thrust transfers between libration-point periodic orbits (Halo, Lissajous), Chinese-language literature has introduced shape functions in which amplitude and phase vary polynomially, embedding the dynamics of the collinear libration neighbourhood (center manifold, zero-velocity surfaces) into the parametrization. These express dynamically meaningful transfers with fewer parameters than generic shapes; the resulting initial guess is refined by Gauss pseudospectral methods or HDDP.

Practical Notes

Tangential thrust is the central simplification. Shape-based methods typically assume $\alpha=\gamma$ , eliminating thrust-direction freedom to obtain a closed form. When the true optimal has significant radial component (e.g. plane changes, de-orbit burns), shape-based solutions can be far from optimal.
Check feasibility first. $|k_1 k_2^2|<1$ is necessary; analogous conditions exist for other shape families.
Most effective as an initial-guess generator. Shape methods are most useful for producing feasible seeds for direct/indirect refinement; do not treat them as final engineering solutions.
Multi-revolution: scan over $k_2$ . Since $k_2$ controls the number of revolutions, global optimization often fixes $k_2$ , optimizes the remaining parameters, then changes $k_2$ and repeats.
Re-derive under CR3BP. Directly importing two-body shapes into the CR3BP distorts the result; a dual-primary correction term (as in Vellutini) and a re-derived thrust profile are required.

References

Petropoulos, A. E., Longuski, J. M., 2004, "Shape-based algorithm for the automated design of low-thrust, gravity-assist trajectories," J. Spacecr. Rockets (the founding paper on exponential sinusoids).
Izzo, D., 2006, "Lambert's problem for exponential sinusoids," J. Guid. Control Dyn., DOI: 10.2514/1.21796 (the multi-revolution low-thrust Lambert problem and the $\tan\gamma_1$ feasibility interval).
Wall, B. J., Conway, B. A., 2010, "Shape-based approach to low-thrust rendezvous trajectory design," J. Guid. Control Dyn. (inverse-polynomial shape functions).
Vellutini, E., Avanzini, G., 2014, "Shape-Based Design of Low-Thrust Trajectories to Cislunar Lagrangian Point," J. Guid. Control Dyn., DOI: 10.2514/1.G000165 (CR3BP-modified exponential sinusoid for Earth–Moon $L_1$ ).
Conway, B. A. (ed.), 2010, Spacecraft Trajectory Optimization (textbook, chapter on shape-based methods).
Research on Gauss pseudospectral methods for low-thrust transfers between libration-point periodic orbits (amplitude-phase shape functions).