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Stress-Sharing: A Bio-Inspired Approach to Decentralized Fault Repair in Modular Spacecraft
Sidhdharth D. Sikka, Yue Shen, Shaoshuai Mou
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 94%
Last extracted: 7/18/2026, 9:59:10 AM
Summary
This paper introduces 'Stress-Sharing', a decentralized, bio-inspired repair policy for modular spacecraft. The approach models the spacecraft as a lattice-constrained graph where surviving modules autonomously migrate toward damaged regions using local distress signals to restore connectivity. The method ensures connectivity preservation through a local criticality test and is evaluated via PyBullet simulations, demonstrating effectiveness in consolidating modules into a single connected component even under high fault densities (up to 30%).
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Relation Signals (9)
Sidhdharth D. Sikka → affiliatedwith → Purdue University
confidence 99% · Sidhdharth D. Sikka is with Manifold Research Group and the School of Aeronautics and Astronautics, Purdue University
Stress-Sharing → appliedto → Modular Spacecraft
confidence 98% · Structural damage in modular spacecraft can disrupt mechanical and communication connectivity... We evaluate the policy... for large modular spacecraft.
Yue Shen → affiliatedwith → Manifold Research Group
confidence 95% · Yue Shen is with Manifold Research Group
Shaoshuai Mou → affiliatedwith → Purdue University
confidence 95% · Shaoshuai Mou is with the School of Aeronautics and Astronautics, Purdue University
Sidhdharth D. Sikka → affiliatedwith → Manifold Research Group
confidence 95% · Sidhdharth D. Sikka is with Manifold Research Group
Stress-Sharing → inspiredby → Biological Wound Healing
confidence 95% · introduce a fully decentralized, asynchronous stress-sharing repair policy inspired by biological wound healing
Stress-Sharing → evaluatedusing → PyBullet
confidence 92% · We evaluate the policy in PyBullet rigid-body simulation
Stress-Sharing → modelssystemas → Lattice-Constrained Graph
confidence 90% · We model the spacecraft as a lattice-constrained graph
Stress-Sharing → usesmechanism → Local Criticality Test
confidence 90% · The policy combines local stress-sharing signal propagation with connectivity-safe pivot motions... We use the conservative local mobility test
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Abstract
Abstract:Structural damage in modular spacecraft can disrupt mechanical and communication connectivity, reducing system capability. Existing approaches rely on redundancy or preplanned reconfiguration and do not enable autonomous repair under local information and physical constraints. We model the spacecraft as a lattice-constrained graph and introduce a fully decentralized, asynchronous stress-sharing repair policy inspired by biological wound healing: local distress signals guide surviving modules toward damaged regions to close fragmented gaps, after which each displaced module locally retraces its own motions to recover the pre-damage shape, using only local information and no absolute position sensing. We evaluate the policy in PyBullet rigid-body simulation across structures of up to 160 modules, three fault densities (10, 20, 30%), and random and localized damage. The policy consolidates the surviving modules into a single connected body: even in the most severe case tested, where 30% of modules fail at random, it gathers roughly 80% or more of the surviving modules into one connected component, and this fraction improves with assembly size, making the approach well suited as a swarm-scale repair policy for large modular spacecraft.
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- Source: https://arxiv.org/abs/2607.13444v1
- Canonical: https://arxiv.org/abs/2607.13444v1
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X .2026.Doi Number Preprint. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. (Corresponding author: Sidhdharth D. Sikka) Sidhdharth D. Sikka is with Manifold Research Group and the School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907 USA (e-mail: sikkas@purdue.edu). Yue Shen is with Manifold Research Group (e-mail: yzs.shen@gmail.com). Shaoshuai Mou is with the School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907 USA (e-mail: mous@purdue.edu). Stress-Sharing: A Bio-Inspired Approach to Decentralized Fault Repair in Modular Spacecraft Sidhdharth D. Sikka Purdue University, West Lafayette, IN, USA Yue Shen Manifold Research Group Shaoshuai Mou Purdue University, West Lafayette, IN, USA (2026) Abstract Structural damage in modular spacecraft can disrupt mechanical and communication connectivity, reducing system capability. Existing approaches rely on redundancy or preplanned reconfiguration and do not enable autonomous repair under local information and physical constraints. We model the spacecraft as a lattice-constrained graph and introduce a fully decentralized, asynchronous stress-sharing repair policy inspired by biological wound healing: local distress signals guide surviving modules toward damaged regions to close fragmented gaps, after which each displaced module locally retraces its own motions to recover the pre-damage shape, using only local information and no absolute position sensing. We evaluate the policy in PyBullet rigid-body simulation across structures of up to 160160 modules, three fault densities (1010, 2020, 30%30\%), and random and localized damage. The policy consolidates the surviving modules into a single connected body: even in the most severe case tested, where 30%30\% of modules fail at random, it gathers roughly 80%80\% or more of the surviving modules into one connected component, and this fraction improves with assembly size, making the approach well suited as a swarm-scale repair policy for large modular spacecraft. IEEEkeywords Bio-inspired robotics, decentralized control, fault tolerance, modular spacecraft, multi-agent systems, self-reconfiguration 1 Introduction Spacecraft on long-duration missions must tolerate failures with no opportunity for manual repair, yet traditional mitigation relies on redundancy and conservative design margins that add mass and cost while remaining vulnerable to unanticipated and cascading faults. In response, the field is shifting toward modular, reconfigurable architectures that can lower lifecycle cost and respond to damage by physically reorganizing their surviving components [22], reflecting the autonomy that on-orbit servicing and in-space assembly will require [2]. However, most distributed-spacecraft and swarm-assembly approaches [7, 9] still rely on attached manipulator arms or dedicated free-flying service robots to reposition modules [19], lacking the intrinsic, highly distributed structural plasticity needed for rapid fault recovery in degraded environments. In this paper, we consider the problem of damage-responsive reconfiguration in modular spacecraft. Following a localized failure, a subset of modules becomes inactive and the remaining structure must restore operational connectivity using only local sensing and physically admissible motions (Fig. 1). Unlike global shape planning, this setting imposes three coupled constraints: (i) motion is limited to lattice-admissible pivot operations, (i) connectivity must be preserved throughout execution, and (i) agents operate without centralized coordination or global state information. These constraints fundamentally limit the applicability of existing reconfiguration and planning approaches. Figure 1: Homeostatic reconfiguration in a modular robotic system: after a localized failure, neighboring modules reposition to restore connectivity using only local information and admissible motions. 1.1 Related Work Modular reconfiguration systems A broad set of literature has studied hardware and algorithms for modular robotic reconfiguration. Lattice-based systems such as M-Blocks demonstrate dynamic pivoting through inertial actuation [11], while ElectroVoxel achieves pivot-based reconfiguration in microgravity [13]. At the algorithmic level, universal reconfiguration results show that arbitrary shapes can be achieved with a constant number of helper modules [1]. More generally, modular reconfiguration has been studied through decentralized control, distributed motion planning, cellular automata, and lattice-based transformation rules [10, 23, 6]. These works establish a rich foundation for shape transformation, but they typically assume nominal operation, without modeling module failure, or focus on achieving a target configuration rather than restoring functionality after damage. Distributed self-repair paradigms Classical distributed self-repair strategies often rely on paradigms that are poorly matched to spacecraft constraints. Scale-independent morphallaxis, as developed for robotic collectives and related regenerative swarm robotics, rebuilds a proportionately scaled version of the structure after damage [12]. Other approaches, including directed-growth and hole-routing style methods for modular robots [16, 10], attempt to migrate empty space or restructure the body so that missing components can be replaced. While effective in many terrestrial settings, these paradigms are less suitable for modular spacecraft, where total mass is fixed, spare modules may not be available, and the objective is often to restore critical connectivity without global reconstruction. Connectivity maintenance and network repair A related literature addresses connectivity preservation and restoration in networked multi-agent systems. In wireless sensor and actor networks, RECRA and related localized motion-based restoration algorithms relocate non-critical nodes to repair broken communication graphs using local information [4, 21]. In multi-robot systems, decentralized connectivity maintenance has been studied through graph-theoretic and control-theoretic methods, including local-global approaches, bounded-input connectivity preservation, and distributed connectivity control [5, 17, 3]. These works provide important primitives for decentralized decision-making, especially bounded-hop tests for determining whether a node can move without disconnecting its neighbors. However, they operate primarily on communication graphs or continuous mobile robot models, and do not map these ideas onto the rigid geometric and kinematic constraints of a pivoting three-dimensional lattice. Biologically inspired and regenerative systems Biological inspiration has also played a central role in self-reconfiguration and repair. Hormone-based coordination has been used to trigger adaptive restructuring in modular robots [14, 20], while regenerative and developmental approaches have explored local message-passing, gradients, and self-organized recovery [12, 16, 18]. More recently, [15] identifies stress-sharing as a mechanism by which local homeostatic error can induce neighboring units to increase plasticity and coordinate restorative restructuring. These ideas are highly suggestive for robotics, but existing implementations do not explicitly enforce connectivity preservation and physically admissible motion during damage-responsive reconfiguration. Gap Taken together, the literature reveals an important gap. Existing approaches either assume global planning and persistent connectivity, operate on abstract graphs without physical motion constraints, or provide bio-inspired coordination mechanisms without explicitly enforcing connectivity preservation during execution. To the best of our knowledge, there is no existing framework for damage-responsive repair in modular robotic spacecraft that simultaneously satisfies the following requirements: strictly local decision-making, lattice-constrained physically admissible motion, and connectivity-safe execution throughout the repair process. To address this gap, we propose a decentralized reconfiguration strategy that combines local stress-sharing signal propagation with connectivity-safe pivot motions, enabling a damaged modular structure to restore connectivity and partially recover its pre-failure organization without centralized planning or external service robots. 1.2 Contributions The main contributions of this paper are as follows: • A fully decentralized, asynchronous repair framework for modular spacecraft that restores connectivity after damage using only local sensing and communication. The framework combines a bio-inspired stress-sharing policy for connectivity restoration with a restructuring phase that partially recovers the pre-damage shape, without centralized coordination or absolute position sensing. • A high-fidelity PyBullet rigid-body simulation framework for evaluating damage-responsive reconfiguration, modeling physically realized pivot motions through rolling-sphere contact dynamics. We evaluate the proposed approach on assemblies of up to 160160 modules under random and localized failures with damage levels of 1010–30%30\%. The remainder of this paper is organized as follows. Section I establishes the graph-theoretic preliminaries, physical constraints, and the damage-responsive reconfiguration problem. Section I presents the distributed stress-sharing repair framework. Section IV evaluates the policy in a PyBullet rigid-body simulator and reports the Monte Carlo results and performance analysis, with the full simulator implementation given in Appendix 6. Section V concludes the paper and outlines future directions. 2 Problem Formulation We model a modular spacecraft as a lattice-connected assembly of interchangeable modules, where we assume only the relative position of the modules is relevant and not their relative attitudes. At time t, the system is represented as a directed graph t=(V,Et,gt),G_t=(V,E_t,g_t), where V is the set of modules (vertices). Throughout the paper, u,v,w∈Vu,v,w∈ V denote arbitrary modules. Et⊆V×VE_t V× V is the set of directed edges representing physical connections between modules, and gt:Et→ℝ3g_t:E_t ^3 assigns to each directed edge (u,v)(u,v) the relative translation from u to v. For each (u,v)∈Et(u,v)∈ E_t, gt(u,v)=uv∈ℝ3,g_t(u,v)= ρ_uv ^3, with inverse gt(v,u)=gt(u,v)−1=−uv=vu.g_t(v,u)=g_t(u,v)^-1=- ρ_uv= ρ_vu. Because all edge transformations are pure translations, composition along any path reduces to vector addition. In particular, for consecutive edges (u,v),(v,w)∈Et(u,v),(v,w)∈ E_t, gt(u,w)=uv+vw.g_t(u,w)= ρ_uv+ ρ_vw. Each relative translation uv ρ_uv is constrained to lie along the unit lattice directions uv∈±^,±^,±^,‖uv‖=1. ρ_uv∈\± x,\,± y,\,± z\, \| ρ_uv\|=1. Let u↝w=(v0,v1,…,vk)u w=(v_0,v_1,…,v_k) denote a path from u to w, where v0=uv_0=u, vk=wv_k=w, and (vi−1,vi)∈Et(v_i-1,v_i)∈ E_t for i=1,…,ki=1,…,k. We define the displacement function dt(u,w)=∑i=1kgt(vi−1,vi).d_t(u,w)= _i=1^kg_t(v_i-1,v_i). This function returns the relative displacement vector between two vertices by composing the lattice translations along the path u↝wu w. We assume cycle consistency, i.e. dt(u,u)=d_t(u,u)= 0, implying that the displacement function is path-independent. 2.1 Active Subgraph and Damage A subset V¯t⊆V V_t V denotes the set of active modules at time t. The corresponding active edge set is E¯t⊆V¯t×V¯t E_t V_t× V_t, and the active subgraph is defined as ¯t=(V¯t,E¯t,g¯t), G_t=( V_t,\, E_t,\, g_t), where g¯t g_t denotes the restriction of the edge transformation map to the active edges. Only modules in V¯t V_t participate in reconfiguration and control. Throughout this paper, we will use u¯,v¯,w¯∈V¯t u, v, w∈ V_t to refer to arbitrary active modules. A damage event at time t induces a new active set V¯t+1=V¯t∖Ft, V_t+1= V_t F_t, where Ft⊆V¯tF_t V_t is the set of damaged or failed modules at that time. All physical connections remain present in the underlying graph EtE_t, but edges incident to inactive vertices are excluded from E¯t+1 E_t+1. As a result, the active subgraph ¯t+1 G_t+1 may become disconnected, necessitating reconfiguration to restore structural or functional connectivity. The neighbor set for a vertex u at time t is defined as Nt(u):=w∣(u,w)∈EtN_t(u):=\\,w (u,w)∈ E_t\,\, and the active neighbor set for a vertex u¯ u is defined as N¯t(u¯):=w¯∣(u¯,w¯)∈E¯t N_t( u):=\\, w ( u, w)∈ E_t\,\. 2.2 Detachability and Attachability A vertex u¯ u may participate in reconfiguration subject to the following rules. Detachability A vertex u¯ u may detach from a neighboring vertex v by removing both directed edges (u¯,v)( u,v) and (v,u¯)(v, u) from EtE_t, resulting in Et+1=Et∖(u¯,v),(v,u¯),E_t+1\;=\;E_t \( u,v),(v, u)\, (1) if and only if |N¯t+1(u¯)|≥1| N_t+1( u)|≥ 1. That is, the vertex initiating detachment must remain connected to at least one other active neighbor after the operation. The vertex v may be either active or inactive. Attachability A vertex u¯ u may attach to a vertex v by adding directed edges (u¯,v)( u,v) and (v,u¯)(v, u) to the graph, Et+1=Et∪(u¯,v),(v,u¯),E_t+1\;=\;E_t∪\( u,v),(v, u)\, (2) if and only if the following conditions hold: 1. the relative displacement u¯v∈±^,±^,± ρ_ uv∈\± x,\,± y,\,± z\, 2. the direction u¯v ρ_ uv is not already occupied by an existing edge incident to v, i.e., u¯v≠xv∀x∈Nt(v). ρ_ uv≠ ρ_xv ∀ x∈ N_t(v). When the attachment occurs, the edge transformations are defined as gt+1(u¯,v)=u¯v,gt+1(v,u¯)=−u¯v.g_t+1( u,v)= ρ_ uv, g_t+1(v, u)=- ρ_ uv. 2.3 Shape Similarity We summarize the shape of the active subgraph at time t by its inter-module distance matrix PtP_t, with entries [Pt]u¯v¯=‖dt(u¯,v¯)‖2,u¯,v¯∈V¯t,[P_t]_ u v=\|d_t( u, v)\|_2, u, v∈ V_t, the pairwise distances between active modules. Because it depends only on relative displacements, PtP_t is invariant to the global pose of the assembly. For two configurations with inter-module distance matrices P and Q, their shape difference is the square-loss Gromov–Wasserstein (GW) discrepancy [8] diff(P,Q)=minπ∈Π(P,Q)(∑i,k,j,l|Pik−Qjl|2πijπkl)1/2,diff(P,Q)= _π∈ (P,Q) ( _i,k,j,l |P_ik-Q_jl |^2\, _ij _kl )^1/2, where Π(P,Q) (P,Q) denotes the set of admissible correspondences between the active vertex sets represented by P and Q. Each correspondence is represented by a nonnegative matrix π whose row and column sums are uniform over the active vertices of the respective configurations, with πij _ij indicating how strongly active vertex i in the first configuration is matched to active vertex j in the second. The GW discrepancy therefore searches for the correspondence that best preserves pairwise inter-module distances between the two active subgraphs. Because it compares only relative distances rather than absolute coordinates, the measure is invariant to global translation, rotation, reflection, and module relabeling. The matrices are normalized by their common maximum, so diff(P,Q)∈[0,1]diff(P,Q)∈[0,1]. Following a damage event at time t, the active subgraph ¯t+1 G_t+1 may be disconnected. We seek a sequence of admissible reconfiguration operations that restores active connectivity while minimizing deviation from the pre-damage shape. Let PtP_t denote the pre-damage shape and PtfP_t_f the terminal shape at time tf≥t+1t_f≥ t+1. The reconfiguration problem is formulated as min¯τ=t+1tfdiff(Pt,Ptf) _\ G_τ\_τ=t+1^t_f \! (P_t,\,P_t_f ) such that ¯tf G_t_f is connected and every intermediate operation obeys the detachability and attachability rules (1)–(2) of Section 2.2. 3 Proposed Approach: Distributed Stress-Sharing Repair 3.1 Homeostasis in Nature Biological tissues repair themselves through local action without centralized control. In wound healing (Fig. 2), cells migrate toward an injury site under local chemical and mechanical cues and coordinate through signaling pathways to close the gap. Shreesha and Levin [15] formalize one such mechanism, stress sharing: when units exchange information about their internal strain, the collective coordinates restructuring through a lightweight local channel, with no central coordinator. This principle motivates the repair policy developed below. Figure 2: Cellular migration during wound healing: tissue damage (left), directional migration toward the wound (middle), and repair (right). Repair proceeds without centralized oversight, through local mechanisms such as chemical gradients and signal propagation. 3.2 Failure Signals and Directional Propagation We propose an asynchronous distributed policy inspired by stress sharing: failure information diffuses through the active subgraph, biasing non-critical modules to migrate toward the fault while preserving connectivity. This coagulation process is detailed in Algorithm 1. The full proposed approach is summarized in Figure 3. Figure 3: Per-agent control flow of the two-phase distributed repair process. Left (coagulation): an agent propagates a distress token and, if a pivot improves alignment with the fault and the bounded-hop criticality test certifies it as movable, executes a connectivity-safe detach–attach toward the fault; otherwise it stays and forwards. Right (restructuring): each displaced agent retraces its coagulation pivots in reverse, reconstructing each source cell relative to the current pose of its bonded neighbor and executing the connectivity-safe reverse pivot. Each ego agent u¯ u maintains a set of distress tokens ℳt(u¯)⊆Ft×ℝ3,M_t( u) F_t×R^3, where a token (f,)(f, ξ) indicates a suspected failed vertex f∈Ftf∈ F_t and a direction vector ξ expressed in the frame of u¯ u. Local detection If u¯ u detects that a former neighbor f is inactive (e.g., heartbeat timeout), then u¯ u generates the token (f,):=u¯f,(f, ξ) ξ:= ρ_ uf, where the right-hand side is interpreted as the last-known displacement prior to deactivation. Propagation When u¯ u receives or creates a token (f,)(f, ξ) from a neighbor, it updates and forwards the token by composing the edge translation: ←w¯u¯+, ξ← ρ_ w u+ ξ, and broadcasts (f,)(f, ξ) to neighbors w¯∈N¯t(u¯) w∈ N_t( u). Because path composition is additive under pure translations, ξ approximates the displacement from neighbors w¯ w toward the fault location. 3.3 Local Criticality Test When ℳt(u¯)M_t( u) is nonempty, the ego agent u¯ u must decide whether it can move without creating a new disconnect in the active graph ¯t G_t. We use the conservative local mobility test from prior network-repair work [21], parameterized by a safety radius rsafe≥2r_safe≥ 2: the largest number of hops over which u¯ u verifies that its neighbors would remain mutually connected after it leaves. Let ¯t−u¯ G_t- u be the active graph with u¯ u and its incident edges deleted, and d¯t−u¯(x¯,v¯)d_ G_t- u( x, v) the shortest-path distance between x¯ x and v¯ v in that graph (∞ if disconnected). The agent u¯ u is declared movable if it is a leaf (|N¯t(u¯)|≤1| N_t( u)|≤ 1) or every pair of distinct neighbors stays connected within rsafer_safe hops after its disconnection, d¯t−u¯(x¯,v¯)≤rsafe∀x¯,v¯∈N¯t(u¯),x¯≠v¯.d_ G_t- u( x, v)≤ r_safe ∀\, x, v∈ N_t( u),\ x≠ v. The test thus asks a purely local question: if u¯ u disappears, can all of its former neighbors still reach one another by short paths? We use rsafe=2r_safe=2 throughout, the minimal admissible radius; the parameter study (Appendix 8) shows reconnection is insensitive to rsafer_safe over values 2, 3 or 4, so the cheapest setting is adopted. The test is conservative, since a failed test may still admit a longer global replacement path. Lemma 1 (Conservative sufficiency of the local mobility test). Let ¯t=(V¯t,E¯t,g¯t) G_t=( V_t, E_t, g_t) be the active graph. If u¯ u passes the rsafer_safe local mobility test, then disconnecting u¯ u does not disconnect the active component containing u¯ u. The converse is false: deleting u¯ u can be globally safe even when the bounded-hop test rejects it. Proof. If |N¯t(u¯)|≤1| N_t( u)|≤ 1, then u¯ u is a leaf or isolated vertex, and removing it cannot disconnect any two other vertices. Now suppose |N¯t(u¯)|>1| N_t( u)|>1. Because u¯ u passes the test, every pair of distinct neighbors x¯,v¯∈N¯t(u¯) x, v∈ N_t( u) is joined by a path x¯↝v¯⊆¯t−u¯ x v G_t- u of length at most rsafer_safe, so all former neighbors of u¯ u lie in one connected component of ¯t−u¯ G_t- u. Any component of ¯t−u¯ G_t- u arising from the original component containing u¯ u must contain at least one former neighbor; since those neighbors are mutually connected, all remaining vertices belong to a single component, and deleting u¯ u creates no new disconnect. For non-necessity, take a simple even cycle of length ℓ and any vertex u¯ u on it. Deleting u¯ u leaves a replacement path of length ℓ−2 -2 between its two former neighbors, so the deletion is globally safe; yet the bounded-hop test accepts it only if ℓ−2≤rsafe -2≤ r_safe, i.e. ℓ≤rsafe+2 ≤ r_safe+2. Thus any even cycle with ℓ>rsafe+2 >r_safe+2 is globally safe but locally rejected (the first rejected cycle is the six-cycle for rsafe=2r_safe=2 and the octagon for rsafe=4r_safe=4). ∎ Remark 1 (Conservatism). The test is conservative: it certifies only replacement paths of length at most rsafer_safe, so moves whose safety rests on longer or nonlocal cycles are rejected even when globally safe. Increasing rsafer_safe admits more moves at the cost of a larger local search; eliminating the conservatism entirely would require global articulation-point information, violating the decentralized assumption of this work. 3.4 Distress-Directed Pivot Selection Once an agent u¯ u is certified movable, it selects the closest target distress token (f⋆,⋆)∈ℳt(u¯)(f , ξ ) _t( u), i.e., (f⋆,⋆)=argmin(f,)∈ℳt(u¯)‖2.(f , ξ )= *arg\,min_(f, ξ) _t( u)\| ξ\|_2. The agent evaluates all physically admissible pivot motions, meaning those satisfying the detachability and attachability constraints in Section 2.2, where each candidate motion induces a local lattice displacement Δ∈ℝ3 p ^3. Let ⟨⋅,⋅⟩ ·\,,· denote the inner product of two vectors in ℝ3R^3. To prioritize movement toward the fault, the agent selects the pivot that maximizes alignment with the target direction: Δ⋆=argmaxΔ⟨Δ,⋆⟩, p = _ p p, ξ , (3) subject to the alignment constraint ⟨Δ⋆,⋆⟩>0 p , ξ >0. Anti-oscillation memory Each agent additionally maintains a small per-agent set ℋt(u¯)H_t( u) of previously visited module pairs. A candidate pivot is rejected if its module pair already lies in ℋt(u¯)H_t( u). This is a strictly local memory and is necessary to break oscillations that can arise on symmetric structures, where two cells repeatedly tie under (3) and the deterministic tie-break alternates between them as the local frame rotates. In sparse topologies where no improving move exists from either cell, ℋt(u¯)H_t( u) also serves as a termination certificate at the agent level: once every admissible pair is exhausted, the agent remains stationary. Stochastic exploration When no admissible pivot strictly improves alignment, the agent admits an exploration step: with probability ε it selects an admissible pivot uniformly at random from those passing the geometric and memory checks, rather than remaining stationary. We set ε=1.0 =1.0 (always explore when no improving move exists), which the parameter study (Appendix 8) shows aids reconnection on dense structures. This is the primary deadlock-breaking mechanism: in dense or symmetric neighborhoods where greedy alignment offers no improving move, random repositioning lets the agent escape the local configuration and re-evaluate from a new cell, with the anti-oscillation memory ℋt(u¯)H_t( u) preventing it from revisiting cells. Because every exploratory pivot is still connectivity-safe (it passes the criticality test) and its outcome is bounded by the per-agent motion budget (Appendix 6), exploration cannot break connectivity or run unbounded. Algorithm 1 Coagulation Policy 1: Input: agent u¯ u; tokens ℳt(u¯)M_t( u); budget amove(u¯),afwd(u¯)a_move( u),\,a_fwd( u) 2: while amove(u¯)a_move( u) and afwd(u¯)>0\,a_fwd( u)>0: 3: for f∈Nt(u¯)f∈ N_t( u) with f∉V¯tf∉ V_t and afwd(u¯)>0a_fwd( u)>0: 4: ℳt(u¯)←ℳt(u¯)∪(f,u¯f)M_t( u) _t( u)∪\(f,\, ρ_ uf)\ 5: if ℳt(u¯)=∅M_t( u)= : 6: continue 7: (f⋆,⋆)←argmin(f,)∈ℳt(u¯)∥ (f , ξ )← *arg\,min_(f, ξ)\,∈\,M_t( u) ξ 8: if amove(u¯)>0a_move( u)>0 and Criticality Test (Sec. 3.3): 9: Δ⋆←argmaxΔ∉ℋt(u¯)⟨Δ,⋆⟩ p ← *arg\,max_ subarrayc p _t( u) subarray p,\, ξ 10: if ⟨Δ⋆,⋆⟩>0 p , ξ >0 or rand()<εrand()< : 11: execute Δ⋆ p about bonded neighbor b; 12: amove(u¯)-=1a_move( u)\! -=\!1; 13: ℋt+1(u¯)←ℋt(u¯)∪Δ⋆H_t+1( u) _t( u)∪\ p \; 14: ℛ(u¯).append(b,)R( u).append(b, s); (Sec. 3.6) 15: continue 16: for w¯∈N¯t(u¯) w∈ N_t( u): 17: ℳt+1(w¯)←ℳt(w¯)∪(f⋆,w¯u¯+⋆)M_t+1( w) _t( w)∪\(f ,\, ρ_ w u+ ξ )\ 18: afwd(u¯)-=1a_fwd( u)\! -=\!1 3.5 Action Budgets and Message Complexity Action budgets Each agent begins the phase with two integer budgets: a motion budget amovea_move and a communication budget afwda_fwd. A movable token-holder spends one amovea_move per executed pivot; if it cannot or does not move, it instead relays the token, spending one afwda_fwd, and a fault-adjacent source likewise stops seeding once its afwda_fwd is spent. The two pools are separate so that relaying never starves motion. Both are finite, which bounds each agent’s total work and drives the phase to termination. Message complexity A forwarding wave at u¯ u emits at most |N¯t(u¯)|| N_t( u)| messages, so a tick in which every holder forwards costs O(|E¯t|)O(| E_t|) token-creations, and the per-agent communication budget caps the cumulative count over a phase at O(|V¯t|afwd)O(| V_t|\,a_fwd). Total messages to convergence are problem-dependent, governed by the number of pivots and the wavefront geometry, so Section 4 reports move counts as the operative measure of repair effort. 3.6 Restructuring After Coagulation After the coagulation phase, the active subgraph may be reconnected but its shape is distorted, since restoring connectivity required agents to migrate away from their pre-damage positions. The second phase, restructuring, partially recovers the pre-damage shape without re-breaking the connectivity established during coagulation. Each displaced agent retraces its own coagulation pivots in reverse, locally undoing the motion it contributed. The procedure is summarized in Algorithm 2. Let tct_c denote the time at which coagulation terminates and restructuring begins. 3.6.1 Pivot history During coagulation, each active vertex u¯ u records the ordered sequence of pivots it executes, ℛ(u¯)=((b1,1),…,(b|ℛ(u¯)|,|ℛ(u¯)|)),R( u)= ((b_1, s_1),…,(b_|R( u)|, s_|R( u)|) ), where, for the k-th pivot, bk∈Vb_k∈ V is the bonded neighbor about which u¯ u rotated, and k∈ℝ3 s_k ^3 is u¯ u’s pre-pivot position expressed in the body frame of bkb_k. A vertex is displaced whenever ℛ(u¯)≠∅R( u)≠ . 3.6.2 Reverse replay Restructuring replays ℛ(u¯)R( u) in reverse, from the most recent pivot k=|ℛ(u¯)|k=|R( u)| down to k=1k=1. For the current step k, the target position is the recorded source cell reconstructed against the bonded neighbor’s current pose, k(u¯)=t(bk)+Rt(bk)k, q_k( u)= p_t(b_k)+R_t(b_k)\, s_k, where t(bk) p_t(b_k) and Rt(bk)∈SO(3)R_t(b_k) (3) are the current position and body-frame orientation of the bonded neighbor. Because k q_k is expressed entirely relative to bkb_k, it requires no global reference and remains correct even if the structure has been rigidly repositioned since the original pivot was made. 3.6.3 Greedy reverse pivot The agent reuses the pivot-selection rule of Section 3.4 with the reverse target in place of a distress direction. Writing the residual rk(u¯):=‖t(u¯)−k(u¯)‖2r_k( u):=\| p_t( u)- q_k( u)\|_2, the agent retains the admissible pivots that strictly decrease it, t(u¯)=Δ:‖t(u¯)+Δ−k(u¯)‖2<rk(u¯),P_t( u)= \ p:\| p_t( u)+ p- q_k( u)\|_2<r_k( u) \, and executes the one of greatest reduction, Δ⋆=argmaxΔ∈t(u¯)(rk(u¯)−‖t(u¯)+Δ−k(u¯)‖2). p = _ p _t( u) (r_k( u)-\| p_t( u)+ p- q_k( u)\|_2 ). When bkb_k is still bonded to u¯ u, the maximizing pivot is the exact inverse of the original and returns u¯ u to k q_k; when the local geometry no longer admits it, the rule selects the closest admissible pivot, leaving u¯ u at an intermediate cell that is nonetheless nearer the source. Once step k completes or admits no improving pivot, the agent advances to step k−1k-1. A step whose bonded neighbor is no longer active, bk∉V¯tb_k∉ V_t (e.g. it was a failed module used only as a passive pivot reference), cannot be anchored locally and is skipped; the agent retraces the remainder. Algorithm 2 Restructuring Policy 1: Input: own pivot history ℛ(u¯)R( u), fault set FtF_t 2: for k∈|ℛ(u¯)|,…1k∈|R( u)|,… 1: 3: if bk∉V¯tb_k∉ V_t: 4: continue 5: if not Criticality Test (Sec. 3.3): 6: continue 7: k←t(bk)+Rt(bk)k q_k← p_t(b_k)+R_t(b_k)\, s_k 8: t(u¯)←P_t( u)← admissible pivots reducing ‖t(u¯)−k‖\| p_t( u)- q_k\|, excluding FtF_t and occupied sites 9: if t(u¯)≠∅P_t( u)≠ : 10: Execute argmax reduction; 11: Re-bond adjacent modules Table 1: Results of Monte Carlo Trials in Simulated Damage Scenarios Tree FC Metric n 10% 20% 30% 10% 20% 30% Reconnection Rate (%) 10 87 71 45 85 68 51 80 56 30 6 45 18 3 160 37 10 1 25 3 0 Restoration (%) 10 96 92 85 96 92 85 80 98 96 91 97 93 84 160 98 97 93 98 95 88 Shape diff. (%) 10 18 19 19 18 19 19 80 5 7 8 6 8 7 160 5 7 7 5 6 – Phase 1 Moves 10 6 7 6 6 7 7 80 75 69 47 85 73 49 160 201 167 99 210 154 94 Figure 4: Reconnection rate versus number of modules n for tree (left) and fully connected (right) structures. Solid curves are random faults, dashed are localized, colored by fault density. Reconnection falls with both density and scale, and at equal density localized faults are markedly easier than random. Figure 5: Restoration (percentage of surviving non-fault modules in the single largest component) versus number of modules n, random faults, colored by density. Dotted curves are the level immediately after damage; solid curves are after the repair policy (phase-1 end); the gap is the policy’s contribution. The achieved level stays ≳80% \!80\% and rises with assembly size, even where full reconnection has fallen. 4 Simulations We evaluate the coagulation–restructuring policy by Monte Carlo on tree (acyclic) and fully connected (FC, dense multi-neighbor) lattice assemblies of n∈[10,160]n∈[10,160] modules (Appendix 7). We sweep three fault densities (10/20/30%10/20/30\%) and two spatial distributions reported in detail, random and localized. Each cell uses 500500 trials. We report four quantities. Reconnection rate is the fraction of trials whose final active subgraph is a single connected component. Restoration is L/AL/A, the fraction of the A active (non-fault) modules lying in the largest active component (L) at phase-1 end: it is 100%100\% when the survivors form one body, and excluding faults from both terms makes it invariant to how many modules were lost. We also report the shape difference diff(Pt,Ptf)diff(P_t,P_t_f) before and after restructuring and mean Phase 1/2 move counts. Only trials whose fault set actually disconnects the post-damage subgraph are counted as meaningful. 4.1 Physical Realization We evaluate in a custom PyBullet rigid-body simulator: the decision-level policy of Section 3 runs exactly as specified, but as a contact-rich rigid-body world rather than kinematic placement. A 90∘90 pivot is a rolling-sphere coupling between an agent and its bonded neighbor (one sphere rolling without slipping around another), so it sweeps exactly the source and target lattice cells and settles into bonding tolerance. Being force-driven, each pivot incurs residual pose error, occasional non-convergence, and attitude drift, contact-dynamics costs a kinematic model omits and under which all results are obtained. Additional detail is available in Appendix 6. Module geometry and the locality assumption Each module is a unit-diameter sphere with cardinal contact connectors, as in M-Blocks [11]. This is load-bearing for locality: the criticality test (Section 3.3) and pivot selection (Section 3.4) consult only a bounded neighborhood, which certifies a safe pivot only because a sphere rotating 90∘90 about a contact neighbor sweeps exactly the source and target lattice cells; rotational symmetry thus collapses the clearance check to a lattice-cell occupancy check among graph-local modules. Non-spherical modules break this (a cube sweeps corners through graph-distant cells), so certifying clearance would need geometric awareness beyond the local radius. We therefore treat cubes, polyhedral voxels, and articulated modules as future work. 4.2 Results Two regimes of connectivity restoration The central finding is that reconnection and restoration behave oppositely with scale (Table 1). Reconnection, the all-or-nothing measure of full reunification, falls steeply with both density and n, and less so on trees than on FC (Fig. 4), reaching ≈0%≈\!0\% for 30%30\% random damage at n=160n=160. Restoration instead stays high, and is best read as the policy’s improvement over the post-damage state (Fig. 5): on trees, where damage shatters the structure, the gain is large and grows with severity (at 30%30\%, n=160n=160 the largest surviving component rises from ≈26%≈\!26\% to ≈93%≈\!93\% of survivors, a +67+67-point gain); dense FC structures fragment less, so the policy’s contribution is smaller but the achieved level is comparable (tree edging FC, e.g. 93%93\% vs. 88%88\% at 30%30\%, n=160n=160). Either way restoration stays ≳80% \!80\% and rises with n, so even a “2%2\%-reconnection” trial has gathered the large majority of survivors into one body: substantial consolidation is the norm, full closure the exception at scale. The residual fragments are a limit of strictly-local greedy repair. Communication saturates, and the last fragments are too far apart for any local pivot to bridge. Phase-1 effort grows with n and density to a few hundred moves at n=160n=160. Spatial distribution dominates fault amount At equal density, the spatial pattern matters more than the amount. Localized faults are far easier than random (at n=160n=160, tree 20%20\%: 38%38\% vs. 10%10\%; 30%30\%: 20%20\% vs. ≈1%≈\!1\%; clustered falls between), because localized damage leaves one large component plus a compact hole the wavefront bridges directly, whereas random damage shatters the structure into many fragments that must each be reattached. This reverses the intuition that scattered failures are benign and concentrated damage catastrophic; under strictly-local repair the opposite holds. Shape recovery in phase 2 Restructuring reduces shape difference modestly, by ≈0.8≈\!0.8 percentage points on average, and not monotonically: it improves about two-thirds of trials and worsens the rest by a comparable amount, a ≈2:1≈\!2:1 frequency gain. The effect is small by design, bounded by the same connectivity constraint as the rest of the policy: the criticality test forbids any displaced module that has become an articulation point from pivoting home, so phase 2 can refine shape only through the non-critical minority, leaving a safety-limited residual. Larger recovery would require relaxing the strict connectivity guarantee (e.g. bounded temporary disconnections), which we leave to future work. Shape values in Table 1 are averaged over fully-reconnected trials only; “–” marks cells where none reconnected. 5 Conclusions We presented a decentralized stress-sharing strategy that lets a damaged modular spacecraft repair itself using only local information and connectivity-safe pivots, validated under rigid-body physics. Three findings stand out. Consolidation, gathering the survivors into a single body, stays high and improves with assembly size, even where full reconnection becomes rare. The spatial concentration of damage, not its magnitude, is the primary determinant of repairability. And the residual failures at high density are a limit of the strictly-local greedy nature of our approach. The locality of the method rests on the rotational symmetry of spherical modules, whose pivot sweep stays within the source and target cells; other geometries would require longer-range clearance certification. The residual-fragment limit points directly at the next step: a lightweight longer-range rendezvous mechanism, invoked only for the few fragments local repair cannot close, could convert high consolidation into full reconnection. Non-spherical and articulated modules, and convergence and shape-recovery guarantees, remain open for future work as well. More broadly, these results suggest that resilience in space robotics may rely less on redundancy and more on distributed reorganization inspired by biological repair, letting modular spacecraft maintain function after unexpected failures without external intervention. 6 Simulator Implementation This appendix details how the decision-level policy of Section 3 is realized in the PyBullet rigid-body simulator used for all results. Dynamics and the rolling-sphere pivot Each module is a rigid unit-diameter sphere (R=0.5R=0.5) of unit mass in a gravity-free world; PyBullet integrates the Newton–Euler dynamics at a fixed timestep Δt=0.05 t=0.05 s with 4040 solver iterations and light damping, all bonds and pivots enforced at the velocity level rather than placed kinematically. A 90∘90 pivot is a rolling-sphere coupling between the agent and its bonded neighbor: at each substep the relative pose is projected onto the manifold of fixed centre separation 2R2R with zero tangential slip (a no-slip roll of one sphere around another, via several Gauss–Seidel iterations), replacing PyBullet’s default joint. This keeps the lattice-cell abstraction faithful, the sweep occupying exactly the source and target cells and settling into bonding tolerance, but, being force-driven, incurs residual pose error, occasional non-convergence, and attitude drift. Pivot primitives, clearance, and radii A movable agent enumerates two candidate families about a bonded neighbor: a corner pivot to an adjacent lattice cell, and a lateral handoff that mid-flight swaps its bond to a second bonded neighbor, preserving the always-bonded invariant of Section 2.2; candidates are scored by alignment (3) with deterministic tie-breaking. Clearance is checked at the lattice level, by target-cell occupancy plus the anti-oscillation memory of Section 3.4, rather than by swept-volume queries; on completing a pivot an agent re-bonds to its new lattice neighbors, the path that yields essentially all reported reconnections (a substep proximity-bond fallback fires rarely). Two independent bounded-hop radii are used: the safety radius rsafe=2r_safe=2 of the criticality test (Section 3.3) and an exclusion radius rexc=4r_exc=4 over which a pivoting agent locks out concurrent nearby pivots. Concurrency and pivot execution Agents pivot concurrently under the rexcr_exc-hop exclusion lock, with a fresh random per-tick visit order and implicit backoff, and no explicit priority or reservation. A start-of-tick cleanup removes spurious lattice triangles (an artifact of bonding in each neighbor’s rotated frame), itself gated by the criticality test, since an ungated cleanup by an articulating agent can increase the component count. A pivot that fails to converge within a timeout is reversed or snapped to the nearest lattice cell; only completed, re-bonded pivots count as moves. Faults, budgets, and termination A fault is retained as a full-mass collision body, so others cannot pass through it, but is excluded from every algorithmic operation (V¯t V_t, token propagation, the connectivity test); it may still act as a passive bonded neighbor for nearby pivots, never a handoff. The per-agent budgets of Section 3.5 (amove=5a_move=5, afwd=50a_fwd=50) bound total work and give a decentralized termination signal: a phase ends on reconnection or at quiescence (every agent budget-exhausted or idle with no actionable token and none pivoting), a genuine fixed point since finite budgets bound moves and forwards and sources stop seeding once their communication budget is spent. Splitting the pools is what prevents communication from draining the budget at scale and starving motion (which would cause a spurious size-dependent collapse), and the tight motion budget doubles as a regularizer, since under a loose budget an agent lacking an improving move keeps taking disruptive exploratory pivots. 7 Structure Generation and Damage Models Structure generation. Both topologies are grown by a random walk on the cubic lattice from a single seed module at the origin: we maintain a frontier of unoccupied cells adjacent to the current structure, then repeatedly pick a frontier cell (uniformly at random), place a module there, and extend the frontier, until n modules are placed. The topologies differ only in bonding. A fully connected (FC) structure bonds each new module to all of its already-placed lattice neighbors, yielding a dense multi-neighbor body with cycles; a tree bonds each new module to a single parent (the frontier cell’s originating neighbor), giving an acyclic spanning structure. Random frontier selection produces varied shapes at each n. Damage models. A damage event selects a fault set FtF_t and is constrained to disconnect the active graph. A random set is a uniform sample of |Ft||F_t| modules; a localized set is a single contiguous region grown by breadth-first search from a random seed module. We use rejection sampling, redrawing the set until an offline connectivity check confirms it disconnects the active subgraph. 8 Parameter Sensitivity and Baselines We vary each policy parameter in isolation, holding the others at the operating point above, on n∈40,80,160n∈\40,80,160\ across all fault densities and spatial patterns (100100 trials per cell); we focus on reconnection rate, the most parameter-sensitive metric, while shape difference stays in a narrow band and restoration stays high throughout. Safety radius Reconnection varies by only a few points over rsafe∈[2,4]r_safe∈[2,4] on both topologies. The conservative mobility test’s depth is thus not a critical parameter, and we adopt the cheapest setting, rsafe=2r_safe=2. Exploration temperature On trees reconnection is insensitive to ε ; on FC it rises monotonically with exploration (from ≈18%≈\!18\% at ε=0 =0 to ≈34%≈\!34\% at ε=1.0 =1.0), since random repositioning helps escape the denser deadlocks. We therefore use ε=1.0 =1.0. Token-emission interval Denser emission improves reconnection, sharply so on FC: it holds near its peak for τgen≤5 _gen≤ 5 but collapses for sparse emission (FC falls to single digits by τgen=20 _gen=20). With the communication budget decoupled from motion the dense setting is affordable, and we use τgen=2 _gen=2. Directionality baseline Because no prior method targets this exact setting (strictly local, lattice-constrained, connectivity-safe repair), we compare against a constructed baseline rather than methods from prior literature. The problem constraints leave little room for a “naive” random policy: a module acts only in response to detected damage, requiring token propagation, and may not break connectivity, requiring the local safety test, so any admissible decentralized policy already contains this substrate. The single free design choice is how a movable module picks among its admissible pivots. Our baseline therefore runs the entire proposed pipeline, fault detection, token propagation, the criticality/safety test, and action budgets, and changes only that one step: it selects a pivot uniformly at random rather than the one best aligned with the fault direction (3). Both score the same already-enumerated candidate set, so using the direction adds essentially no computation. This isolates the contribution of the stress-sharing gradient: dropping it lowers reconnection by ≈5≈\!5 percentage points on both topologies (tree 38%→33%38\%\!→\!33\%, FC 33%→27%33\%\!→\!27\%) with restoration essentially unchanged (≤2≤\!2 points). The gradient is thus a near-free increment on a substrate that itself carries the bulk of the recovery. 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