Envisioning an Optimal Network of Space-Based Lasers for Orbital Debris Remediation
David O. Williams Rogers Matthew C. Fox Paul R. Stysley Hang Woon Lee





West Virginia University, Morgantown, WV, 26506

NASA Goddard Space Flight Center, Greenbelt, MD, 20771

West Virginia University, Morgantown, WV, 26506

Envisioning an Optimal Network of Space-Based Lasers for Orbital Debris Remediation
    Hang Woon Lee[Assistant Professor, Department of Mechanical, Materials and Aerospace Engineering; hangwoon.lee@mail.wvu.edu. Member AIAA (Corresponding Author).]
    
School of Physics, Nankai University, Tianjin, 300071, China 

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§ ABSTRACT 
The rapid increase in resident space objects, including satellites and orbital debris, threatens the safety and sustainability of space missions. This paper explores orbital debris remediation using laser ablation with a network of collaborative space-based lasers. A novel delta-v vector analysis framework quantifies the effects of multiple simultaneous laser-to-debris (L2D) engagements by leveraging a vector composition of imparted delta-v vectors. The paper introduces the Concurrent Location-Scheduling Problem (CLSP), which optimizes the placement of laser platforms and schedules L2D engagements to maximize debris remediation capacity. Due to the computational complexity of CLSP, it is decomposed into two sequential subproblems: (1) optimal laser platform locations are determined using the Maximal Covering Location Problem, and (2) a novel integer linear programming-based approach schedules L2D engagements within the network configuration to maximize remediation capacity. Computational experiments are conducted to evaluate the proposed framework’s effectiveness under various mission scenarios, demonstrating key network functions such as collaborative nudging, deorbiting, and just-in-time collision avoidance. A cost-benefit analysis further explores how varying the number and distribution of laser platforms affects debris remediation capacity, providing insights into optimizing the performance of space-based laser networks.




§ NOMENCLATURE


@l @  =   l@
    L2D     laser-to-debris 

    DVA     Δ v vector analysis

    CLSP     concurrent location-scheduling optimization problem

    MCLP     maximal covering location problem

    L2D-ESP     laser-to-debris engagement scheduling problem

    𝒯     mission time horizon (index t)

    𝒟     set of debris (index d) 

    𝒥     set of debris orbital slots (index j)

    𝒫     set of space-based laser platforms (index p) 

    𝒮     set of laser platform orbital slots (index s)

    𝒦     set of valuable assets in space

    R     debris remediation capacity reward 

    π     network topology reward 

    c_m     momentum coupling factor
    
    φ     on-debris delivered laser fluence

    ρ     debris density

    m     debris mass

    r     position vector

    v     velocity vector

    W     laser-to-debris engagement boolean parameter

    z     laser platform location decision variable

    y     laser-to-debris engagement decision variable

    x     debris relocation decision variable





§ INTRODUCTION
 
The number of resident space objects is rapidly increasing, largely due to the emergence of small satellites, owing to standardized manufacturing processes, advancements in technology, and lower costs for sharing space on launch vehicles. This increase in the number of satellites is correlated with the significant growth in orbital debris, posing a serious risk to both manned and unmanned missions as well as active satellites that form critical infrastructure, yielding an escalation in conjunction alerts (i.e.,, satellites approaching within 3km <cit.>) and an increased probability of in-space collisions, which can create debris clouds of varying sizes.

Large space debris, objects with characteristic lengths greater than 10cm, is identified as the primary source of new debris <cit.> and consists mostly of defunct satellites, payloads, and rocket stages. Its presence in orbit increases the cost of operations, requiring active satellites to perform collision avoidance maneuvers, thereby demanding additional fuel consumption, the loss of mission objectives due to maneuvering, and increased labor costs for planning these maneuvers. Additionally, large debris restricts access to space for new satellites or payloads, affecting activities that heavily rely on space infrastructure, such as telecommunications, financial exchanges, and climate monitoring <cit.>.

Conversely, small debris comprises objects with characteristic lengths between 1 and 10cm. The challenges of tracking small debris pieces, coupled with their significant population close to 500000 objects <cit.>, present a continuous threat of collisions in space. Even with their relatively small mass, these fragments have the potential to inflict hypervelocity impacts, penetrating spacecraft shielding and risking the success of the mission. The origins of small debris include loose parts from operational or defunct satellites, payloads, or rocket stages, such as sodium-potassium droplets, solid rocket motor slag and dust, multi-layer insulation, ejecta, and paint flakes <cit.>. However, a significant part of the population has been generated by anti-satellite tests <cit.>, accidental explosions of satellites and rocket bodies <cit.>, and debris-to-debris collisions <cit.>.
 
Considering the substantial size of the debris population and the looming threat of triggering the Kessler syndrome <cit.>, strategies limited to the prevention of new debris formation, such as implementing regular conjunction assessments for active spacecraft <cit.> and enforcing restrictions on mission operations and end-of-life procedures <cit.> are insufficient for addressing the debris problem comprehensively. In response to this, a range of promising and innovative debris remediation technologies has been proposed to reduce the debris population.

The literature on debris remediation introduces contact-based methods as strategies for eliminating orbital debris. Chaser spacecraft equipped with tethered nets are considered an effective debris remediation mechanism <cit.> for addressing irregularly shaped debris.
Space balls and space winches are mechanisms designed to tackle the problem of debris with high angular momentum. These methods are capable of exerting a retarding torque on debris, aiding in its stabilization. However, they require an accurate activation at the time of the engagement <cit.>.
For relatively stable (i.e., detumbled)  objects, debris remediation mechanisms such as grappling can be implemented for targeted and controlled removal. For instance, spacecraft equipped with robotic arms can accomplish this task <cit.>, by leveraging a rendezvous maneuver to capture the target and subsequently relocating it into a disposal orbit or into an orbit where atmospheric effects induce reentry. However, this technique is highly dependent on the physical properties of the debris, such as its shape and surface texture. It also necessitates precise attitude adjustment to counteract the forces exerted by the robotic arm's movements <cit.>. Hooks and harpoons are penetrating mechanisms used for debris remediation; however, they require proximity operations, endangering the success of the mission, and during penetration, they can trigger an explosion <cit.> generating more debris pieces.

Unlike the contact-based debris remediation mechanisms described above, which are targeted at large debris, remediating small debris requires specific methods compatible with their size. Novel tethered plate systems, capable of remediating debris pieces without damaging the plate's structure, have been proposed <cit.>. Furthermore, perimeter-ring-truss systems can handle small debris while ensuring scalability and practicality in the design phase <cit.>. However, the cost of remediation per debris piece is analyzed to be the highest in comparison with other methods <cit.>.

Ground-based lasers have emerged as a promising, cost-effective debris remediation solution to address the growing orbital debris population while circumventing the challenges and limitations of contact-based DR methods. 
The change in velocity, Δ v, required to reduce the target's orbit altitude can be imparted by a ground-based laser leveraging photon pressure or laser ablation mechanisms. Photon pressure-based lasers exert a small force on the target object sufficient to induce small orbit changes. Mason et al. <cit.> present the use of two photon pressure ground-based laser systems, to engage debris multiple times and change its orbit due to the applied perturbation, thereby ultimately avoiding collisions with operational satellites.
However, given the laser's wavelength, debris temperature and material properties there is a threat of generating specular reflections on debris surface (i.e., Iridium flares), which scale linearly with the laser power <cit.>. In light of the small forces that photon pressure imparts on the target, their application is constrained to collision avoidance only <cit.>. 
Alternatively, laser ablation mechanisms rely on the laser's high energy to rapidly melt the target's surface and generate a material jet that produces a reactive momentum, typically higher than the one produced by photon pressure-based lasers <cit.>. On the one hand, continuous wave heating produces irregular melt ejection, which can generate more debris. In the case of a tumbling target, the average momentum transfer can be nullified due to the slow heating and decay characteristic of the imparted thrust <cit.>.
On the other hand, pulsed lasers appear as a mechanism suitable for collision avoidance and deorbiting debris, given that the per pulse energy delivered from the ground-based station is enough to change debris orbit <cit.>.

Despite their promise as debris remediation solutions, ground-based laser debris remediation systems face significant technical challenges. These systems are subject to numerous atmospheric constraints, including aerosol attenuation, cloud cover, scintillation effects, and turbulence, all of which affect beam quality <cit.>. Moreover, ground-based lasers have limitations in range and angles and thus require strategic positioning of their ground stations to maximize efficiency while considering various civil and operational constraints (i.e., ideally situated away from airports and air routes) <cit.>. 
Ultimately, given its ground-based nature, their laser engagement opportunities depend on debris passing over its operational range, constraining ground-based lasers to preventive just-in-time collision avoidance without the ability to tackle immediate collision threats <cit.>.

Against this backdrop, the idea of space-based lasers has garnered increased attention over the past decade, owing to their capacity to address debris of varying sizes (both large and small) and their potential to overcome the inherent challenges associated with ground-based lasers <cit.>. Space-based lasers possess several advantages compared to their ground-based counterparts, including more efficient energy delivery and beam quality <cit.>, increased access to debris (contact duration and opportunities) <cit.>, and reduced risk of collateral damage due to more precise control enabled by shorter ranges <cit.>. 
According to the NASA Cost and Benefit Analysis of Orbital Debris Remediation report <cit.>, ground-based and space-based laser systems are the only debris remediation methods that can handle both trackable and non-trackable debris pieces with the best cost-to-benefit remediation ratio. Moreover, when not used for remediation purposes, the operator can rely on them to track and characterize debris objects <cit.>.

The state-of-the-art literature on space-based lasers successfully validates the concept of small debris remediation by leveraging laser ablation mechanisms and exploring different laser parameters. Regarding large debris, literature presents how just-in-time collision avoidance actions can be effectively taken by changing the debris orbit due to imparted momentum. Yet, constraining the number of platforms to one or two cannot scale up efficiently to a more extensive debris population, which is rapidly expanding every year and is highly heterogeneous in materials, size, density, and orbital parameters. Further, the effectiveness of laser ablation is heavily influenced by the irradiation distance, angle, and revisit time, posing a challenge for a single or two laser platforms.
In addition, while nudging debris or placing it on a descent trajectory can alleviate immediate threats to specific valuable assets, it may inadvertently create hazardous conjunction events for other objects in space.

In response to these challenges, we envision a network of interconnected, collaboratively working space-based lasers, optimally designed to maximize the debris remediation capacity, that is, the ability to nudge and deorbit debris and perform just-in-time collision avoidance. Aiming to materialize this concept, we address the following research questions in this paper:


  * “Where do we optimally locate the space-based laser platforms with respect to each other and the debris field?”

  * “How do we optimally schedule laser-to-debris engagements such that the debris remediation capacity is maximized?”


We advance the state of the art in several ways. First, we propose leveraging the formulation of the Maximal Covering Location Problem (MCLP) <cit.>, typically found in the literature on facility location problems, to design an optimal network topology that maximizes laser engagement rewards with the debris field. In its original domain, MCLP aims to find the optimal locations for a set of facilities to maximize coverage over a set of customers. 
We exploit the similarities between satellite constellation pattern design and facility location problems discussed in Ref. <cit.> and interpret the facilities as laser platforms and the debris as customers. Second, we study the effect of multiple laser engagements on target debris by conducting a vector analysis of the Δ v's imparted by the laser platforms, which we name Δ v Vector Analysis (DVA). Employing multiple lasers increases the degrees of freedom in the debris control mechanism for space-based laser debris remediation missions. This allows for finer control over the target debris resultant Δ v magnitude and direction, offering a wider range of potential paths for its subsequent descent or collision avoidance trajectory. Moreover, leveraging this new framework, we design a novel integer linear programming (ILP)-based laser-to-debris (L2D) engagement scheduling problem (ESP) that dictates which combination of lasers should engage with which debris while considering a reward function that accounts for debris mass, change in periapsis radius, and the risk of potential conjunctions with operating satellites before and after the engagements. It is important to remark that we are introducing, for the first time, an optimization framework that considers a network of space-based lasers to engage space debris of different sizes. Finally, we present several case studies, considering small, and large debris; further, we set a case with a mixed debris field along with 10 valuable assets. For all three case studies, the framework presented in this paper outperforms Walker-Delta <cit.>-based networks with the same number of platforms. Additionally, we conduct a cost and benefit analysis to characterize the impact of varying the number of laser platforms in the debris remediation capacity.

The rest of the paper is structured as follows: Sec. <ref> describes the technical details of the laser ablation process and DVA. Section <ref> presents the parameter generation and the mathematical optimization formulation. Section <ref> illustrates the case studies with debris fields of diverse sizes, and Sec. <ref> presents the conclusions of this paper.

 

§ LASER-TO-DEBRIS ENGAGEMENT MODELING

In this paper, space-based laser platforms engage debris by irradiating a laser beam, inducing a laser ablation process that alters the debris orbit. The governing dynamics of the laser ablation process are described in Sec. <ref> and the DVA mechanism is presented in Sec. <ref>. 

The L2D framework is built upon several assumptions. First, debris is considered to be a perfect sphere. Second, all L2D engagements are deterministic, meaning that the position and velocity of debris after the engagement are known. Third, each laser platform has complete knowledge of the characteristics of debris (e.g., material, mass, orbit) and whether they endanger the safe operation of valuable in-space assets, for instance, the International Space Station, and in-service telecommunications and Earth-observation satellites.
Finally, we assume that the space-based laser network operates as a centralized system with global knowledge. In this system, each platform is aware of the magnitude and direction that all other platforms in the network can impart over debris, as well as their orbital states during the mission.

We define the mission time horizon as a set of uniformly discretized time steps 𝒯 = {t_0, …, t_T-1}, with index t and cardinality T, where t_0 corresponds to the epoch.
The laser platform set is defined as 𝒫={p_1,…,p_P} with index p and cardinality P, and the debris set 𝒟={d_1,…,d_D} with index d and cardinality D.



 §.§ Laser Ablation Principle

Pulsed laser ablation is the process by which a solid ejects plasma from its surface due to the action of a short, intense laser pulse <cit.>. At regimes of high irradiance, the vaporized surface material becomes ionized and begins to absorb the incident laser beam leading to vapor breakdown and plasma formation <cit.>. The ablation rate, defined as the thickness of the ablated material per laser pulse, is inversely proportional to the square root of the beam diameter, and negatively correlated with laser wavelength given the reduced optical absorptivity and high reflectivity of targets at large wavelengths <cit.>. Given the velocity of the ejected material, and by the principle of momentum conservation, the ejected mass of the plasma generates a net impulse on the object <cit.>. 

Pulsed laser ablation can be leveraged to impart an impulse to debris, causing a change in velocity that alters the size and/or shape of its resulting orbit. The magnitude of this perturbation is determined by laser parameters such as wavelength, output power, pulse energy, beam quality, and the length and frequency of pulses <cit.>, as well as by the properties of the target debris, including its mass, density, and surface material composition <cit.>. Moreover, the geometric relationship between the laser and debris determines the direction and magnitude of the velocity perturbation applied to the debris, influencing the resulting orbit.

The momentum coupling factor c_m relates the imparted impulse on debris with the used laser's pulse optical energy <cit.>, and is defined as <cit.>:

    mΔ v = c_mE

where E is the laser energy delivered to the debris by a single pulse and m is the debris mass. Further, the on-debris delivered laser fluence φ is defined as <cit.>:

    φ = 4ED_eff^2T_tot/π B^4ζ^2λ^2u^2

with D_eff^2 being the effective illuminated beam diameter, T_tot the total system loss factor, B the beam quality factor, ζ a constant that regulates diffraction, λ the wavelength, and u the range between the platform and debris. Consequently, the per-pulse L2D Δ v delivered on debris with surface mass ρ is given as:

    Δ v = η c_mφ/ρ

where η is the impulse transfer efficiency, which takes into account shape effects, tumbling, improper thrust direction on debris, and other factors <cit.>.



 §.§ Δ v Vector Analysis
 
In this section, we present the DVA, a novel framework to quantify the effects of multiple simultaneous L2D engagements on target debris by leveraging a vector composition of the imparted Δv vectors.

We let r_td^- and v_td^- indicate the position and velocity vectors, respectively, of debris d at time step t immediately before an L2D engagement. Similarly, r_td^+ and v_td^+ are the position and velocity vectors, respectively, of debris d at time step t immediately after an L2D engagement.

We assume that at time step t an L2D engagement induces an instantaneous change in the velocity of target debris but its position remains unchanged. Hence, we have:

    
           v_td^+ = v_td^- + Δv_tpd

           r^-_td = r^+_td=r_td
    

where Δv_tpd represents the change in velocity experienced by debris d due to an L2D engagement from laser platform p.

At time step t, given position vectors r_td and r_tp of debris d and platform p, respectively, the relative position vector that points from platform p to debris d is given as u_tpd = r_td - r_tp; its unit vector is given as û_tpd = u_tpd /u_tpd, with u_tpd=‖u_tpd‖_2. The total change in velocity imparted from laser platform p to debris d at time step t is calculated by generalizing Eq. (<ref>) for every time step t, laser platform p, and debris d. Equation (<ref>) presents the vector Δv_tpd from laser platform p to debris d at time step t given unit vector û_tpd.

    Δv_tpd = N_dη_pdφ_pd c_m,pd/ρ_dû_tpd

where N_d is the number of laser pulses per time step t on debris d, computed from the time step size and the laser pulse repetition frequency (PRF), which denotes the number of laser pulses per second.

The DVA framework captures multiple, simultaneous L2D engagements and represents them as a total effective single L2D engagement. The total Δv_td over debris d at time step t due to multiple L2D engagements can be represented as follows:

    Δv_td = ∑_p∈𝒫_tdΔv_tpd

where 𝒫_td represents the set of laser platforms that engage debris d at time step t.
To determine how Δv_td affects debris trajectory, the new debris orbital parameters (i.e., semi-major axis, eccentricity) can be analytically computed using r_td and v_td^+ = v_td^- + Δv_td and compared with those before the L2D engagement.

The significance of DVA lies in its ability to enable a higher degree of control over debris dynamics by controlling the imparted Δv in both magnitude and direction, thus leading to a new debris orbit. Figure <ref> illustrates an example involving two laser platforms acting on the same debris at a given time step. In this specific scenario, there are three different perturbed orbits available for the debris: (1) one acted solely by platform p_1, (2) another acted solely by platform p_2, and (3) one acted collaboratively by both platforms simultaneously, resulting from a combined Δv_t_1d = Δv_t_1p_1d + Δv_t_1p_2d. The availability of multiple L2D engagement opportunities is advantageous, providing the space-based laser network with greater flexibility to achieve more effective debris remediation.





§ SPACE-BASED LASERS NETWORK DESIGN AND OPERATIONS OPTIMIZATION

The debris remediation capacity of space-based networks is correlated with the location of laser platforms. Given the distribution of the debris field and its concentration over specific altitude bins, the network's target access rate depends on the orbits taken by the platforms. Further, the laser ablation mechanism depends on the relative geometry between the platforms and debris. Hence, an optimal network topology can increase the effectiveness of laser ablation, leading to a higher debris remediation capacity.

During the debris remediation mission, each platform can present multiple L2D engagement opportunities. The dynamic scheduling of resources enables the network to select the platforms that present the most favorable relative geometry with respect to debris, such that the debris remediation capacity is maximized due to the L2D ablation mechanism.

The remainder of Sec. <ref> is organized as follows. In Sec. <ref>, we exhibit the generation of parameters involved in the optimization, and in Sec. <ref> the debris remediation capacity reward is introduced. Section <ref> materializes the proposed location-scheduling optimization problem, and Secs. <ref> and <ref> present the decoupled location and scheduling optimization problems, respectively.



 §.§ Network Optimization Parameters

As a consequence of an L2D engagement, debris can relocate to a new orbit. We define 𝒥_td as the set of orbital slots for debris d at time step t. Each orbital slot j ∈𝒥_td is associated with the position and velocity vectors defined at time step t to fully define the state of debris orbit. Laser platform orbital slots are defined in set 𝒮, indexed by s and with cardinality S. Each laser platform is assumed to take an orbital slot in 𝒮 and maintain it during the entire mission. Moreover, we define a set of valuable assets 𝒦 with cardinality K.

The feasibility of the L2D engagements is encoded with Boolean parameter W_tsd, given as:

    W_tsd=
            1,    if a platform located at orbital slot s can engage debris d at time step t
    
            0,    otherwise


To determine if an L2D engagement is feasible, two conditions must be satisfied. First, debris d has to be in the line-of-sight of a laser platform located at orbital slot s at time step t. Given the relative position vector u_tsd and the range u_tsd = ||u_tsd||_2, Eq. (<ref>) computes the line-of-sight indicator. 

    q_tsd = ((r_ts)^2-(R_⊕+ϵ)^2)^1/2+((r_td)^2-(R_⊕+ϵ)^2)^1/2-u_tsd

where R_⊕ is the radius of the Earth and ϵ is a bias parameter. If q_tsd>0 debris is in the line-of-sight of a platform and q_tsd≤0 otherwise. Second, as defined in Ref. <cit.>, an L2D engagement can happen if u_tsd lies within the maximum and minimum operational ranges defined as u_max^d and u_min^d, respectively. The upper bound is determined by the given laser specifications, and the lower bound is set by considering the safety and operational aspects of the system. 
Leveraging parameter W_tsd, it is possible to generate for each time step t the set of laser platform orbital slots 𝒮_tdj, with index s and cardinality S_tdj, that engage debris d ∈𝒟 and generate debris orbital slot j ∈𝒥_t+1,d.



 §.§ Laser-to-Debris Engagement Reward

We introduce debris remediation capacity reward R_tdij to quantify the value of relocating debris d from its current orbital slot i ∈𝒥_td to a new orbital slot j ∈𝒥_t+1,d at time step t, as a consequence of an L2D engagement while considering potential conjunctions with the set of valuable assets 𝒦. It is defined as:

    R_tdij = C^0_td + C_tdij + αΔ h_tdij  + βM_td


First, C^0_td is an incentive term that accounts for the conjunction analysis between debris d and the set of valuable assets 𝒦 during the entire mission scenario assuming no L2D engagements, by checking if debris orbit lies inside the deterministic conjunction ellipsoid of any valuable asset in 𝒦. We define C^0_td as:

    C^0_td = 
        
            G_0,    if t ∈ [t_min, t_max]
    
            0,    otherwise

If a conjunction, or multiple ones, is feasible between debris d and any valuable asset 𝒦,  C^0_td = G_0 for all t ∈ [t_min, t_max] where G_0 is a large positive constant that incentivizes the network to engage debris during the time window [t_min, t_max]. The length of the interval and its starting point are parameters to be defined by a user considering the characteristics of the space-based laser, its capability for changing debris orbits, and the desired miss distance. Given t_c, the time step at which the first conjunction occurs (if debris d is to have multiple conjunctions with the same valuable asset or with several ones), t_max should be defined such that t_max≤ t_c.
Figure <ref> illustrates a scenario where reward C^0_td is not activated since there is no feasible conjunction. Figure <ref> outlines the case where a conjunction is imminent and the reward is activated.


Second, C_tdij is a look-a-head penalty term that accounts for whether relocating debris d from orbital slot i to orbital slot j due to an L2D engagement at time step t triggers at least one conjunction with any valuable asset in 𝒦.
Once all debris' candidate orbital slots are generated for step t + 1, each orbital slot is propagated from t+1 to t+1+τ, where τ represents the number of look-a-head time steps. Further, we compare for every t ∈ [t+1, t+1 +τ] if the range between debris d at slot j with any valuable asset is less than the range threshold. If a specific L2D engagement with debris generates a conjunction with any valuable asset defined in 𝒦, we set C_tdij=-G to ensure that the network is discouraged from relocating debris d to orbital slot j, as illustrated in Fig. <ref>. Conversely, Fig. <ref> highlights two simultaneous L2D engagements where the resultant debris orbital slot does not invade the valuable asset's conjunction ellipsoid; hence C_tdij is not activated.



Third, Δ h_tdij is a look-a-head incentive term that compares the periapsis radius of debris d orbital slot j after an L2D engagement at time step t with the periapsis radius threshold h^*, at which we consider that debris is governed by atmospheric drag and subject to induced natural orbital decay. Once this decay is initiated, the debris is considered deorbited within the framework. Given h_t+1,dj, the periapsis radius of the new orbital slot j for debris d after an L2D engagement, the reward term is defined as:

    Δ h_tdij = γ_tdij(h^*/h_t+1,dj)^3

where γ_tdij is a variable that compares the periapsis radius of debris d between initial orbital slot i at time step t and after the L2D engagement at final orbital slot j at time step t+1, defined as:

    γ_tdij = 
        
            -G_h,    if h_t+1,dj≥ h_tdi
    
            1,    otherwise

If an L2D engagement increases the periapsis radius, then a large negative constant G_h is assigned to γ_tdij to discourage the engagement.
Furthermore, given that Δ h_tdij is a parameter that captures the ratio between the periapsis radius threshold and debris orbit periapsis radius after the engagement, a value of Δ h_tdij≥ 1 implies that the object is deorbited; hence, we constrain it to be at most one to avoid obtaining large rewards due to the nature of the terms' formulation. Additionally, we scale it by α in Eq. (<ref>) giving more flexibility to the user.

Lastly, M_td accounts for debris d mass, which is activated if debris is engaged at time step t, and whose objective in the reward formulation is two-fold. 
First, debris of large mass is identified as the main source of new debris in case of a collision, and therefore in alignment with Refs. <cit.> and <cit.>, which assign a debris remediation capacity reward to debris proportional to its mass, we aim to assign a bigger reward for engaging debris of larger mass.
Second, given Eq. (<ref>), it is clear that for the same laser and c_m parameters, debris of larger mass will have a smaller Δ v than those of smaller mass, making it harder to achieve significant changes in its trajectory, resulting in a smaller Δ h_tdij. To address the two key points stated above, we introduce the mass reward term, defined as:

    M_td = 
        
            m_d/m_max,    if debris d is engaged at step t
    
            0,    otherwise

where m_max is the maximum debris mass in 𝒟. We normalize the mass of debris d dividing it by the maximum debris mass to be consistent in terms of magnitude with Δ h_tdij. Further, we introduce β in Eq. (<ref>) to scale it with respect to other parameters.



 §.§ The Concurrent Location-Scheduling Optimization Problem

The optimal mission design and operation of a network of space-based lasers is tackled by proposing the Concurrent Location-Scheduling optimization Problem (CLSP). This problem seeks to determine the optimal location for a set of platforms while considering an optimal L2D engagement scheduling to maximize the debris remediation capacity.



  §.§.§ Decision Variables

We model the placement of laser platforms using the following platform location decision variables:

    z_s = 
        1,    if a laser platform is located at orbital slot s
    
        0,    otherwise

Similarly, we leverage the following L2D engagement decision variables:

    y_tsd = 
        1,    if a laser platform located at orbital slot s engages debris d at time step t
    
        0,    otherwise


At each time step t, debris d is enforced to take a new orbital slot j ∈𝒥_t+1,d. The set of debris orbital slots can be subdivided into two subsets. First, 𝒥_t+1,d⊆𝒥_t+1,d with index j and cardinality J_t+1,d contains the orbital slots that encode no changes in the current orbit for debris d since it is not engaged, and consequently they have no associated reward. Second, the set of new orbits, each one with its corresponding reward R_tdij, is encoded as 𝒥_t+1,d∖𝒥_t+1,d with index j. In CLSP, debris relocation decision variables are given as:

    x_tdij = 
        1,    if debris d relocates from orbital slot i to orbital slot j at time step t
    
        0,    otherwise




  §.§.§ Constraints and Objective Function

We introduce constraints (<ref>) to enforce that a laser platform located at orbital slot s engages debris d at time step t only if it satisfies the L2D engagement requirements encoded in parameter W_tsd. Constraints (<ref>) enforce that each platform has at most one L2D engagement per time step.


   W_tsdz_s ≥y_tsd,      ∀t∈𝒯, ∀s ∈𝒮,∀d∈𝒟

   ∑_d∈𝒟 y_tsd ≤1,      ∀t∈𝒯, ∀s ∈𝒮 



We define path contiguity constraints to ensure the correct flow of debris throughout the entire mission horizon. First, we require all debris to occupy a new orbital slot at time step t_1 with constraints (<ref>). Second, constraints (<ref>) ensure that debris d follows a continuous path between steps t-1 and t+1 with respect to the intermediate reference slot i ∈𝒥_td at time step t.

    
           ∑_j∈𝒥_t_1d x_t_0di_0j = 1,      ∀d ∈𝒟

           ∑_j∈𝒥_t+1,d x_tdij -  ∑_υ∈𝒥_t-1,d x_t-1,dυi = 0,      ∀t∈𝒯∖{t_0,t_T-1}, ∀d ∈𝒟, ∀i ∈𝒥_td 
    


Constraints (<ref>) are linking constraints that couple debris relocation decision variables x_tdij and L2D engagement decision variables y_tsd. A relocation to orbital slot j ∈𝒥_t+1,d is only possible if debris d is engaged by a set of laser platforms located at orbital slots in 𝒮_tdj, which enable such a relocation. We relax 𝒥_t+1,d given that it has an associated 𝒮_tdj=∅ and debris is enforced to occupy only one new orbital slot imposed by constraints (<ref>) and (<ref>).

    ∑_s∈𝒮_tdj y_tsd≥ S_tdjx_tdij,   ∀ t∈𝒯∖{t_T-1},∀ d∈𝒟, ∀ i ∈𝒥_td,∀ j ∈𝒥_t+1,d∖𝒥_t+1,d


Additionally, constraint (<ref>) enforces the number of platforms in the network to be equal to P.

    ∑_s∈𝒮 z_s = P


The domains of all decision variables are given as follows:


    z_s ∈{0,1},     ∀s∈𝒮  

    y_tsd ∈{0,1},       ∀t∈𝒯,∀s∈𝒮, ∀d ∈𝒟 

    x_tdij ∈{0,1},       ∀t∈𝒯∖{t_T-1},∀d ∈𝒟, ∀i ∈𝒥_td,∀j ∈𝒥_t+1,d  



Lastly, the model's objective is to maximize the remediation capacity of the network, which is given as:

    ∑_t∈𝒯∖{t_T-1}∑_d∈𝒟∑_i∈𝒥_td∑_j∈𝒥_t+1,d R_tdijx_tdij




  §.§.§ Mathematical Formulation

Piecing everything together, the mathematical formulation for CLSP is given as follows:

max      Objective function (<ref>) 

s.t.       Constraints (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>)


We report all sets, parameters, and variables used in the CLSP mathematical formulation in Table <ref>.





  §.§.§ Illustrative Example

To illustrate CLSP, we present a small-scale instance with S=3, T=3, D=3, and P=2.
Figure <ref> outlines the tree structure of the problem with platform orbital slots s represented with rhombus, and debris orbital slots i,j with squares. The magenta lines represent feasible decision variables y_tsd, the gray lines are debris d feasible relocation variables x_tdij at time step t from orbital slot i∈𝒥_td to the set of orbital slots j ∈𝒥_t+1,d, and the dark lines are debris feasible relocation variables x_tdijto orbital slots j ∈𝒥_t+1,d∖𝒥^t+1_d. Unfeasible decision variables are given a large negative reward R_tdij and are not displayed in the figure.



In Fig. <ref> debris is initially located at orbital slot i ∈𝒥_t_0d and has an associated virtual orbital slot j ∈𝒥_t_1,d, which represents no change in orbit if no L2D engagement occurs, and consequently R_t_0dij=0. The subset of debris orbital slots that represent new orbits depends on the number of feasible L2D engagements, for instance, while the leftmost debris can be engaged only by the leftmost platform orbital slot, debris located in the center orbital slot can be engaged by at most two platforms, hence it has three new orbits, all of them with its corresponding reward R_t_0dij. Given that, at time step t_1 the number of total debris orbital slots is eight and that the number of possible L2D engagements is seven, the total number of debris orbital slots at time step t_2 expands to 18.

The problem has 94 integer binary variables and 85 constraints and is solved using the Gurobi optimizer 11, which retrieves an objective value (i.e., debris remediation capacity reward obtained) of 7.50 with a duality gap of 0.00%. In Fig. <ref> the selected platform orbital slots are colored in dark gray, the debris orbital slots in blue, and decision variables y_tsd=1 and x_tdij=1 are straight lines, conversely, decision variables equal to zero are represented with dashed lines. 

Even though the problem is presented for a small number of time steps and orbital slots, the number of decision variables and constraints grows exponentially. Further, Fig. <ref> accurately showcases the tree structure of the problem and how it rapidly expands. Scaling this formulation for larger mission horizons with more debris and platform orbital slots makes the problem computationally prohibitive, restricting its application to small-scale problems.

To overcome this challenge, we propose two separate optimization formulations. First, we find the optimal laser network topology design using MCLP, then, we leverage it to optimize the debris remediation capacity using the L2D-ESP.



 §.§ Optimal Laser Network Topology Design

The problem of designing an optimal laser network topology is tackled under the assumption that debris does not relocate; consequently, decision variables x_tdij are relaxed, that is, removed, along with constraints (<ref>), (<ref>), (<ref>) and (<ref>). The resulting formulation inherits the location-scheduling structure from CLSP with location decision variables z_s, L2D engagement decision variables y_tsd and an orbital slot-independent reward, given as:

    R_td = C^0_td + M_td


The network topology reward R_td accounts for feasible conjunctions between debris d and any valuable asset in 𝒦 leveraging parameter C^0_td, and for the mass of the engaged debris d with parameter M_td, both introduced in Eq. (<ref>).
However, the new reward R_td cannot assess whether that L2D potential interaction will effectively lower debris periapsis radius because the Δ h_tdij term from Eq. (<ref>) is relaxed. To address this problem, we introduce Boolean parameter W_tsd', defined as:

    W'_tsd=
            1,    if W_tsd=1 and h_t+1,dj≤ h_tdi
    
            0,    otherwise


Given the nature of reward R_td, the new formulation can prioritize the selection of platform orbital slots closer to larger debris and neglect orbital slots that can engage small debris objects since the change in debris periapsis radius is not considered in the reward.
Further, the resulting topology which is concentrated over larger debris, can considerably reduce the L2D engagement opportunities during the L2D-ESP described in Sec. <ref>.
To tackle this problem, we propose to leverage MCLP to design an optimal laser network topology. We refer the reader to Appendix A for an in-depth justification.



  §.§.§ Decision Variables

To formulate our problem as an MCLP, we keep platform location decision variables z_s and introduce new decision variables x_td defined as:

    x_td = 
        1,    if debris d is engaged at time step t
    
        0,    otherwise


These new decision variables are distinct from CLSP L2D engagement decision variables y_tsd in a manner that they encode if debris is engaged or not, and do not discriminate which platforms perform L2D ablation over debris.



  §.§.§ Constraints and Objective Function

In light of the fact that no L2D engagement can occur between a platform located at orbital slot s and debris d if orbital slot s is not taken, linking constraints (<ref>) between decision variables z_s and x_td are imposed. Decision variables x_td are activated if at least S_td∈ℤ_≥1 platforms engage debris d at time step t.

    ∑_s∈𝒮 W'_tsdz_s≥ S_td x_td,   ∀ t∈𝒯, ∀ d ∈𝒟

Here, S_td is a parameter inspired from the cardinality of set 𝒮_tdj used in constraints (<ref>) which imposes the required platforms to engage debris d in order to relocate it to orbital slot j. In this formulation, S_td aims to impose a lower bound on the number of platforms required to engage debris d at time step t.
This requirement can be beneficial in scenarios where large debris is targeted and a large Δv is required to generate a significant change in debris orbit.

The number of platforms to be used in the network is enforced by constraints (<ref>), and the decision variables domain in the location formulation are defined as:


        z_s ∈{0,1},     ∀s∈𝒮  

        x_td ∈{0,1},       ∀t∈𝒯, ∀d ∈𝒟 



The MCLP formulation aims to maximize the realized network topology reward of the laser network during the mission time horizon 𝒯, which is encoded in objective function (<ref>):

    π = ∑_t∈𝒯∑_d∈𝒟R_td x_td

If an L2D engagement exists for debris d at time step t, then x_td=1 and reward R_td associated is obtained.  It is important to mention that if R_td = 1, ∀ t ∈𝒯, ∀ d ∈𝒟, the problem will yield the total number of potential engagements that the network has at the beginning of the L2D-ESP.



  §.§.§ Mathematical Formulation

Piecing all constraints and the objective function together, the resulting MCLP optimization formulation that determines the location of the laser platforms to maximize the network topology reward is given as:

max      Objective function (<ref>) 

s.t.       Constraints (<ref>), (<ref>), (<ref>) and (<ref>)

The nature of the objective function and the lack of L2D engagement constraints similar to constraints (<ref>) incentivize the network to locate platforms at orbital slots that have a larger number of debris in range. Lastly, Table <ref> presents the new parameters and variables introduced for the MCLP formulation.





 §.§ Optimal Laser-to-Debris Engagement Scheduling Problem

In Sec. <ref>, we presented a framework to obtain the optimal location of the laser platforms to maximize the network topology rewards. In this section, we build upon the assumption that the location of platforms is given and we maintain the engagement constraints from CLSP to derive the L2D-ESP. 

Relaxing constraints (<ref>) and (<ref>), the L2D-ESP is:

max      Objective function (<ref>) 

s.t.       Constraints (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>)


Even though the constraints that involve location decision variables z_s are relaxed, the scheduling problem continues to suffer from an exponential expansion of the solution space. To overcome this challenge, we propose a trade-off between the optimality of the solution and the computational runtime by implementing a myopic policy algorithm. The algorithm consists of breaking the problem into a set of coupled subproblems and solving them in a sequential manner such that the solution of a subproblem is used as initial conditions for the immediately subsequent subproblem. For the L2D-ESP, we parameterize the time step t and partition it into T coupled subproblems, where each subproblem solves the L2D-ESP for a time step t, therefore, each one of them has a solution space smaller than the original scheduling problem, making it lighter to solve.



  §.§.§ Decision Variables

To solve the L2D-ESP by leveraging myopic policy, we parameterize time step t present in decision variables (<ref>) and (<ref>). Additionally, orbital slot index i ∈𝒥_td is relaxed from constraints (<ref>) since the initial orbital slot for each subproblem is known. The new decision variables for the subproblem are defined as:

    
       y_pd(t) = 
        1,     if platform p engages debris d 

        0,     otherwise
    

       x_dj(t, i) = 
        1,     if debris d relocates to orbital slot j 

        0,     otherwise
    
    




  §.§.§ Constraints and Objective Function

To define the engagement constraints, we parameterize time step t in constraints (<ref>) and (<ref>). Furthermore, since no platform orbital slots are considered in the framework, we introduce the set of platforms 𝒫_dj(t) ⊆𝒫 with index p and cardinality P_dj(t), that generate orbital slot j ∈𝒥_d(t) for debris d ∈𝒟. Then, the engagement and linking constraints between variables for the subproblem are:


    ∑_d∈𝒟y_pd(t) ≤1,       ∀p∈𝒫

    ∑_p∈𝒫_dj(t) y_pd(t) ≥P_dj(t)x_dj(t, i),      ∀d∈𝒟, ∀j ∈𝒥_d(t) 



In addition to constraints (<ref>), we need to enforce each debris to relocate to at most one new orbital slot. To achieve this, we introduce constraints (<ref>).

    ∑_j∈𝒥_d(t) x_dj(t, i) ≤ 1,   ∀ d ∈𝒟


The domains of the decision variables are given as:


    x_dj(t,i) ∈{0,1},       ∀d∈𝒟, ∀j∈𝒥_d(t)  

    y_pd(t) ∈{0,1},       ∀p∈𝒫, ∀d∈𝒟



To account for the debris remediation capacity of the network for each subproblem t, we derive the reward for every time step t from Eq. (<ref>) by parameterizing indices t and i, given as:

    R_dj(t, i) = C^0_d(t) + C_dj(t,i) + αΔ h_dj(t,i)  + βM_d(t)

albeit we are solving subproblem t, we let C^0_d(t)=C^0_td and C_dj(t,i)=C_tdij with C^0_td and C_tdij defined as in Eq. (<ref>). Therefore, objective function (<ref>) encodes the debris remediation capacity of the network for subproblem t.

    V(t) = ∑_d∈𝒟∑_j∈𝒥_d(t)R_dj(t,i)x_dj(t,i)




  §.§.§ Mathematical Formulation

The optimization problem for a single subproblem t is formulated as:

max      Objective function (<ref>)

s.t.       Constraints (<ref>), (<ref>) (<ref>), (<ref>) and (<ref>)

Table <ref> presents the sets, parameters, and decision variables specific to subproblem t.



As a result of implementing the myopic policy algorithm, we can obtain the network's debris remediation capacity over the entire mission horizon by summing up objective function (<ref>) for all subproblems, as defined in Eq. (<ref>).

    V = ∑_t∈𝒯V(t)




§ CASE STUDIES
 
To demonstrate the extension and flexibility against different mission environments of the proposed formulations, we present three case studies involving debris remediation missions for (1) small, (2) large, and (3) mixed debris populations (comprising both small and large debris) in the presence of 10 valuable assets. For each case study, we determine the optimal network topology of 10 laser platforms based on the given debris field using the MCLP, obtain its network topology reward, and then evaluate its debris remediation capacity using the L2D-ESP.

The results obtained for the optimal network of 10 space-based laser platforms are benchmarked against two baseline cases: (1) a single platform case and (2) a 10-platform case in an optimized Walker-Delta network. The first benchmark, against a single platform system, helps reveal the extent to which debris remediation capacity improves with additional platforms. The second benchmark is intended to compare the effectiveness of asymmetry in the network topology in the debris remediation capacity. Walker-Delta <cit.> is a symmetrical constellation pattern that enforces satellites to be placed in circular orbits, all of them having the same altitude and inclination, making them beneficial for missions that require global coverage. The Walker-Delta pattern is denoted as P/O/F, where P stands for the number of satellites (platforms), O for the number of orbital planes, and F for the phasing factor between satellites. To identify the optimized 10-platform Walker-Delta network, we performed an enumeration of every possible pattern given 10 platforms, resulting in 18 patterns. Additionally, we randomly selected 20 pairs of semi-major axes and inclinations from the set of platform's orbital slots and generated all possible combinations of Walker-Delta configurations using 10 satellites, resulting in 360 networks. Using a brute-force algorithm, we obtain from the pool of generated Walker-Delta networks the best-performing one (i.e., the one that collects the highest network topology reward) and obtain its debris remediation capacity leveraging the L2D-ESP. The orbital elements of the simulated debris remediation networks in the case studies along with a three-dimensional visualization of the platform's orbit and debris field are presented in Appendix B.

We adopt the laser and debris parameters from Ref. <cit.> (except for the perigee threshold). The laser system considered is a laser-diode-pumped solid-state oscillator-amplifier with an amplifier medium of Nd:YAG or ND:glass, which operates using 100ps ultraviolet pulses at the 3rd harmonic of neodymium at a wavelength of 355nm.
In line with the adopted value of c_m, we assume that the material composition of all debris is aluminum.
Furthermore, we propagate the orbits of laser platforms and debris considering up to the J_2 perturbation. The epoch t_0 is set to February 26, 2024, at 04:30:51 Coordinated Universal Time. After an L2D engagement, we assume the debris periapsis radius remains constant, neglecting the influence of J_2 to reduce the computational runtime when computing C_tdij. Lastly, to avoid a computationally prohibitive optimization problem, we propose a trade-off between the number of time steps and debris considered, that is, a larger population will be simulated with fewer time steps and vice versa.
Table <ref> summarizes the parameters shared across all case studies.





 §.§ Case Study 1: Small Debris Field

We consider a seven-day mission time horizon, uniformly discretized with a time step size of 130s, adopted from Ref. <cit.>. The time step size is determined by the L2D engagement duration of 10s, with a PRF of 56Hz, plus an additional 120s for the laser equipment's cooling. Table <ref> presents the remaining laser and debris parameters.



The small debris field is initialized based on results from a simulation conducted using ESA's Meteoroid And Space debris Terrestrial Environment Reference (MASTER-8) model <cit.>. The simulation considers small debris over an altitude range of 186km to 2000km, which accounts for explosion/collision fragments, sodium-potassium droplets, solid rocket motor slag, and multi-layer insulation. The MASTER-8 outputs are provided as relative frequencies across 100 altitude bins that discretized the low Earth orbit (LEO) altitude range of 100km to 2000km into bins having a width of 18.14km. To initialize a representative small debris field across the LEO altitude band, 820 samples are drawn from the respective relative frequencies for each altitude bin. Figure <ref> shows the small debris population per altitude bin. Each small debris object is assumed to be in a circular orbit with uniformly distributed values for the argument of periapsis, the argument of latitude, and right ascension of the ascending node (RAAN), bounded between 0 and 360deg, and inclination ranging from 0 to 180deg.

Next, we generate a set of orbital slots that laser platforms can take by defining an altitude range from 400km to 1100km, equally spaced in nine altitude layers. We adopt this range as it embraces the peak of operating satellites, outlined in Fig. <ref> and the small debris population. Further, we define 10 steps for the argument of latitude and RAAN, uniformly spaced between 0deg and 360deg. In addition, we define 9 uniformly spaced steps for the inclination between 35deg and 90deg. All orbits are assumed to have an eccentricity of zero at the epoch.



For this specific case study, we are not considering valuable assets operating in the environment; consequently, C^0_td = C_tdij = 0 for all t ∈𝒯, d ∈𝒟, i ∈𝒥_td, j ∈𝒥_t+1,d. Second, M_td is assumed to be equal for all d ∈𝒟 due to the assumption that all debris has equal density.



  §.§.§ Results and Discussions

The network topology reward obtained is π^∗= 9269, and the retrieved optimal network topology consists of 10 platforms asymmetrically distributed. The optimal L2D-ESP is implemented with the optimal network topology obtained, retrieving a debris remediation capacity reward of V^∗=3132.34. Figure <ref> provides a snapshot of the debris field with 10 laser platforms at time step t = 11, and Fig. <ref> visualizes an L2D engagement over the Antarctic region at that same time step.



From the L2D optimization results it is possible to derive certain metrics that are useful for characterizing the mission performance, even though they are not the objective of the optimization.
First, the network engages 599 small debris throughout the seven-day mission time horizon, representing the 73.04% of the total population considered. Second, the number of debris deorbited during this time horizon is 422, or 51.46% of the total population. Third, we understand debris nudging as the sum of the differences between the periapsis radius at the epoch and the end of the simulation for all engaged but not deorbited debris, yielding a value for the optimal 10-platform network of 64707.56km.

The single platform case collects a network topology reward of π^∗ = 1038 and has a debris remediation capacity of V^∗ = 722.08. The total number of engaged debris is 313 deorbiting a total of 29 and nudging debris 95321.31km. The adoption of 10 platforms over one increases by 88.80% and 76.94% the network topology reward π and the debris remediation capacity V, respectively. Considering the derived metrics, increasing the number of platforms leads to 93.12% more debris deorbited. The single platform mission nudges debris 95321.31km outperforming the 10-laser platform by 43.31%, given that the former is less successful in deorbiting objects, it has more debris objects contributing to this metric.

Lastly, we obtain a Walker-Delta network with pattern 10/1/0, an altitude of 7303.14km and inclination of 48.75deg. The network topology reward obtained by the Walker-Delta network is π= 7354 and its debris remediation capacity is V=2927.24. The derived metrics for this network exhibit that 538 debris are engaged, 382 are deorbited, and the network nudges debris a total of 63801.59km.

The debris remediation capacity of the network depends not only on the number of platforms but also on their distribution. Breaking the symmetry in the distribution of lasers leads to an increase of 20.66% and 6.54% in the network topology reward and the debris remediation capacity, respectively. Additionally, the MCLP-based network outperforms the symmetrical Walker-Delta network on the derived metrics as it deorbits 9.47% more debris, and presents an increase of 1.40% on the total debris periapsis radius difference with respect to the epoch.



 §.§ Case Study 2: Large Debris Field

The mission time horizon is set to 31 days, uniformly discretized with a time step of 160s. Each L2D engagement lasts 40s, followed by a cooling period of 120s.
For the debris population, we adopt the 50 statistically most concerning debris objects as identified by Ref. <cit.>. For simplicity, we assume that all debris objects have the same cross-sectional area while retaining their actual masses as reported in Ref. <cit.>. With this approach, the magnitude of the delivered Δ v is not constant among the considered debris field since it is inversely proportional to debris mass.
To define the platform orbital slots, we maintain the values defined for Sec. <ref> except for the altitude range, since the L2D range is larger we extend the altitude upper bound to 1400km. We report the specific parameters used in this case study in Table <ref>.





  §.§.§ Results and Discussions

The network topology reward obtained is π^∗=20330.23 with an asymmetrical platform distribution, and its debris remediation capacity is V^∗=26826.56. Figure <ref> displays the large debris field and the 10 laser platforms at time step t=1314 whereas Fig. <ref> shows two L2D engagements. The first one consists of a single L2D engagement from one platform to debris; the second one is a collaborative L2D engagement from two platforms to one debris, highlighting the significance of DVA and how the network leverages this concept to maximize rewards.



Similarly to the analysis carried out in Sec. <ref>, we derive metrics from the obtained results. The total number of L2D engagements during the operation of the mission is 20225 and the debris that is engaged the most has a mass of 9000kg. Considering that it is less likely to induce significant periapsis radius reductions due to L2D engagements at each individual time step given the large mass of debris, the term that accounts for debris mass in Eq. (<ref>) weighs the most in the reward, which is reflected in the number of engagements that debris of higher mass receive. Further, the network nudges debris 941.54km.

Changing the cardinality of the network and imposing only one platform into the MCLP problem formulation retrieves a network topology reward π^∗=2916.35 and a debris remediation capacity of V^∗=6526.54. The significant reduction in the number of platforms directly impacts the results by reducing 85.65% and 75.67% the network topology reward and the debris remediation capacity, respectively. The single platform performs 4163 L2D engagements, with the same most engaged debris object as the 10-platform network has. The difference with respect to the 10-platform network is 79.41%. The single platform nudges debris 214.70km signifying a decrement of 77.19% compared to the 10-platform network.

The best-performing Walker-Delta network, with a pattern of 10/10/0, an altitude of 7040.64km and an inclination of 76.25km, achieves a network topology reward of π=11255.10 and a debris remediation capacity V=22651.94. The Walker-Delta network underperforms compared to the asymmetrical MCLP-based network by 44.63% and 15.56% for the network topology reward and debris remediation capacity, respectively. The total number of L2D engagements for the Walker-Delta network is 17632, which is 12.82% less than that of the optimal MCLP-based network, but it continues to have the most L2D engagements to a 9000kg debris object. Lastly, the Walker-Delta network nudges debris 975.45km, compared to the 941.54km obtained by leveraging the optimal network. This phenomenon can be associated with the fact that neither the MCLP formulation nor the L2D-ESP directly optimizes the difference in debris periapsis radius with respect to the epoch.



 §.§ Case Study 3: Mixed Debris Field With Valuable Assets

In this case study, the mission horizon of seven days is uniformly discretized with a time step size of 160s to encompass the L2D engagement and cooling times for small and large debris. Further, we maintain the laser operational ranges for small and large debris as defined in Sec. <ref> and Sec. <ref>, respectively.

The small debris field is generated as in Sec. <ref> and we define a population of 820 objects. The large debris field is the same from Sec. <ref> in addition to defunct satellite COSMOS 2221 (NORAD ID: 22236) with an associated reward of G_0=e6 during the network topology optimization, and G_0=e4 for the L2D-ESP. Lastly, we define a set of 10 valuable assets 𝒦 = {k_1, …, k_10} and Table <ref> presents their respective orbital elements defined at epoch t_0, with ω and ν as the argument of periapsis and true anomaly, respectively. The conjunction ellipsoid for all valuable assets is assumed to be a perfect sphere of radius 10km.



In addition to the 10 valuable assets, the mixed debris field study includes a known close conjunction event on February 28, 2024. The conjunction event was between the defunct satellite 22236 and the non-maneuverable operational NASA TIMED satellite (NORAD ID: 26998) with a reported miss distance of 20m at the time of closest approach (TCA) <cit.>. In the simulated conjunction event within the mixed debris field, a miss distance of 2.30km is observed at the TCA, corresponding to time step t=1081.



  §.§.§ Results and Discussions

The optimal 10-platform network achieves a network topology reward of π^∗=7.10e7 and has a debris remediation capacity obtained of V^∗=2.86e5. Figure <ref> displays the mixed debris field and the 10 laser platforms at time step t=7. Figure <ref> is a detailed view of the same scenario pinpointing into three L2D engagements. First, two platforms engage two large debris objects, second, a single platform engages small debris from a closer range.



The optimal 10-platform network engages 546 debris objects of various sizes, successfully deorbiting 141 small debris, and nudging debris 43510.04km. 
The conjunction event expected to occur at time step t=1081 in this scenario is successfully avoided, with the miss distance between  NASA satellite 26998 and defunct satellite 22236 increasing from 2.30km to 2104.47km after 17 L2D engagements during the imposed time step window t∈[500, 622]. The new minimum miss distance between the two objects is 545.91km at time step t=103 when the debris has not been engaged yet.

The baseline single platform case collects a network topology reward π^∗=1.60e7 and has a debris remediation capacity of V^∗= 69211.24. Compared to the optimal 10-platform network, these values represent significant reductions of 77.46% in network topology reward and 75.83% in debris remediation capacity. The single platform engages with 228 debris, successfully deorbits 24 small debris, and nudges debris 25733.06km. These metrics present reductions of 82.97% in the number of total debris deorbited and 40.85% in the total distance nudged compared to the 10-platform network. The new closest approach between defunct satellite 22236 and satellite 26998 has a distance of 57.43km at time step t=936, and a new miss distance of 411.71km at the original TCA corresponding to t=1081.

The best-performing Walker-Delta network has a pattern 10/5/2 with an orbital altitude of 6953.14km and an inclination of 76.25deg. It achieves a network topology reward π=2.98e5 and a debris remediation capacity of V=1.23e5. When the same number of laser platforms is used but constrained to fit the symmetrical Walker-Delta pattern, there is a substantial reduction in both the network topology reward and debris remediation capacity by 99.58% and 56.90%, respectively. The Walker-Delta network engages with 441 debris objects from which it successfully deorbits 129 of them, resulting in 8.51% fewer than the optimal 10-platform network. The minimum miss distance between the active and defunct satellite is 545.94km at time step t=103, and even though the Walker-Delta does not engage the defunct satellite at the imposed time step window t∈[500, 622], it does so before it, consequently, the relative distance between the two objects at time step t=1081 is 2605.02km.



 §.§ General Discussions

Even though the case studies presented in this paper do not span all possible mission scenarios, we handpicked the most representative ones for debris remediation missions. Each mission scenario is uniquely characterized by a set of variables that condition the outcome of the space-based laser debris remediation mission. For the three different scenarios, we tested a triplet of networks, two of them optimally locate their one and 10 platforms exploiting the MCLP formulation considering the debris field distribution, and the remaining one is an optimized Walker-Delta network from a pool of 360.

Increasing the number of platforms from one to 10 leads to a significant rise in the debris remediation capacity of the network and its derived metrics. In all case studies, the debris remediation capacity improves close to 75% when the number of platforms increases. The results obtained in all three case studies enable us to conclude that further increments in the debris remediation capacity and derived metrics can be achieved using the same number of platforms by breaking the symmetry of the network and determining the optimal location of platforms leveraging MCLP to deliver the additional increments. The MCLP provides flexibility in the location of laser platforms as it accounts for the mission environment defined and seeks to select the orbital slots, which are not constrained to follow a symmetrical pattern, that maximizes the network topology reward. The outcome of this formulation significantly improves the obtained debris remediation capacity, given by the L2D-ESP, as the platforms are better located relative to each other, the debris field considered, and, if any, the satellite-debris conjunction events.



§ COST AND BENEFIT ANALYSIS

In this section, a cost and benefit analysis is performed for different optimal networks, varying the number of platforms P from one to 10. Additionally, a 10-platform Walker-Delta network is included in the study. We define the cost of a network as the number of platforms used, and the benefit as the total rewards π^∗ and V^∗ obtained. All networks are simulated in the same mission environment adopted in Sec. <ref>. 

Figure <ref> highlights the rewards obtained by each network in the optimization problems, where 10-WD stands for the 10-platform Walker-Delta network. The leftmost bar plot of Fig. <ref> presents the network topology reward π^∗, demonstrating a positive correlation between the increment in the number of platforms and the achieved reward. Further, all MCLP-based optimal networks obtain a higher reward with respect to the 10-platform Walker-Delta network.
The center of Fig. <ref> outlines the total debris remediation capacity reward V^∗ obtained leveraging the L2D-ESP. Concerning this metric, the Walker-Delta network is outperformed by MCLP-based networks with at least three platforms. Focusing on MCLP-based networks only, the obtained V^∗ does not maintain a monotonic increase with the increment in the number of platforms since adopting 8 platforms retrieves the highest reward. The rightmost bar plot of Fig. <ref> breaks down the total debris remediation capacity reward by each term of Eq. (<ref>); C_tdij term has been omitted since no networks perform L2D engagements that would trigger the penalty. First, C^0_td, the reward term that accounts for conjunctions if no L2D engagements occur, is obtained by all MCLP-based networks since the flexibility gained in the design space allows the network to engage defunct satellite 22236 in the specified time window, conversely, the Walker-Delta network is not able to collect any C^0_td reward. Moreover, the latter outperforms MCLP-based networks analyzing individual reward terms αΔ h_tdij, which accounts for the ratio between debris periapsis radius after the L2D engagement and periapsis radius threshold, and βM_td, the term that accounts for engaged debris mass, however, it is not conveyed in the total reward given its lack to obtain C^0_td, the term that weighs the most in Eq. (<ref>).



Figure <ref> presents relevant mission metrics derived from the L2D-ESP, however, it is noteworthy to mention that objective function (<ref>) is not designed to tackle any of them directly. The left bar plot of Fig. <ref> presents the number of small debris deorbited per network, exhibiting a positive correlation between the number of deorbited objects and the number of platforms. The right bar plot of Fig. <ref> reports the magnitude of nudged debris per network, where The Walker-Delta network is outperformed by all MCLP-based asymmetrical networks, and the 9-platform network provides the highest debris nudging value of 46484.58km. 



In summary, this section outlines the performance of different mission architectures varying the number of space-based lasers used and their network topology. The results demonstrate the necessity of breaking the inherent symmetry of Walker-Delta networks. MCLP-based optimal networks with at least three platforms outperform these symmetrical networks in terms of debris remediation capacity. Adopting MCLP expands the network topology design space, allowing for the retrieval of probably optimal network topologies that do not adhere to the symmetrical Walker-Delta pattern.



§ CONCLUSIONS
 
In this paper, we addressed the problem of how to efficiently tackle the remediation of orbital debris of different sizes. We presented an optimization framework suitable for the design and operation of a network of space-based lasers capable of deorbit and nudge debris, and to perform just-in-time collision avoidance. 

Introducing the MCLP formulation to design debris remediation networks, for the first time, allows us to obtain its network topology considering several variables of the mission environment, mainly, the characteristics of the targeted debris field, the number and specifications of the laser platforms, and the operation of valuable assets in space. We propose an innovative L2D-ESP that seeks to maximize the debris remediation capacity of the network during the operation of the mission. The latter leverages the novel concept of DVA, a collaborative engagement framework that enables to achieve of a higher degree of control over the L2D engagements.

In this research, we propose two sets of experiments. The first one corresponds to the case studies where the main objective is to exhibit the applicability and extension of the optimization formulations proposed. Further, the different case studies enable us to conclude that the debris remediation capacity can be increased if (i) the number of platforms used in the network is incremented and (ii) given the same number of laser platforms, breaking the symmetry of Walker-Delta networks by leveraging the MCLP formulation.
The second set of experiments corresponds to the cost and benefit analysis which ranges the number of platforms from one to 10 and includes a 10-platform Walker-Delta network. The results obtained outline that increasing the number of platforms can be beneficial up to a certain cardinality, which after it either the debris remediation capacity or certain metrics of interest tend to decrease.

There are several avenues for future research. First, consider the impact of distance on the delivered fluence to debris and include stochasticity in L2D ablation processes. Second, enhance the flexibility of the network by allowing platforms to reposition into different orbits, leveraging the concept of satellite constellation reconfiguration to maximize debris remediation capacity in a dynamic debris environment. Lastly, explore the design of a novel laser platform, taking into account laser specifications, onboard hardware characteristics, and its mass and shape, enabling an economical cost-benefit analysis of space-based laser networks for orbital debris remediation.



§ ACKNOWLEDGMENT

This work was supported by an Early Career Faculty grant from NASA’s Space Technology Research Grants Program under award No. 80NSSC23K1499.



§ APPENDIX A: LOCATION OPTIMIZATION FORMULATION

In this section, we present a step-by-step justification for the adoption of MCLP to tackle the optimal design of the space-based laser network topology. The CLSP formulation introduced in Sec. <ref> is given as:


max     ∑_t∈𝒯∖{t_T-1}∑_d∈𝒟∑_i∈𝒥_td∑_j∈𝒥_t+1,d R_tdijx_tdij <ref>

    s.t.      W_tsdz_s ≥y_tsd,      ∀t∈𝒯, ∀s ∈𝒮,∀d∈𝒟<ref>

       ∑_d∈𝒟 y_tsd ≤1,      ∀t∈𝒯, ∀s ∈𝒮  <ref>

       ∑_j∈𝒥_t_1d x_t_0di_0j = 1,      ∀d ∈𝒟<ref>

       ∑_j∈𝒥_t+1,d x_tdij -  ∑_υ∈𝒥_t-1,d x_t-1,dυi = 0,      ∀t∈𝒯∖{t_0,t_T-1}, ∀d ∈𝒟, ∀i ∈𝒥_td<ref>

       ∑_s∈𝒮_tdj y_tsd ≥S_tdjx_tdij,      ∀t∈𝒯∖{t_T-1},∀d∈𝒟, ∀i ∈𝒥_td,∀j ∈𝒥_t+1,d∖𝒥_t+1,d<ref>

       ∑_s∈𝒮 z_s = P <ref>

        z_s ∈{0,1},     ∀s∈𝒮 <ref> 

        y_tsd ∈{0,1},       ∀t∈𝒯,∀s∈𝒮, ∀d ∈𝒟 <ref>

        x_tdij ∈{0,1},       ∀t∈𝒯∖{t_T-1},∀d ∈𝒟, ∀i ∈𝒥_td,∀j ∈𝒥_t+1,d <ref>



As described in Sec. <ref>, the tree structure in the problem's solution space makes it computationally prohibitive. To overcome this problem, we adopt a series of assumptions that impact decision variables, parameters, and sets to obtain a space-based platform location optimization formulation.

The core assumption to overcome the tree structure is that debris maintains its initial orbit defined at t_0 during the entire mission. The consequences of this statement are several; first, debris cannot be deorbited, which makes the debris field constant over time. Second, debris orbital slot indices i,j can be dropped, consequently constraints (<ref>) and (<ref>) are relaxed. Third, given that orbital slot indices are dropped, CLSP decision variables x_tdij are now defined as:

    x_td = 
        1,    if debris d is engaged at time step t
    
        0,    otherwise


As a result of dropping relocation indices, reward R_tdij defined in Eq. (<ref>) is recasted as R_td=C^0_td + M_td. Removing Δ h_tdij forbids the optimization model to be informed about the post-L2D engagement behavior of debris (i.e. whether debris periapsis radius increases or decreases). To overcome this situation, we redefine W_tsd as W_tsd' where the latter incorporates as an additional requirement that debris periapsis radius after an L2D engagement has to be lower than its initial one. 

With the described assumptions and relaxations, the derived formulation is:    


max     ∑_t∈𝒯∑_d∈𝒟R_tdx_td <ref>

    s.t.      W'_tsdz_s ≥y_tsd,      ∀t∈𝒯, ∀s ∈𝒮,∀d∈𝒟

       ∑_d∈𝒟 y_tsd ≤1,      ∀t∈𝒯, ∀s ∈𝒮  <ref>

       ∑_s∈𝒮_td y_tsd ≥S_tdx_td,      ∀t∈𝒯,∀d∈𝒟

       ∑_s∈𝒮 z_s = P,<ref>

        z_s ∈{0,1},     ∀s∈𝒮 <ref> 

        y_tsd ∈{0,1},       ∀t∈𝒯,∀s∈𝒮, ∀d ∈𝒟 <ref>

        x_td ∈{0,1},       ∀t∈𝒯,∀d ∈𝒟<ref>



Even though relocation constraints are relaxed, the structure of the problem inherits the location-scheduling structure of the CLSP, since y_tsd are the L2D engagement scheduling decision variables that dictate whether an L2D engagement occurs from a platform located at slot s to debris d at time step t. Consequently, despite the fact that the new formulation does not have a tree structure, the dimension of decision variables y_tsd∈{0,1}^T × S × D can make the problem computationally prohibitive.
On top of that, given that the reward does not account for debris orbit changes, the formulation will tend to assign to each platform those orbital slots that maximize engagement with debris of larger mass.
Given two debris d_1 and d_2 with masses m_d_1 < m_d_2 and C^0_td_1=C^0_td_2=0, then M_td_1 < M_td_2 and consequently R_td_1 < R_td_1. However, throughout the L2D-ESP where changes in debris periapsis are considered, if Δ h_td_1ij≫Δ h_td_2ij, then:

    
        αΔh_td_1ij  + βM_td_1 > αΔh_td_1ij  + βM_td_1

        R_td_1ij > R_td_2ij
    

outlining that the outcome, which corresponds to the optimal network that maximizes reward R_td, can overlook platform orbital slots that collect higher debris remediation rewards during the L2D-ESP.

To tackle this problem, and considering that it is not proper to perform L2D assignments if debris change in orbit is neglected, we propose to relax CLSP L2D engagement scheduling decision variables y_tsd. Consequently, constraints (<ref>), (<ref>), (<ref>) and (<ref>) are dropped. The resulting optimization problem is given as:


max     ∑_t∈𝒯∑_d∈𝒟R_tdx_td <ref>

    s.t.      ∑_s∈𝒮 z_s = P,<ref>

        z_s ∈{0,1},     ∀s∈𝒮 <ref>

        x_td ∈{0,1},       ∀t∈𝒯,∀d ∈𝒟<ref>


However, the new problem lacks coupling constraints that link location decision variables z_s and decision variables x_td such that L2D engagements with debris d occur only if a platform is occupying an orbital slot such that W_tsd=1. Inspired by constraints (<ref>) and (<ref>), we introduce the new coupling constraints given as:

    ∑_s∈𝒮 W'_tsdz_s≥ S_td x_td,   ∀ t∈𝒯, ∀ d ∈𝒟<ref>

where x_td are activated if at least S_td platforms perform L2D ablation on debris d at time step t. Hence, the formulation imposes a minimum number of platforms required to obtain debris' reward R_td at time step t. Consequently, the optimal laser network topology design problem is given as:


    max     ∑_t∈𝒯∑_d∈𝒟R_td x_td<ref>

    s.t.      ∑_s∈𝒮 z_s = P<ref>

       ∑_s∈𝒮 W'_tsdz_s≥S_td x_td,    ∀t∈𝒯, ∀d ∈𝒟<ref>

        z_s ∈{0,1},     ∀s∈𝒮<ref>

        x_td ∈{0,1},       ∀t∈𝒯, ∀d ∈𝒟<ref>


where its structure resembles the well-known MCLP <cit.>.



§ APPENDIX B: 3D ILLUSTRATIONS AND CASE STUDIES RESULT'S ORBITAL ELEMENTS

We report the corresponding orbital elements of the results obtained in Sec. <ref> along with the 3D visualization of the debris remediation network with the platform's orbits in the ECI reference frame. Figures <ref>, <ref> and <ref> display for P=1, 10 and the Walker-Delta network the debris remediation networks for the small, large and mixed debris field cases, respectively. Tables <ref>, <ref> and <ref> present the orbital elements for the P=1, 10 and Walker-Delta networks for the small, large, and mixed debris field cases, respectively.

 








