Trajectory Constraints

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

Definition

Trajectory constraints are the restrictions a spacecraft trajectory must satisfy beyond the equations of motion when posed as an optimal control problem (or NLP after transcription). They encode mission feasibility: where the vehicle must be at certain times, what it must avoid along the way, and which state or control bounds cannot be violated. A trajectory that satisfies dynamics plus all constraints is called feasible; the optimizer seeks the feasible trajectory that minimizes the performance index (Betts 1998; Conway 2010).

Constraints fall into four families by where and how they act:

Terminal / boundary constraints - conditions at the initial and/or final time of a phase (rendezvous, insertion, interception).
Path constraints - inequalities that must hold at every instant along a phase (eclipse avoidance, heating, thrust magnitude).
State constraints - path constraints that involve only the state, not the control (perigee altitude, maximum distance); they induce costate jumps in indirect methods.
Event / phase constraints - conditions at internal time instants (phase-angle matching at a transfer interface) or linking variables across phases.

Inter-arc continuity conditions at patch points are sometimes called defect constraints in direct transcription; they are covered in Defect Constraint and inter-arc matching conditions in Multi-arc Optimal Control.

Mathematical Formulation

A single-phase optimal control problem with full constraint types has the form (Betts 1998):

\min_{\boldsymbol{u}(\cdot),\,t_f} J = \Phi[\boldsymbol{x}(t_0),\boldsymbol{x}(t_f),t_f] + \int_{t_0}^{t_f} L\,\mathrm{d}t,

subject to

\dot{\boldsymbol{x}} = \boldsymbol{f}(\boldsymbol{x},\boldsymbol{u},t),

\boldsymbol{\psi}[\boldsymbol{x}(t_0),\boldsymbol{x}(t_f),t_f] = \boldsymbol{0} \quad \text{(terminal/boundary)},

\boldsymbol{g}_l \leq \boldsymbol{g}[\boldsymbol{x}(t),\boldsymbol{u}(t),t] \leq \boldsymbol{g}_u \quad \text{(path)},

\boldsymbol{x}_l \leq \boldsymbol{x}(t) \leq \boldsymbol{x}_u \quad \text{(simple state bounds)}.

Equality is recovered by setting $g_l = g_u$ for some component. Betts (1998) also distinguishes quadrature functions $\int q\,\mathrm{d}t$ , used to bound cumulative quantities (e.g., total eclipse duration).

In the indirect formulation, terminal constraints couple to the costate via multipliers $\boldsymbol{\nu}$ : $\boldsymbol{\lambda}(t_f) = \partial\Phi/\partial\boldsymbol{x}(t_f) + (\partial\boldsymbol{\psi}/\partial\boldsymbol{x}(t_f))^{T}\boldsymbol{\nu}$ . Together with the unknown $\boldsymbol{\lambda}(t_0)$ this yields a TPBVP.

State Constraints and Costate Jumps

When a pure state inequality $S(\boldsymbol{x}) \leq 0$ becomes active on a boundary arc $[t_1,t_2]$ , the standard first-order necessary conditions must be augmented with a multiplier $\eta(t) \geq 0$ supported on that arc. The costate then satisfies the jump condition

\boldsymbol{\lambda}(t_i^{+}) = \boldsymbol{\lambda}(t_i^{-}) - \eta(t_i)\,\nabla S[\boldsymbol{x}(t_i)],

at the entry/exit time $t_i$ of the boundary arc. For state constraints of order $p$ (i.e., $S^{(p)}$ depends explicitly on $\boldsymbol{u}$ ), jumps appear in $\boldsymbol{\lambda}^{(p-1)}$ rather than $\boldsymbol{\lambda}$ itself. This non-smoothness is the main reason indirect methods struggle with active state constraints; direct methods absorb it implicitly through the NLP active-set machinery.

Typical Cislunar Instances

Terminal: multi-revolution Lambert boundaries

Multi-revolution Lambert problems add a revolution number $N$ to the classical two-body transfer between fixed $\boldsymbol{r}_0$ , $\boldsymbol{r}_f$ over flight time $T$ . Duan et al. (2025) show that orbit-dynamical boundary constraints reshape the admissible solution manifold across $N$ : as $N$ grows, a growing fraction of sampled $(\boldsymbol{r}_0,\boldsymbol{r}_f,T)$ triples have no feasible solution at that revolution, so the per-revolution share of admissible solutions decreases with $N$ . The phenomenon is geometric - at fixed transfer angle, additional revolutions force the semi-major axis into a narrow band - and must be accounted for when initializing a shooting method or building surrogate models for uncertain Lambert inputs.

Path: eclipse avoidance

For solar-electric missions, thrust is unavailable (or degraded) inside the umbral/penumbral cone of an occulting body. Sowell & Taheri (2024) encode this as a smooth throttle factor on the propulsion acceleration,

\boldsymbol{\Delta}_{\text{prop}} = \frac{T_{\max}}{m}\,\delta\,\bar{\delta}_e\,\hat{\boldsymbol{\alpha}}, \quad \bar{\delta}_e = \tfrac{1}{2}\left[1 + \tanh(S_e/\rho_e)\right],

where $S_e = a_D - a_{RS} - a_{RB}$ is the eclipse switching function built from apparent radii and separation; $\rho_e$ controls smoothing. Embedding $\bar{\delta}_e \in [0,1]$ directly into the dynamics avoids a multiphase transcription with explicit eclipse entry/exit events, at the cost of one extra parameter ( $\rho_e$ ) that trades smoothing against event fidelity. The same construction generalizes to limiting eclipse duration by adding a quadrature penalty $\int (1 - \bar{\delta}_e)\,\mathrm{d}t$ .

Event: system-to-system transfer timing

Transfers between the Sun-Earth and Earth-Moon CR3BP systems are located geometrically by intersections of unstable/stable manifold tubes on a Poincare section. In the ephemeris model those intersections occur only at particular lunar phase orientations. Howell & Kakoi (2006) parameterize the relative orientation of the Earth-Moon and Sun-Earth rotating frames by a 3-1-3 Euler sequence $(\alpha, i, \beta)$ with $i \approx 5^{\circ}$ fixed, plus a Poincare-section plane angle $\psi$ . The timing constraint reduces to selecting $(\alpha, \beta, \psi)$ - linearly or quadratically fitted over the transfer epoch - so that the geometric intersection survives in the real ephemeris. Co-rotating $\alpha$ and $\beta$ (increase $\alpha$ , decrease $\beta$ by the same amount) preserves the intersection location while rephasing the epoch, an inexpensive way to scan the lunar month for low-cost windows.

Handling Techniques

Direct transcription turns each constraint into NLP rows: defect equations per collocation node (see Defect Constraint), plus path constraint evaluations at nodes and centers. Sparsity-exploiting SQP/IP solvers (SNOPT, IPOPT) handle $10^4$ - $10^6$ constraints routinely.
Soft constraints / relaxation replace a hard equality $\boldsymbol{X}_P = \boldsymbol{X}_Q$ (e.g., a 4D Poincare-section intersection between a perturbed manifold and a target orbit) by a closest-point search $\min \|\boldsymbol{X}_P - \boldsymbol{X}_Q\|$ followed by a Lambert arc that bridges the residual. Qiao & Yang (2024) apply this to Earth-Moon $L_1$ Halo-to-GEO transfers: the strict section-intersection condition is over-constrained in 4D and blocks optimization; relaxing it to a Lambert $\Delta V$ term in the objective turns the problem into a smoothly parametrized search. The relaxation preserves feasibility (every Lambert bridge is a valid transfer) at the price of letting $\Delta V$ absorb the geometric mismatch.
Smoothing (hyperbolic tangent / CSC) regularizes piecewise-defined constraints (eclipse on/off, bang-bang control) so indirect integrators and NLP Jacobians remain smooth; see also the composite smooth control used alongside PMP.
Augmented Lagrangian / penalty methods move active inequality constraints into the objective; see Augmented Lagrangian Method.

Practical Notes

Terminal constraints dominate the shooting method Jacobian structure; tighter terminal boxes shrink the convergence basin.
Path constraints that change activity frequently (e.g., eclipses on a low-thrust spiral) are the main driver of mesh refinement in pseudospectral methods.
Phase-linking constraints in multi-body regimes are the bridge to Multi-arc Optimal Control theory.

References

Betts, J. T. (1998). Survey of numerical methods for trajectory optimization. Journal of Guidance, Control, and Dynamics, 21(2), 193-207.
Conway, B. A. (Ed.). (2010). Spacecraft Trajectory Optimization. Cambridge University Press.
Howell, K. C., & Kakoi, M. (2006). Transfers between the Earth-Moon and Sun-Earth systems using manifolds and transit orbits. Acta Astronautica, 59(1-5), 367-380.
Sowell, S., & Taheri, E. (2024). Eclipse-conscious low-thrust trajectory optimization using pseudospectral methods and control smoothing techniques. Journal of Spacecraft and Rockets. doi:10.2514/1.A35789
Qiao, C., & Yang, L. (2024). Design and optimization of Earth-Moon L1 low-energy transfer orbit. Systems Engineering and Electronics, 46(10). (in Chinese)
Duan, Y., Zhang, Y., & Liu, Y. (2025). Adaptive polynomial chaos expansion method for uncertain multiple-revolution Lambert problem. Celestial Mechanics and Dynamical Astronomy.
You, S., & Dai, R. (2022). Trajectory optimization with state constraints. Journal of Guidance, Control, and Dynamics. doi:10.2514/1.G006815