Reconfigurable Laser-to-Debris Scheduling

• The paper introduces a comprehensive ILP model combining laser ablation physics, platform reconfiguration, and orbital mechanics to maximize debris remediation rewards.
• It employs a receding-horizon approach to decompose the scheduling problem, achieving near-optimal performance with significant capacity gains from dynamic platform maneuvers.
• Empirical results show that reconfigurability boosts remediation capacity by up to 40% and deorbit counts by up to 32%, critical for rapid-response debris mitigation.

The Reconfigurable Laser-to-Debris Engagement Scheduling Problem (R-L2D-ESP) arises in the context of coordinating multiple space-based lasers for orbital debris remediation tasks. R-L2D-ESP formalizes the optimal spatiotemporal allocation of laser engagements and platform maneuvers—encompassing both the scheduling of debris “nudging” via laser ablation and the dynamic reconfiguration of the laser constellation itself—to maximize debris remediation capacity. This capacity includes mission objectives such as directed deorbiting, collision avoidance, and dynamic response to new debris events (Rogers et al., 2024, Rogers et al., 16 Dec 2025). The problem is typically captured as a large-scale 0-1 integer linear program (ILP), incorporating both the physics of momentum transfer and the operational constraints of space-based constellations.

1. Formal Problem Definition and Notation

R-L2D-ESP considers a discretized planning horizon $\mathcal{T} = \{0,1,\ldots,T-1\}$, a constellation of $P$ reconfigurable laser platforms $\mathcal{P} = \{0,\ldots,P-1\}$, and a debris catalogue $\mathcal{D} = \{0,\ldots,D-1\}$. Each platform at each timestep may occupy one of multiple orbital slots, change its slot (via plane/altitude/phasing maneuvers), or engage debris using pulsed-laser ablation. Debris objects evolve on a time-expanded graph of orbital slots, with feasible transitions determined by the possible impulsive maneuvers from laser engagements.

Key notation:

Symbol	Description
$S^p_t$	Orbital slots available for platform $p$ at time $t$
$J_{t,d}$	Debris slots available for debris $d$ at time $t$
$x_{t\,d\,i\,j}$	Binary: debris $d$ moves $i \to j$ at $t$
$y^p_{t\,s\,d}$	Binary: platform $p$, slot $s$ engages debris $d$ at $t$
$z^p_{t\,s\,w}$	Binary: platform $p$ transfers slot $s \to w$ at $t$
$c^p_{t\,s\,w}$	Δv cost for transfer $s \to w$ by platform $p$ at $t$
$c^p_{\max}$	Δv budget for platform $p$
$R_{t\,d\,i\,j}$	Reward for debris $d$: slot $i\to j$ at $t$
$(PS)_{t\,d\,j}$	Minimal set of platform-slot pairs required for $i\to j$ for debris $d$

These elements capture the essential decision variables and constraints governing both the combinatorial assignment of engagements and the kinematic evolution of both platforms and debris.

2. Mathematical Programming Model

The R-L2D-ESP employs a comprehensive ILP encompassing platform maneuvers, engagement scheduling, orbital mechanics, and operational resources. The canonical objective is to maximize cumulative remediation reward:

$$\max_{x, y, z} \sum_{t=0}^{T-2} \sum_{d=0}^{D-1} \sum_{i \in \mathcal{J}_{t,d}} \sum_{j \in \mathcal{J}_{t+1,d,i}} R_{t\,d\,i\,j} x_{t\,d\,i\,j}$$

Subject to constraints:

• Network flow for contiguous debris slot assignment per $d$ (conserving flow over the debris slot tree).
• Coalition engagement linking: a debris slot transfer $i\to j$ at $t$ is only possible if all required platform-slot pairs participate ($y^p_{t\,s\,d}$).
• Platform operational coupling and exclusive assignment constraints (a platform can only perform one engagement or slot transfer each timestep).
• Δv budget limits per platform: $$\sum_{t,s,w} c^p_{t\,s\,w} z^p_{t\,s\,w} \le c^p_{\max}$$.
• Binary domains for all decision variables.

Feasibility of each engagement is enforced via visibility and range constraints, with debris and platform slot graphs following a time-expanded directed acyclic graph model.

3. Physical Modeling and Delta-v Vector Composition

Laser-to-debris engagements are modeled via momentum transfer from pulsed laser ablation. The instantaneous velocity impulse is:

$$\Delta\bm v_{t\,p\,d} = n_{\mathrm{L2D}} \frac{\eta_1 c_m \varphi_{t\,p\,d}}{\mu_d} \hat u_{t\,p\,d}$$

$$\varphi_{t\,p\,d} = \eta_2 \frac{E}{\pi} \left( \frac{2 \phi}{M^2 a \lambda u_{t\,p\,d}} \right)^2$$

where $\mu_d$ is debris surface density, $c_m$ is the momentum coupling coefficient, $n_{\mathrm{L2D}}$ the number of pulses, $\hat u_{t\,p\,d}$ the line-of-sight unit vector, and $\varphi_{t\,p\,d}$ the delivered fluence. Multiple platforms can deliver coordinated multi-laser impulses, with the resultant $$\Delta\bm v_{t\,d} = \sum_{(p,s)\in(PS)_{t\,d\,j}} \Delta\bm v_{t\,p\,d}$$. Slot transitions for debris are analytically propagated via state updates on orbital elements.

Rewards $R_{t\,d\,i\,j}$ encode both penalties (for generating conjunctions) and positive incentives (associated with periapsis reduction and outright deorbit). (Rogers et al., 2024, Rogers et al., 16 Dec 2025)

4. Solution Strategies and Complexity

The full-horizon ILP is NP-hard, with exponential growth in $T,\,P,\,D$. Direct solution is intractable for operationally relevant scenarios. R-L2D-ESP is commonly addressed by decomposing the scheduling horizon into shorter rolling (receding) windows of $L\ll T$:

1. At time $t$, formulate the ILP for timesteps $t,\,\dots,\,t+L-1$.
2. Apply resulting controls to the first step only, update platform and debris slot assignments, and increment $t$.
3. Repeat with a window shifted by one.

This receding-horizon approach maintains provable optimality within each window and achieves near-global performance within a few percent of the unconstrained optimum when using $L=3$–5, while drastically reducing computational burden. Δv-budget ($c^p_{\max}$) and window length $L$ are critical design parameters: higher values yield greater scheduling flexibility but increase onboard resource consumption and computation time (Rogers et al., 16 Dec 2025).

5. Sensitivity Analysis and Computational Results

Case studies quantify the advantages of constellation and platform reconfigurability:

• With $P=6,\,D=395$ over 2 days (3-minute steps), static configurations yield a remediation capacity of 20,890.17 and 104 deorbits. Plane-change reconfiguration increases capacity to 27,617.32 (+32.2%) and 137 deorbits (+31.7%). Altitude-change scenarios show similar gains (+32.6% capacity, +25% deorbits) (Rogers et al., 16 Dec 2025).
• For rapid breakup mitigation (1 day, $P=4$): without reconfiguration, 801.86 capacity and 6 deorbits. With altitude + phasing reconfiguration, capacity rises to 9,021.26 (+1,025%) and 47 deorbits (+683%).
• Capacity gains scale with receding-horizon depth: $L=2$ delivers ~6% improvement, rising to ~40% for $L=5$. Altitude-change is especially effective at larger $L$.
• Δv-budget sensitivity: at 0.5 km/s, plane-change reconfiguration gives the largest relative gain (23%), but altitude-change provides monotonic improvement up to +32.6% at 2 km/s.

A plausible implication is that reconfigurability is especially critical during time-critical events, with gains in deorbit and remediation capacity exceeding an order of magnitude in acute scenarios (Rogers et al., 16 Dec 2025).

6. Architectural Insights and Design Recommendations

Key findings on laser constellation design and scheduling regime are as follows:

• Most remediation gains from increasing the number of platforms saturate at $P\approx8$.
• Breaking spatial symmetries (e.g., Walker–Delta lattices) via optimized placement (solved as a Maximal Covering Location Problem) yields 6–20% additional performance.
• The ability to dynamically reassign engagement pairings (“reconfigurability” in assignment and platform slot) is essential, both for enabling collaborative multi-laser delta-v summation and effective collision-avoidance during conjunction windows.
• System designers should typically allocate at least 1.5 km/s Δv per satellite and a horizon $L\approx4$ to capture most practical gains (Rogers et al., 2024, Rogers et al., 16 Dec 2025).

7. Context and Implications for Orbital Debris Remediation

R-L2D-ESP enables co-optimization of on-orbit constellation maneuvers and laser-to-debris engagements, extending beyond static assignment and scheduling paradigms. It underpins the operational viability of next-generation debris remediation systems by integrating kinematic maneuvering, constraint-driven scheduling, and physical ablation models. Empirical results confirm that reconfigurable constellations outperform static deployments, achieving 30%–40% higher remediation capacity and 25%–32% more deorbits in baseline scenarios, with up to 10× gains in rapid-response environments. This suggests that system-level reconfigurability is a primary lever for scalable and adaptive orbital debris remediation (Rogers et al., 16 Dec 2025, Rogers et al., 2024).