Control Parametrization, B-Spline, Spherical Variables and Throttle

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

Definition

Control parametrization expresses the infinite-dimensional continuous-thrust control $\boldsymbol{u}(t)$ as a finite set of parameters. Writing $\boldsymbol{u}(t) = \sum_{i} c_i\,\phi_i(t)$ where $\{\phi_i\}$ is a chosen basis family and $\{c_i\}$ are coefficients to be optimized, the optimal control problem becomes a nonlinear program (NLP) in $\{c_i\}$ . Common basis families are polynomials, Fourier series, and B-splines. The throttle factor / throttle function is a scalar/functional parametrization specific to low-thrust engines, describing how thrust magnitude varies with time.

Polynomial and Fourier Bases

The simplest parametrizations. The polynomial basis $\phi_i(t) = t^i$ is suitable for smooth control but becomes ill-conditioned at high order; Chebyshev or Legendre polynomials improve conditioning. The Fourier basis $\phi_i(t)=\cos(i\omega t), \sin(i\omega t)$ is well suited to periodic control such as libration-point orbit keeping. Both are globally supported: changing one coefficient affects the entire control profile, making local features such as thrust switches hard to express.

B-Spline Parametrization

B-splines use piecewise polynomials with compactly supported basis functions and are the most common parametrization in trajectory optimization. Write

\boldsymbol{u}(t) = \sum_{i} \boldsymbol{P}_i\,B_{i,q}(t),

where $\boldsymbol{P}_i$ are control points (optimization variables) and $B_{i,q}(t)$ are $q$ -th order ( $q-1$ degree) B-spline basis functions. Key properties:

Compact support. Each $B_{i,q}$ is nonzero only on $q$ knot intervals; adjusting one control point affects the profile only locally — an advantage over global bases when the control contains switches.
Continuity order. With non-repeated interior knots, a $q$ -th order B-spline is $C^{q-1}$ ; repeating a knot lowers continuity at that point, useful for representing discontinuities such as mode switches in multi-mode propulsion.
Convex hull. The curve lies within the convex hull of its control points, which makes thrust-magnitude constraints easy to impose.

Sanchez et al. (2020) apply B-splines to trajectory optimization and attitude control, often combined with direct collocation or HDDP.

Spherical Control Variables

For CR3BP low-thrust optimization, HDDP (Aziz et al. 2019) uses a spherical parametrization, decomposing thrust into magnitude $T$ and two steering angles $\alpha,\beta$

\boldsymbol{u} = [T,\alpha,\beta]^T,

with angles defined in a radial–transverse–normal (RSW) frame and mapped to the synodic frame by

\begin{bmatrix}T_r\\T_s\\T_w\end{bmatrix} = \begin{bmatrix} T\sin\alpha\cos\beta\\T\cos\alpha\cos\beta\\T\sin\beta\end{bmatrix},\qquad \boldsymbol{T}_{syn} = [\hat{\boldsymbol{r}}_2\ \hat{\boldsymbol{s}}\ \hat{\boldsymbol{w}}]\boldsymbol{T}_{RSW}.

Advantages: decoupling magnitude from direction, simple magnitude constraint $0\le T\le T_{\max}$ , independent trust-region scaling for magnitude (normalized by $T_{\max}$ ) and angles (larger bounds).

Frame degeneracy: when the position vector $\boldsymbol{r}_2$ is collinear with velocity, the RSW basis is undefined; switch to Cartesian control $\boldsymbol{u}=[T_x,T_y,T_z]^T$ in those cases.

Throttle Factor and Throttle Function

The throttle factor is the nondimensional thrust-magnitude variable $u\in[0,1]$ of a low-thrust engine, with $u=0$ for coast and $u=1$ for full thrust, decoupled from the steering angles. Its optimal value is determined by the switching function given by the Pontryagin Minimum Principle:

Time-optimal control: throttle is constantly 1 (always full thrust).
Fuel-optimal control: throttle is bang-bang, set to 0 or 1 by the sign of the switching function; equivalent to the limit of the energy-optimal homotopy.
Energy-optimal control: throttle takes continuous values, determined jointly by the costate and the homotopy parameter.

Du et al. (2024), Tu et al. (2025), and others give throttle profiles for CR3BP low-thrust collision avoidance and Halo orbit transfers.

The throttle function generalizes the throttle factor to multi-mode propulsion. Zhang Z et al. (2026) use two throttle functions $\delta_{P1}(t),\delta_{P2}(t)\in\{0,1\}$ for a dual-mode electric propulsion system, with bang-bang solutions derived from the Pontryagin Minimum Principle. In the indirect method, the throttle function is an explicit output of the optimal control, not an externally imposed parametrization.

Practical Notes

Match basis to physics. Transfers with thrust switches favour B-splines (local support) or discretized switching functions; smooth continuous low thrust favours Fourier/Chebyshev.
Spherical vs Cartesian. Spherical variables are friendly to directional trust regions and natural for magnitude constraints but require handling RSW degeneracy; Cartesian variables avoid degeneracy but have a non-convex magnitude constraint $\|\boldsymbol{T}\|\le T_{\max}$ .
Throttle as an optimization variable. In indirect methods the throttle is determined analytically by the switching function and is not an optimization variable; in direct methods it is an independent optimization variable at each collocation point, subject to the path constraint $0\le u_k\le 1$ .
B-spline order. Low thrust typically uses order 3–5 ( $C^2$ – $C^4$ ); too high invites numerical ill-conditioning, too low cannot represent the required thrust rate.
Knot multiplicity for discontinuity. Repeated knots at mode switches lower continuity and allow discontinuities.

References

de Boor, C., 1978, A Practical Guide to Splines (the mathematical foundation of B-splines).
Sanchez, P., et al., 2020 (B-spline control parametrization for trajectory optimization and attitude control).
Aziz, J. D., Scheeres, D. J., Lantoine, G., 2019, "Hybrid Differential Dynamic Programming in the CR3BP," JGCD (spherical control variables, RSW degeneracy switching, trust-region scaling).
Colagrossi, A., et al., 2021, "Guidance, navigation and control for 6DOF rendezvous in cislunar multi-body environment" (control parametrization for 6DOF rendezvous).
Du, L., et al., 2023, "Two trajectory configurations for the low-thrust transfer between northern and southern halo orbits" (throttle factor applied to Halo transfers).
Du, L., et al., 2024, "A novel calculation method for low-thrust transfer trajectories in the Earth-Moon restricted three-body problem" (throttle parameter and switching function).
Tu, X., et al., 2025, "Optimal control for low-thrust collision avoidance in CRTBP" (throttle factor for collision avoidance).
Zhang, Z., et al., 2026, Space Sci. Technol. 6:0441 (throttle functions for dual-mode electric propulsion and their bang-bang solutions).