Communication-Efficient Digital-Twin Coordination for Heterogeneous LLM Embodied Agents over Computing Power Networks

# Communication-Efficient Digital-Twin Coordination for Heterogeneous LLM Embodied Agents over Computing Power Networks
Source: [https://arxiv.org/html/2607.09330](https://arxiv.org/html/2607.09330)
Nuocheng Yang,*Student Member, IEEE*, Sihua Wang, Zihan Chen,*Member, IEEE*, Tony Q\. S\. Quek, , and Changchuan Yin,*Senior Member, IEEE*N\. Yang, S\. Wang, and C\. Yin are with the Beijing Laboratory of Advanced Information Network, and the Beijing Key Laboratory of Network System Architecture and Convergence, Beijing University of Posts and Telecommunications, Beijing 100876, China \(emails: \{yangnuocheng, sihuawang, ccyin\}@bupt\.edu\.cn\)\.Z\. Chen and T\. Q\. S\. Quek are with the Information Systems Technology and Design Pillar, Singapore University of Technology and Design, 487372, Singapore \(emails: zihan\_chen@mymail\.sutd\.edu\.sg, tonyquek@sutd\.edu\.sg\)\.

###### Abstract

Embodied agent teams powered by heterogeneous large language models \(LLMs\) are being widely deployed in physical artificial intelligence such as smart factories, warehouses, and service robotics\. To enable collaboration among such an agent team, efficient coordination mechanisms that operate reliably under limited network resources are required\. However, existing heterogeneous LLM\-agent coordination frameworks that rely on multi\-round natural\-language\-based conversations introduce three coupled challenges\. First, inter\-agent dialogue incurs communication overhead that grows rapidly with team size\. Second, the quality of coordination is constrained by the heterogeneous capabilities of the agent team’s LLMs\. Third, agents may suffer from action delays due to iterative negotiation\. To address these challenges, we propose LDT\-Coord, a networked coordination framework built upon a lightweight digital twin \(DT\)\. Specifically, each agent independently selects its intended action and reports both the action decision and a structured temporal constraint over shared resources to the DT server, thereby decoupling coordination performance from natural\-language reasoning ability\. Then, DT executes a training\-free, rule\-based orchestrator algorithm to resolve cross\-agent conflicts and returns coordination instructions to prevent such conflicts\. To further reduce communication overhead, we formulate agent reporting control as a constrained partially observable Markov decision process \(C\-POMDP\) and solve it with the PPO\-Lagrangian algorithm\. Simulation results show that LDT\-Coord achieves a task success rate comparable to conventional coordination methods while reducing communication overhead by more than70×70\\timesand maintaining robustness under LLM heterogeneity\.

## IIntroduction

Large language models \(LLMs\) have shown strong capability in understanding, reasoning, and generation\[[1](https://arxiv.org/html/2607.09330#bib.bib1),[2](https://arxiv.org/html/2607.09330#bib.bib2),[3](https://arxiv.org/html/2607.09330#bib.bib3)\], which drives an embodied agents framework that can perceive, reflect, and act in a physical artificial intelligence world\[[4](https://arxiv.org/html/2607.09330#bib.bib4),[5](https://arxiv.org/html/2607.09330#bib.bib5),[6](https://arxiv.org/html/2607.09330#bib.bib6)\]\. Compared with the conventional centralized framework, where a single server controls each agent, the embodied agents framework can enable distributed intelligence, thereby improving responsiveness, scalability, and robustness in dynamic environments\. As these agents scale from single models to teams, members may run LLMs with heterogeneous capabilities to fit each agent’s task and resource budget\[[7](https://arxiv.org/html/2607.09330#bib.bib7),[8](https://arxiv.org/html/2607.09330#bib.bib8),[9](https://arxiv.org/html/2607.09330#bib.bib9),[10](https://arxiv.org/html/2607.09330#bib.bib10)\]\. To enable efficient coordination of such a heterogeneous agent team, several works focus on facilitating information exchange through multiple rounds of natural language \(NL\) dialogue\[[11](https://arxiv.org/html/2607.09330#bib.bib11),[12](https://arxiv.org/html/2607.09330#bib.bib12),[13](https://arxiv.org/html/2607.09330#bib.bib13)\]\. However, building coordination on top of mutual understanding of each other’s NL output exposes three issues\. First, at the communication level, the payload of multi\-round NL negotiation grows rapidly with team size and number of rounds, which introduces an expensive communication cost\. Second, at the heterogeneity level, the quality of NL dialogue suffers from a short\-board effect, which may be significantly affected by the participant with the weakest LLM\. Finally, at the mechanism level, agents are required to dialogue before acting, thereby introducing cooperation latency\.

Existing work addresses these issues along three technical routes\. To address the communication bottleneck, the authors in\[[14](https://arxiv.org/html/2607.09330#bib.bib14),[15](https://arxiv.org/html/2607.09330#bib.bib15),[16](https://arxiv.org/html/2607.09330#bib.bib16)\]proposed a shared\-state and blackboard method that transforms mesh dialogue into star\-shaped reads and writes over a shared memory\. However, the shared medium remains NL text, so vague descriptions written by agents with the weakest LLMs may continue to pollute the shared state, thereby reducing collaborative efficiency and performance\. To address the heterogeneity bottleneck, the authors in\[[17](https://arxiv.org/html/2607.09330#bib.bib17),[18](https://arxiv.org/html/2607.09330#bib.bib18),[19](https://arxiv.org/html/2607.09330#bib.bib19),[20](https://arxiv.org/html/2607.09330#bib.bib20)\]studied multi\-agent reinforcement learning communication routes that learn compact message vectors to bypass NL\. For the mechanism bottleneck, the authors in\[[21](https://arxiv.org/html/2607.09330#bib.bib21),[22](https://arxiv.org/html/2607.09330#bib.bib22),[23](https://arxiv.org/html/2607.09330#bib.bib23)\]introduce a centralized LLM planner that generates joint actions for all agents\. Although this architecture can avoid action conflicts through a single strong coordinator, it shifts distributed intelligence toward centralized coordination, which suffers from scalability difficulties as the number of agents grows\[[24](https://arxiv.org/html/2607.09330#bib.bib24)\]\.

Meanwhile, the digital twin \(DT\) serves as a high\-fidelity digital mirror of a physical entity in the information space\[[25](https://arxiv.org/html/2607.09330#bib.bib25),[26](https://arxiv.org/html/2607.09330#bib.bib26),[27](https://arxiv.org/html/2607.09330#bib.bib27)\]\. Through continuous state synchronization and bidirectional interaction, it has become a unified carrier for sensing, prediction, and control in networked systems\[[28](https://arxiv.org/html/2607.09330#bib.bib28),[29](https://arxiv.org/html/2607.09330#bib.bib29)\]\. In multi\-agent settings, it has been further used as a centralized mirror and reconstruction tool for the team state, aggregating scattered local observations and action into a consistent global view\[[30](https://arxiv.org/html/2607.09330#bib.bib30),[31](https://arxiv.org/html/2607.09330#bib.bib31)\]\. However, existing DT\-based schemes\[[30](https://arxiv.org/html/2607.09330#bib.bib30),[31](https://arxiv.org/html/2607.09330#bib.bib31)\]still adopt a central\-server architecture in which the DT actively generates a plan for each agent similar to\[[21](https://arxiv.org/html/2607.09330#bib.bib21),[22](https://arxiv.org/html/2607.09330#bib.bib22),[23](https://arxiv.org/html/2607.09330#bib.bib23)\]\. This introduces uncontrollable latency and computational overhead, as DT must re\-understand the physical world from scratch, even though agents have already formed a local understanding of their surroundings\.

To fill this gap, this paper proposes Lightweight Digital\-Twin Coordination \(LDT\-Coord\), which employs a lightweight DT as a coordination middleware to enable efficient collaboration among embodied agent teams driven by heterogeneous LLMs\. Specifically, each embodied agent autonomously perceives, reflects, acts, and reports its chosen action, together with structured temporal constraints on shared resources, to the DT for coordination\. Thus, the lightweight DT does not need to understand the environment from scratch as the traditional method, thereby saving computational and communication resources\. Instead, it uses a unified lightweight orchestrator that operates over a shared set of structured coordination primitives, detects and avoids potential conflicts in the cooperation process by rules, and returns efficient coordination\. The main contributions of this paper are summarized as follows\.

- •Lightweight\-DT coordination framework with structured communication primitives\.We use a lightweight DT as the coordination middleware for heterogeneous LLM agents and define structured primitives, namely the action tuple, the typed constraint declaration, and a short downlink instruction, which turn coordination from mutually understanding language into reporting structured constraints and decouple coordination quality from the language ability of the reporter\.
- •Training\-free unified rule\-based orchestrator for atomic\-task conflict avoidance\.We formalize the mutual\-exclusion, synchronization, and dependency conflict types into typed coordination rules and design a unified orchestrator applied until convergence to find the maximal consistent executable set without any training\.
- •Learned communication\-selection layer for latency\-constrained reporting\.We model the decision of which agents report at each step as a constrained partially observable Markov decision process \(C\-POMDP\) under a per\-step latency constraint and solve it with PPO\-Lagrangian, which compresses state\-reporting communication substantially while keeping coordination quality nearly lossless\.

Experiments show that, compared with the traditional NL\-dialogue coordination strategy among agents, LDT\-Coord attains a comparable success rate while reducing communication by more than 70×\\timeson the Confined\-Space Sorting task, and it stays robust across heterogeneous team configurations of different scales\.

## IIRelated Work

### II\-ANatural\-Language Dialogue\-Based Multi\-Agent Coordination

Recently, a number of works that employ the NL dialogue route have been studied to enable LLM agents to cooperate\. The authors in\[[11](https://arxiv.org/html/2607.09330#bib.bib11)\]proposed a dialogue\-based scheme that lets multiple robotic arms discuss task assignment and collision\-free paths through multi\-round NL exchange\. The authors in\[[12](https://arxiv.org/html/2607.09330#bib.bib12)\]introduced a conversable\-agent framework that orchestrates multi\-agent dialogue flows through a group\-chat manager\. The authors in\[[13](https://arxiv.org/html/2607.09330#bib.bib13)\]proposed a procedure\-driven scheme that encodes standardized operating procedures into the communication among agents to reduce cascading errors\. The authors in\[[14](https://arxiv.org/html/2607.09330#bib.bib14)\]systematized a shared\-memory architecture so that agents coordinate by reading and writing a blackboard rather than talking pairwise\. The authors in\[[32](https://arxiv.org/html/2607.09330#bib.bib32)\]drove two\-agent cooperation through a role\-playing dialogue mechanism\. The authors in\[[33](https://arxiv.org/html/2607.09330#bib.bib33)\]organized software development as a multi\-role dialogue pipeline\. These frameworks coordinate from the perspective of agents understanding one another through language outputs\. Under heterogeneous deployment, the communication quality is dictated by the weakest party\. Moreover, intent is not shared explicitly, so each agent must infer the plans of others from their text\.

To improve the dialogue itself, another line introduces reflection, debate, and interleaved reasoning\. The authors in\[[34](https://arxiv.org/html/2607.09330#bib.bib34)\]proposed a self\-reflective scheme that lets an agent iteratively improve its decisions through verbalized self\-feedback\. The authors in\[[35](https://arxiv.org/html/2607.09330#bib.bib35)\]proposed a multi\-party argument scheme that improves factuality and reasoning through debate among agents\. The authors in\[[36](https://arxiv.org/html/2607.09330#bib.bib36)\]proposed a reasoning\-acting scheme that interleaves reasoning with action and environment feedback to stabilize embodied decisions\. The authors in\[[37](https://arxiv.org/html/2607.09330#bib.bib37)\]proposed a memory\-driven scheme that simulates believable behavior through NL memory and reflection\. The authors in\[[38](https://arxiv.org/html/2607.09330#bib.bib38)\]introduced a curriculum scheme for continual skill acquisition in an open world\. The authors in\[[39](https://arxiv.org/html/2607.09330#bib.bib39)\]proposed a dynamic agent network that forms teams by selecting agents according to contribution\. These methods raise dialogue quality but spend many time steps on negotiation that yields no motion, while weak agents keep polluting the shared context\. More fundamentally, these frameworks generally adopt a mesh dialogue topology\. As a result, the negotiation links, payload, and latency grow rapidly with team size, which is especially acute in heterogeneous, link\-constrained deployment\.

### II\-BCentralized and Hierarchical Multi\-Robot Task Coordination

To circumvent the inherent defect of mesh dialogue, a number of works turn to centralized and hierarchical coordination\. A central planner then generates actions for all agents in a unified fashion\. The authors of\[[21](https://arxiv.org/html/2607.09330#bib.bib21)\]proposed a centralized orchestration scheme that decomposes and dispatches subtasks across a heterogeneous team of agents\. The authors in\[[40](https://arxiv.org/html/2607.09330#bib.bib40)\]introduced a hybrid scheme that couples an LLM with a classical symbolic planner to obtain feasible long\-horizon plans\. The authors in\[[41](https://arxiv.org/html/2607.09330#bib.bib41)\]proposed a language\-to\-motion scheme that generates executable motion sequences from NL instructions\. These works remove conflicts from the perspective of centralized planning, but their coordination quality depends entirely on the capability of the central model\. They also require uploading all observations to the center, so communication and computation scale poorly with team size\.

Classical multi\-robot task allocation instead characterizes coordination from the perspective of combinatorial optimization and scheduling\. The authors in\[[42](https://arxiv.org/html/2607.09330#bib.bib42)\]introduced a formal taxonomy and analysis of multi\-robot task allocation\. The authors in\[[43](https://arxiv.org/html/2607.09330#bib.bib43)\]proposed a human\-robot collaborative assembly planning method based on LLM agents\. These methods assume homogeneous capability and sufficient communication by default and do not target training\-free coordination of heterogeneous LLM teams under constrained links\. Therefore, the centralized route trades a single strong center for coordination quality, which runs counter to the decentralized autonomy and heterogeneity\-robust objectives pursued in this paper\.

### II\-CMulti\-Agent Communication and Learned Coordination

To circumvent the bloated negotiation payload and the message meaning that fluctuates with the sender model, multi\-agent reinforcement learning has widely explored the learned communication route\. The authors in\[[44](https://arxiv.org/html/2607.09330#bib.bib44)\]proposed a differentiable communication scheme that learns continuous messages by back\-propagating across a shared channel among agents\. The authors in\[[45](https://arxiv.org/html/2607.09330#bib.bib45)\]proposed a value\-decomposition scheme that supports centralized training with decentralized execution through monotonic mixing\. The authors in\[[46](https://arxiv.org/html/2607.09330#bib.bib46)\]introduced a multi\-level communication scheme that lets agents exchange messages in a determined order to coordinate their decisions\. These works improve coordination by end\-to\-end learning of compact protocols\. However, their messages and policies must be jointly trained and lack interpretability, so they are difficult to plug into a training\-free heterogeneous LLM team\.

The theoretical frameworks of decentralized partially observable coordination further characterize the difficulty of this problem\. The authors in\[[47](https://arxiv.org/html/2607.09330#bib.bib47)\]introduced a systematic formalization that reveals the intrinsic complexity of solving joint policies under partial observability\. The authors in\[[48](https://arxiv.org/html/2607.09330#bib.bib48)\]proposed a group\-aware coordination graph that characterizes the coordination structure among agents\. These methods rigorously characterize coordination among multiple agents from a re\-learning perspective, but they still presuppose a joint policy obtained through retraining\. This introduces additional training overhead and is difficult to combine quickly with existing teams\.

### II\-DDistributed Consistency and Consensus Control

Concurrency control and consensus in distributed systems provide classical mechanisms for multiple nodes to stay consistent without a central arbiter\. The authors in\[[49](https://arxiv.org/html/2607.09330#bib.bib49)\]proposed an optimistic concurrency control strategy that lets concurrent transactions execute first and validate conflicts at commit time with rollback as needed\. The authors in\[[50](https://arxiv.org/html/2607.09330#bib.bib50)\]introduced a logical\-clock scheme that characterizes the causal precedence among distributed events through a happens\-before partial order\. The authors in\[[51](https://arxiv.org/html/2607.09330#bib.bib51)\]proposed a consensus protocol for strongly consistent replication across unreliable nodes\. The authors in\[[52](https://arxiv.org/html/2607.09330#bib.bib52)\]proposed conflict\-free replicated data types that converge through algebraic structure without explicit coordination\. These mechanisms guarantee correctness from the perspective of reaching agreement on shared state, but their nodes are homogeneous and faithful\. Moreover, the objects they coordinate are data reads and writes or replica state, so they neither carry the action intent produced by heterogeneous LLMs nor handle physical\-reachability constraints\.

Multi\-agent consensus from a control\-theoretic perspective is closer to networked coordination\. The authors in\[[53](https://arxiv.org/html/2607.09330#bib.bib53)\]proposed a secure consensus protocol that gives robust convergence under mixed attacks\. The authors in\[[54](https://arxiv.org/html/2607.09330#bib.bib54)\]proposed a double\-layer finite\-time consensus scheme for heterogeneous dynamics\. The authors in\[[55](https://arxiv.org/html/2607.09330#bib.bib55)\]proposed a blockchain\-based scheme for trustworthy collaboration in digital\-twin edge networks\. These works characterize the convergence of continuous physical quantities to a common value under disturbance and assume homogeneous agents by default\. This focus is orthogonal to the discrete action conflicts of heterogeneous LLMs over discrete shared resources targeted here\.

### II\-EDigital Twin for Networked Coordination

The DT provides a digital replica of the physical world for networked systems\[[28](https://arxiv.org/html/2607.09330#bib.bib28),[29](https://arxiv.org/html/2607.09330#bib.bib29)\]\. A recent line of work treats the DT as a coordination layer for multi\-agent and multi\-robot teams\. The authors in\[[56](https://arxiv.org/html/2607.09330#bib.bib56)\]proposed a digital\-twin scheme that trains a multi\-agent scheduler to orchestrate heterogeneous edge\-end devices and finish deadline\-constrained jobs\. The authors in\[[57](https://arxiv.org/html/2607.09330#bib.bib57)\]proposed a digital\-twin scheme that acts as a coordination layer to assign and reassign tasks across a multi\-UAV fleet\. The authors in\[[31](https://arxiv.org/html/2607.09330#bib.bib31)\]introduced a digital\-twin scheme for dynamic manufacturing\-workshop scheduling based on multi\-agent deep reinforcement learning\. These works let the DT allocate tasks or schedule jobs across a team\. However, the twin coordinates assignment decisions rather than the discrete action intent that heterogeneous agents have already expressed in structured form\.

Another line turns the DT into an active control layer that directly drives online physical\-world decisions\. The authors in\[[25](https://arxiv.org/html/2607.09330#bib.bib25)\]proposed an end\-edge collaborative framework that uses a spatio\-temporal twin to online command a UAV team to adjust its neighbor topology for multi\-target tracking\. The authors in\[[58](https://arxiv.org/html/2607.09330#bib.bib58)\]introduced a digital\-twin edge air\-ground scheme that jointly schedules UAV trajectories, device association, and task offloading through the twin’s real\-time prediction\. The authors in\[[59](https://arxiv.org/html/2607.09330#bib.bib59)\]proposed a smart\-mobility platform that actively coordinates the routes of connected automated vehicles through cloud twin state\. The authors in\[[60](https://arxiv.org/html/2607.09330#bib.bib60)\]introduced an internet\-of\-vehicles twin network that drives adaptive resource allocation and task offloading\. These works confirm the feasibility of a twin actively driving physical\-world decisions\. However, the objects they coordinate are continuous network resources, trajectories, or offloading ratios, and their mechanism is centralized optimization of a single global objective\. As a result, they neither target the discrete action conflicts of heterogeneous LLM decision\-makers nor arbitrate the action intent that each agent has already expressed in structured form\. At a wider scale, the computing power network \(CPN\) paradigm casts coordination as the joint orchestration of heterogeneous cloud\-edge\-end computing resources together with the network that connects them\[[61](https://arxiv.org/html/2607.09330#bib.bib61)\]\. The authors in\[[62](https://arxiv.org/html/2607.09330#bib.bib62)\]proposed an intent\-aware scheduling framework that matches application demands to heterogeneous computing nodes and network paths\. The digital\-twin schedulers discussed above\[[56](https://arxiv.org/html/2607.09330#bib.bib56),[57](https://arxiv.org/html/2607.09330#bib.bib57)\]already act as computing\-power\-aware orchestration planes at the edge\-end tier\. However, these works orchestrate divisible computing jobs and continuous resources under a single global objective, rather than the discrete action intent of heterogeneous LLM decision\-makers\. Despite this progress, no existing route can actively coordinate the discrete action intent of heterogeneous LLM agents under training\-free, low\-communication, and heterogeneity\-robust conditions, which is precisely the gap targeted by the system model and problem formulated in the next section\.

Table[I](https://arxiv.org/html/2607.09330#S2.T1)organizes the comparison around the novelty of this paper and contrasts the key differences between existing routes and LDT\-Coord\.

TABLE I:Comparison of Related Work

## IIISystem Model and Problem Formulation

![Refer to caption](https://arxiv.org/html/2607.09330v1/RoCo-1.jpg)Figure 1:The six multi\-arm collaboration tasks considered in this work, spanning sequential coordination and concurrent execution under increasing workspace overlap\[[11](https://arxiv.org/html/2607.09330#bib.bib11)\]\.![Refer to caption](https://arxiv.org/html/2607.09330v1/x1.png)Figure 2:Illustration of the considered system model\.We consider a collaborative team ofnnheterogeneous LLM\-driven embodied agent arms𝒩=\{1,…,n\}\\mathcal\{N\}=\\\{1,\\dots,n\\\}that jointly complete a taskggin a shared physical workspace \(e\.g\., sort cubes, pack groceries\) as shown in Fig\.[1](https://arxiv.org/html/2607.09330#S3.F1)\. Each armiiis powered by an LLMMiM\_\{i\}whose capability may differ across agents and that can decide its own action individually based on observing only a local projection of the physical state\. To enable efficient collaboration between arms, a coordination middleware DT𝒟\\mathcal\{D\}is introduced\. It derives all of its state observations from structured reports on arms and issues conflict\-free plans to agents through a rule\-based policy\. Specifically, at each discrete time steptt, each arm and𝒟\\mathcal\{D\}execute the closed\-loop of the following four stages, as shown in Fig\.[2](https://arxiv.org/html/2607.09330#S3.F2)\.

1. 1\.State acquisition\.Each armiiobtains its observable state and concatenates it with the task objectiveggto form its local context\.
2. 2\.Action selection\.Each armiiautonomously selects an action within its legal action set based on its local context\. It also generates the constraint declaration that the selected action imposes on the operated objects \(e\.g\., whether the action requires other arms to cooperate with or to avoid it\)\.
3. 3\.Intention reporting\.Each armiipackages its intended action together with the constraint declaration into a structured message and reports it to the𝒟\\mathcal\{D\}\. Each armiiexecutes the selected action simultaneously\.
4. 4\.Conflict avoidance\.After aggregating the received messages,𝒟\\mathcal\{D\}computes the maximal consistent executable action set following conflict\-avoiding orchestration rules of thennarms in this slot\. Then, DT sends termination instructions to the vetoed arms so as to avoid conflicts\.

These four stages repeat until the task is completed\.

### III\-ALocal State Acquisition

Coordination among the arms takes place in a shared physical workspace with a set of operable entitiesℰ=\{1,⋯,E\}\\mathcal\{E\}=\\\{1,\\cdots,E\\\}\. Each objecte∈ℰe\\in\\mathcal\{E\}position on a three\-dimensional positionxt​\(e\)∈ℝ3x\_\{t\}\(e\)\\in\\mathbb\{R\}^\{3\}at time slottt\. The task objectiveggis to gradually transport every object from its initial position to its respective target positionxe⋆∈ℝ3x\_\{e\}^\{\\star\}\\in\\mathbb\{R\}^\{3\}\. We introduce the discrete placement points where an object can rest into a set of shared resources \(e\.g\., workstations, handover panels, slots, and other placement positions\)\. This placement set is denoted as𝒫\\mathcal\{P\}, where each resourcep∈𝒫p\\in\\mathcal\{P\}corresponds to a reference three\-dimensional positionx¯p\\bar\{x\}\_\{p\}\. For transport tasks that require several relay steps, the resources naturally carry an order along the transport flow\. We record the step position of resourceppin this flow as the integerpos​\(p\)∈ℤ\\mathrm\{pos\}\(p\)\\in\\mathbb\{Z\}\. We can also define the resource currently occupied by objecteeis the resource closest to its position, given byℓt​\(e\)=arg⁡minp∈𝒫⁡‖xt​\(e\)−x¯p‖\\ell\_\{t\}\(e\)=\\arg\\min\_\{p\\in\\mathcal\{P\}\}\\\|x\_\{t\}\(e\)\-\\bar\{x\}\_\{p\}\\\|\. Symmetrically, its target resource is the resource closest to the target positionxe⋆x\_\{e\}^\{\\star\}, derived aspe⋆=arg⁡minp∈𝒫⁡‖xe⋆−x¯p‖p\_\{e\}^\{\\star\}=\\arg\\min\_\{p\\in\\mathcal\{P\}\}\\\|x\_\{e\}^\{\\star\}\-\\bar\{x\}\_\{p\}\\\|\. One transport task thus amounts toℓt​\(e\)\\ell\_\{t\}\(e\)advancing along the order towardpe⋆p\_\{e\}^\{\\star\}until the endpoint is reached\. Besides the position state, we further denote logical condition state \(e\.g\., cabinet door open or closed\) asuk∈\{0,1\}u\_\{k\}\\in\\\{0,1\\\}, whereuk=1u\_\{k\}=1means the condition holds, anduk=0u\_\{k\}=0otherwise\. All logical conditions form the set𝒳\\mathcal\{X\}, whose true members at timettform the current logical state setWt⊆𝒳W\_\{t\}\\subseteq\\mathcal\{X\}\. Each armii’s reach bandℬi⊆𝒫\\mathcal\{B\}\_\{i\}\\subseteq\\mathcal\{P\}is the subset of resources that its end effector can touch, with its gripper holding at most one object per step\. Then we can define the shared band reachable by two or more arms as

𝒫sh=\{p∈𝒫∣\|\{i∣p∈ℬi\}\|≥2\}\.\\mathcal\{P\}^\{\\mathrm\{sh\}\}=\\\{p\\in\\mathcal\{P\}\\mid\|\\\{i\\mid p\\in\\mathcal\{B\}\_\{i\}\\\}\|\\geq 2\\\}\.\(1\)𝒫sh\\mathcal\{P\}^\{\\mathrm\{sh\}\}can be the panels on which arms hand objects over in relay sorting, and the common grasp positions of two arms co\-lifting a rope\. The local observation of each agentiiis given by a deterministic partially observable projection operatorΠi\\Pi\_\{i\}\. This operator keeps only the objects falling within its reach band that have not yet reached their target resource, together with the resources they occupy, which is expressed as

Πi​\(st\)=\{\(e,ℓt​\(e\),Wt\)∣ℓt​\(e\)∈ℬi,ℓt​\(e\)≠pe⋆\}\.\\Pi\_\{i\}\(s\_\{t\}\)=\\big\\\{\(e,\\ell\_\{t\}\(e\),W\_\{t\}\)\\mid\\ell\_\{t\}\(e\)\\in\\mathcal\{B\}\_\{i\},\\ \\ell\_\{t\}\(e\)\\neq p\_\{e\}^\{\\star\}\\big\\\}\.\(2\)The observation of armiiused for action decision is the local context and the task objective, which is given by

ci,t=\(Πi​\(st\),g\),c\_\{i,t\}=\(\\Pi\_\{i\}\(s\_\{t\}\),g\),\(3\)wherests\_\{t\}is the global physical state at timettthat aggregates the three\-dimensional positions of the above objects, the gripper holdings, and the logical conditions\. The task objectiveg=\{ςe\}e∈ℰ⋆g=\\\{\\varsigma\_\{e\}\\\}\_\{e\\in\\mathcal\{E\}^\{\\star\}\}is a set of per\-object success criteria, whereℰ⋆⊆ℰ\\mathcal\{E\}^\{\\star\}\\subseteq\\mathcal\{E\}collects the objects involved in the task objective\. Whether objecteeis transported successfully is judged directly by whether its three\-dimensional position has reached the expected position, which is given by

ςe​\(st\)=\{1,‖xt​\(e\)−xe⋆‖≤ϵpos,∀e∈ℰ⋆,0,otherwise,\\varsigma\_\{e\}\(s\_\{t\}\)=\\begin\{cases\}1,&\\\|x\_\{t\}\(e\)\-x\_\{e\}^\{\\star\}\\\|\\leq\\epsilon\_\{\\mathrm\{pos\}\},\\ \\forall e\\in\\mathcal\{E\}^\{\\star\},\\\\\[2\.0pt\] 0,&\\text\{otherwise\},\\end\{cases\}\(4\)whereϵpos\\epsilon\_\{\\mathrm\{pos\}\}tolerance ball around its target positionxe⋆x\_\{e\}^\{\\star\}\.

To further describe the actions of the embodied robots, we decompose the multi\-embodied\-robot collaborative task into the three most basic classes of atomic tasks111In this paper an atomic task refers to the most basic and indivisible temporal coordination relation between two shared\-resource actions, and this decomposition of the pair of basic control\-flow relations follows the characterization of workflow patterns\[[64](https://arxiv.org/html/2607.09330#bib.bib64)\]\., which named mutual exclusion, synchronization, and dependency as follows\.

1. 1\.Mutual exclusion \(e\.g\., sorting\)\.The task objective is to move each entity toward its target position via several cross\-arm relays\. When two arms contend for the entity on the same panel in the same slot, a mutual\-exclusion conflict arises, so that𝒟\\mathcal\{D\}admits the agent with higher priority to avoid this conflict\.
2. 2\.Synchronization \(e\.g\., rope co\-lifting\)\.The task objective is to lift the different sides of the object by the arms simultaneously and place them at a designated position\. In each slot, the arms must ensure that the other arms are in the same lifting group\.
3. 3\.Dependency \(holding to fetch\)\.The task objective is to take a target object under a specific state\. To achieve this, one arm maintains the required logical state, while the other fetches the target object\. The fetching action is valid only when the required logical state holds \(i\.e\.,uk=1u\_\{k\}=1\)\. In each slot, the state\-maintaining arm produces and maintains the logical stateuku\_\{k\}, while the fetching arm declares a persistent dependency on this state\. Then𝒟\\mathcal\{D\}admits the fetching action only when another arm maintains the required logical state in the same slot, which imposes an enforced ordering with in\-between timing\.

### III\-BAutonomous Action Selection

In each slot, every embodied robotic arm must select a suitable action according to its local observationci,tc\_\{i,t\}\. The full action space is𝒪=\{a0\}∪\(ℰ×𝒫\)\\mathcal\{O\}=\\\{a\_\{0\}\\\}\\cup\(\\mathcal\{E\}\\times\\mathcal\{P\}\), wherea0a\_\{0\}denotes staying still, that is, initiating no action in this slot\. The pair\(e,p\)\(e,p\)denotes grasping entityeeand placing it onto resourcepp\. Constrained by partial observability and the reach band, the legal action set of the arm is given by a deterministic mask𝒪ilegal​\(st\)=\{a0\}∪\{\(e,p\)∣ℓt​\(e\)∈ℬi,p∈ℬi,ν​\(e,p\)=1\}\\mathcal\{O\}\_\{i\}^\{\\mathrm\{legal\}\}\(s\_\{t\}\)=\\\{a\_\{0\}\\\}\\cup\\\{\(e,p\)\\mid\\ell\_\{t\}\(e\)\\in\\mathcal\{B\}\_\{i\},\\ p\\in\\mathcal\{B\}\_\{i\},\\ \\nu\(e,p\)=1\\\}\. Here, the task\-validity predicateν∈\{0,1\}\\nu\\in\\\{0,1\\\}is given explicitly by

ν​\(e,p\)\\displaystyle\\nu\(e,p\)=𝟙\[p=pe⋆∨\(p∈𝒫sh\\displaystyle=\\mathds\{1\}\\Big\[\\,p=p\_\{e\}^\{\\star\}\\ \\lor\\ \\big\(p\\in\\mathcal\{P\}^\{\\mathrm\{sh\}\}\\\(5\)∧\|pos\(p\)−pos\(pe⋆\)\|<\|pos\(ℓt\(e\)\)−pos\(pe⋆\)\|\)\]\.\\displaystyle\\wedge\\ \|\\mathrm\{pos\}\(p\)\-\\mathrm\{pos\}\(p\_\{e\}^\{\\star\}\)\|<\|\\mathrm\{pos\}\(\\ell\_\{t\}\(e\)\)\-\\mathrm\{pos\}\(p\_\{e\}^\{\\star\}\)\|\\big\)\\,\\Big\]\.ν​\(e,p\)=1\\nu\(e,p\)=1imply that move objecteeto planeppis available, andν​\(e,p\)=0\\nu\(e,p\)=0otherwise\.ν​\(e,p\)=1\\nu\(e,p\)=1ifppis exactly the target resource ofee, orppis a shared relay resource that lies strictly closer to the targetpe⋆p\_\{e\}^\{\\star\}in order distancepos\\mathrm\{pos\}\. The reference is the distance of the resourceℓt​\(e\)\\ell\_\{t\}\(e\)currently occupied byeetope⋆p\_\{e\}^\{\\star\}, andν​\(e,p\)=0\\nu\(e,p\)=0otherwise\. On this legal action set, armiifurther uses its own LLMMiM\_\{i\}together with the local contextci,tc\_\{i,t\}to score each feasible action\. It takes the highest\-scoring one as the action selected in this step, which is given by

ai,t=arg⁡maxa∈𝒪ilegal​\(st\)⁡1\|a\|​∑klog⁡pMi​\(ak∣ci,t\),a\_\{i,t\}=\\arg\\max\_\{a\\in\\mathcal\{O\}\_\{i\}^\{\\mathrm\{legal\}\}\(s\_\{t\}\)\}\\ \\frac\{1\}\{\|a\|\}\\sum\_\{k\}\\log p\_\{M\_\{i\}\}\\\!\\left\(a\_\{k\}\\mid c\_\{i,t\}\\right\),\(6\)wherepMi\(⋅∣ci,t\)p\_\{M\_\{i\}\}\(\\cdot\\mid c\_\{i,t\}\)is the conditional token likelihood of the LLMMiM\_\{i\}and\|a\|\|a\|is the action token length\. Since the scoring is performed over the enumerated legal set rather than by free generation, no illegal action can be produced\. The action depends only onMiM\_\{i\}andci,tc\_\{i,t\}, and heterogeneity is introduced by the capability differences among\{Mi\}\\\{M\_\{i\}\\\}\. After selecting the suit actionai,ta\_\{i,t\}, each armiiexecutes the selected action immediately\.

### III\-CStructured Intention Reporting

After selecting its own action, each armiimust further impose constraints on the objects to be operated on to ensure that the task can be completed correctly\. To this end, we design three principles to address the respective coordination needs of the mutual\-exclusion, synchronization, and dependency atomic tasks\.

Mutual\-exclusion field\.Coordinationξi,t⊆ℰ∪𝒫\\xi\_\{i,t\}\\subseteq\\mathcal\{E\}\\cup\\mathcal\{P\}lists the entities and resources that this step’s action needs to occupy exclusively\. This means that no two admitted actions may occupy the same element ofξ\\xiin the same slot\. It grounds the relation that two actions cannot proceed at the same time into an occupancy declaration on concrete shared elements\.

Synchronization field\.Coordinationζi,t⊆𝒩\\zeta\_\{i,t\}\\subseteq\\mathcal\{N\}lists the members of the joint group to which this step’s action belongs, namely the roster of initiators of a set of actions that must execute simultaneously\. The group takes effect as a whole if and only if the group members consistently name each other in this slot and all their actions are admitted; otherwise, a collision occurs\. This expresses the relation that they must act simultaneously\.

Dependency field\.An arm characterizes its production of and demand for the shared logical states𝒳\\mathcal\{X\}with two pairs of state fields\. On the supply side,ψi,t⊆𝒳\\psi\_\{i,t\}\\subseteq\\mathcal\{X\}is the state instantaneously produced by this step’s action andχi,t⊆𝒳\\chi\_\{i,t\}\\subseteq\\mathcal\{X\}is the state continuously held by this step’s action\. On the demand side,αi,t⊆𝒳\\alpha\_\{i,t\}\\subseteq\\mathcal\{X\}is the prerequisite state that this step’s action needs\. This state must be produced in the same step by some action of this step, that is an instantaneous dependency\. Correspondingly,ωi,t⊆𝒳\\omega\_\{i,t\}\\subseteq\\mathcal\{X\}is the state that this step’s action needs and that must be held in the same step by some action of this step, that is a persistent dependency\. This pair of demand and supply declarations expresses the relation that ordering and in\-between timing must hold\.

Combining the above information, we define the constraint declaration set as

𝒞i,t=\{ξi,t,ζi,t,αi,t,ωi,t,ψi,t,χi,t\}\.\\mathcal\{C\}\_\{i,t\}=\\big\\\{\\xi\_\{i,t\},\\ \\zeta\_\{i,t\},\\ \\alpha\_\{i,t\},\\ \\omega\_\{i,t\},\\ \\psi\_\{i,t\},\\ \\chi\_\{i,t\}\\big\\\}\.\(7\)Based on𝒞i,t\\mathcal\{C\}\_\{i,t\}armiican clarify its needs for the current action and then packages together with actionai,ta\_\{i,t\}into a structured messagemi,t=⟨ai,t,𝒞i,t⟩m\_\{i,t\}=\\langle a\_\{i,t\},\\mathcal\{C\}\_\{i,t\}\\rangleand reports it to𝒟\\mathcal\{D\}for avoiding collisions\. In our proposed scheme, the agent message is carried as structured key\-value pairs rather than the traditional NL\-based agent dialogue strategy\. The DT𝒟\\mathcal\{D\}can thus obtain information that is stripped of the reporter’s language ability\. Further accounting for the limited communication resources, we defineσi,t∈\{0,1\}\\sigma\_\{i,t\}\\in\\\{0,1\\\}as an indicator to account for whether this report takes place in the time slottt\. Hereσi,t=1\\sigma\_\{i,t\}=1means that armiihas one uplink or downlink interaction with𝒟\\mathcal\{D\}atttandσi,t=0\\sigma\_\{i,t\}=0means no interaction\. Accordingly, the set of agents that report structured information at timettis denoted as𝒰t=\{i∣σi,t=1\}\\mathcal\{U\}\_\{t\}=\\\{i\\mid\\sigma\_\{i,t\}=1\\\}\.

To carry out reporting, we consider a shared uplink\. The end effector of armiiis located atqi∈ℝ3q\_\{i\}\\in\\mathbb\{R\}^\{3\}and the DT𝒟\\mathcal\{D\}is deployed at a fixed positionq𝒟∈ℝ3q\_\{\\mathcal\{D\}\}\\in\\mathbb\{R\}^\{3\}\. The uplink channel gain between the two attenuates with distance according to path loss, given byhi=h0​‖qi−q𝒟‖−μh\_\{i\}=h\_\{0\}\\,\\\|q\_\{i\}\-q\_\{\\mathcal\{D\}\}\\\|^\{\-\\mu\}, whereh0h\_\{0\}is the reference gain andμ\>0\\mu\>0is the path\-loss exponent\. With transmit powerPiP\_\{i\}, link bandwidthBB, and noise power spectral densityN0N\_\{0\}, the achievable uplink rate from armiito𝒟\\mathcal\{D\}is

vi,t=B​log2⁡\(1\+Pi​hiN0​B\)\.v\_\{i,t\}=B\\log\_\{2\}\\\!\\Big\(1\+\\frac\{P\_\{i\}\\,h\_\{i\}\}\{N\_\{0\}\\,B\}\\Big\)\.\(8\)The per\-message delay needed to report one structured messagemi,tm\_\{i,t\}can be expressed as

δi,t​\(qi,\|mi,t\|,Pi,B\)=\|mi,t\|B​log2⁡\(1\+Pi​h0​‖qi−q𝒟‖−μN0​B\),\\delta\_\{i,t\}\(q\_\{i\},\|m\_\{i,t\}\|,P\_\{i\},B\)=\\frac\{\|m\_\{i,t\}\|\}\{B\\log\_\{2\}\\\!\\Big\(1\+\\dfrac\{P\_\{i\}\\,h\_\{0\}\\,\\\|q\_\{i\}\-q\_\{\\mathcal\{D\}\}\\\|^\{\-\\mu\}\}\{N\_\{0\}\\,B\}\\Big\)\},\(9\)where\|mi,t\|\|m\_\{i,t\}\|is the bit length\. All reports share the same uplink to𝒟\\mathcal\{D\}\. The total delay of coordination communication in this slot is therefore the sum of the delays of all reporters in this step, given by

Dt=∑i∈𝒰tδi,t\.D\_\{t\}=\\sum\_\{i\\in\\mathcal\{U\}\_\{t\}\}\\delta\_\{i,t\}\.\(10\)

### III\-DCoordination and Execution at the Digital Twin

After aggregating the messages\{mi,t\}\\\{m\_\{i,t\}\\\}of all acting arms, the DT𝒟\\mathcal\{D\}does not need to understand the task semantics\. It relies only on the declared constraints\{𝒞i,t\}\\\{\\mathcal\{C\}\_\{i,t\}\\\}and the current logical state setWtW\_\{t\}to act\. From the acting set𝒩tact=\{i∣ai,t≠a0\}\\mathcal\{N\}\_\{t\}^\{\\mathrm\{act\}\}=\\\{i\\mid a\_\{i,t\}\\neq a\_\{0\}\\\}, it selects by rules a maximal consistent arm set𝒜t⋆\\mathcal\{A\}\_\{t\}^\{\\star\}whose actions do not conflict with each other, so as to avoid conflicts\. To characterize consistency, we first consider any acting\-arm subset𝒜⊆𝒩tact\\mathcal\{A\}\\subseteq\\mathcal\{N\}\_\{t\}^\{\\mathrm\{act\}\}, each memberiiof which carries this step’s intended actionai,ta\_\{i,t\}\. The action of armiishould be consistent with the actions of the other members of the set𝒜\\mathcal\{A\}and with the current stateWtW\_\{t\}\. This holds when it simultaneously passes the three checks of mutual exclusion, synchronization, and dependency\. We next give the consistency subconditions for these three coordination relations separately, and then conjoin them into a consistency criterionη​\(i∣𝒜,Wt\)∈\{0,1\}\\eta\(i\\mid\\mathcal\{A\},W\_\{t\}\)\\in\\\{0,1\\\}\.

Mutual\-exclusion consistency\.Armiicontends for no exclusive element with any other member of𝒜\\mathcal\{A\}, given by

ηξ\(i∣𝒜\)=𝟙\[∀j∈𝒜∖\{i\}:ξi,t∩ξj,t=∅\]\.\\eta\_\{\\xi\}\(i\\mid\\mathcal\{A\}\)=\\mathds\{1\}\\big\[\\,\\forall j\\in\\mathcal\{A\}\\setminus\\\{i\\\}:\\ \\xi\_\{i,t\}\\cap\\xi\_\{j,t\}=\\varnothing\\,\\big\]\.\(11\)
Synchronization consistency\.The members of the joint groupζi,t\\zeta\_\{i,t\}to which armiibelongs all fall within𝒜\\mathcal\{A\}and name the same group as each other, given by

ηζ\(i∣𝒜\)=𝟙\[ζi,t⊆𝒜∧∀m∈ζi,t:ζm,t=ζi,t\]\.\\eta\_\{\\zeta\}\(i\\mid\\mathcal\{A\}\)=\\mathds\{1\}\\big\[\\,\\zeta\_\{i,t\}\\subseteq\\mathcal\{A\}\\ \\wedge\\ \\forall m\\in\\zeta\_\{i,t\}:\\ \\zeta\_\{m,t\}=\\zeta\_\{i,t\}\\,\\big\]\.\(12\)
Dependency consistency\.The instantaneous prerequisite states needed by armiiare produced in this step by some action in𝒜\\mathcal\{A\}or are already true inWtW\_\{t\}\. Likewise, the persistent states it needs are held in this step by some action in𝒜\\mathcal\{A\}or are already true inWtW\_\{t\}, given by

ηω\(i∣𝒜,Wt\)=𝟙\[αi,t⊆⋃j∈𝒜\(ψj,t∪χj,t\)\\displaystyle\\eta\_\{\\omega\}\(i\\mid\\mathcal\{A\},W\_\{t\}\)=\\mathds\{1\}\\big\[\\,\\alpha\_\{i,t\}\\subseteq\\textstyle\\bigcup\_\{j\\in\\mathcal\{A\}\}\(\\psi\_\{j,t\}\\cup\\chi\_\{j,t\}\)\(13\)∪Wt∧ωi,t⊆⋃j∈𝒜χj,t∪Wt\]\.\\displaystyle\\cup W\_\{t\}\\ \\wedge\\ \\omega\_\{i,t\}\\subseteq\\textstyle\\bigcup\_\{j\\in\\mathcal\{A\}\}\\chi\_\{j,t\}\\cup W\_\{t\}\\,\\big\]\.
The action of armiiis consistent with𝒜\\mathcal\{A\}only when the three checks pass simultaneously, so the consistency criterion is the conjunction of the three, given by

η​\(i∣𝒜,Wt\)=ηξ​\(i∣𝒜\)∧ηζ​\(i∣𝒜\)∧ηω​\(i∣𝒜,Wt\),\\eta\(i\\mid\\mathcal\{A\},W\_\{t\}\)=\\eta\_\{\\xi\}\(i\\mid\\mathcal\{A\}\)\\ \\wedge\\ \\eta\_\{\\zeta\}\(i\\mid\\mathcal\{A\}\)\\ \\wedge\\ \\eta\_\{\\omega\}\(i\\mid\\mathcal\{A\},W\_\{t\}\),\(14\)which takes the value11if and only if the mutual\-exclusion, synchronization, and dependency declarations of armiiare satisfied at the same time\. Accordingly, the executable set optimization method is given by

𝒜t⋆\\displaystyle\\mathcal\{A\}\_\{t\}^\{\\star\}=arg​max𝒜⊆𝒩tact⁡\|𝒜\|,\\displaystyle=\\operatorname\*\{arg\\,max\}\_\{\\mathcal\{A\}\\subseteq\\mathcal\{N\}\_\{t\}^\{\\mathrm\{act\}\}\}\\ \|\\mathcal\{A\}\|,\(15\)s\.t\.η​\(i∣𝒜,Wt\)=1,∀i∈𝒜\.\\displaystyle\\text\{s\.t\.\}\\quad\\eta\(i\\mid\\mathcal\{A\},W\_\{t\}\)=1,\\ \\forall i\\in\\mathcal\{A\}\.\(16\)The subset of maximal cardinality under the premise that every member of the set satisfies its own declared constraints\. Consider an agent armi∉𝒜t⋆i\\notin\\mathcal\{A\}\_\{t\}^\{\\star\}whose action is vetoed by the DT during optimziation of𝒟\\mathcal\{D\}\. To this agent,𝒟\\mathcal\{D\}sends back an short instructiondi,t∈\{d0,d1\}d\_\{i,t\}\\in\\\{d\_\{0\},d\_\{1\}\\\}, whered0d\_\{0\}means yield and stop andd1d\_\{1\}means wait\. The vetoed party backs off autonomously while𝒟\\mathcal\{D\}does not specify a replacement action, so that the arm generates an action again at the next step\. The admitted set is realized as a physical advance through a shared execution mapΦ\\Phi, given byst\+1=Φ​\(st,\{ai,t\}i∈𝒜t⋆\)s\_\{t\+1\}=\\Phi\(s\_\{t\},\\\{a\_\{i,t\}\\\}\_\{i\\in\\mathcal\{A\}\_\{t\}^\{\\star\}\}\), and this map is consistent for all arms\.

### III\-EProblem Formulation

We formalize an optimization problem whose goal is to maximize the expected cumulative task\-objective reward under the communication constraint, which is formulated as

maxπc,𝒰t\\displaystyle\\max\_\{\\pi^\{c\},\\,\\mathcal\{U\}\_\{t\}\}\\quad𝔼​\[∑t=0Tγt​Rt\]\\displaystyle\\mathbb\{E\}\\\!\\left\[\\sum\_\{t=0\}^\{T\}\\gamma^\{t\}R\_\{t\}\\right\]\(17\)s\.t\.Dt≤D¯,∀t,\\displaystyle D\_\{t\}\\leq\\bar\{D\},\\ \\forall t,\(17a\)η​\(i∣𝒜t⋆,Wt\)=1,∀i∈𝒜t⋆,∀t,\\displaystyle\\eta\(i\\mid\\mathcal\{A\}\_\{t\}^\{\\star\},W\_\{t\}\)=1,\\ \\forall i\\in\\mathcal\{A\}\_\{t\}^\{\\star\},\\ \\forall t,\(17b\)whereRt=∑e∈ℰ⋆\(ςe​\(st\+1\)−ςe​\(st\)\)R\_\{t\}=\\sum\_\{e\\in\\mathcal\{E\}^\{\\star\}\}\(\\varsigma\_\{e\}\(s\_\{t\+1\}\)\-\\varsigma\_\{e\}\(s\_\{t\}\)\)is the per\-step reward\. The decision variable is the communication\-selection policyπc\\pi^\{c\}, while the per\-arm action and constraint declaration\(ai,t,𝒞i,t\)\(a\_\{i,t\},\\mathcal\{C\}\_\{i,t\}\)are produced locally by eachMiM\_\{i\}and are treated as given rather than optimized\. The policyπc\\pi^\{c\}decides the per\-step reporting indicators\{σi,t\}\\\{\\sigma\_\{i,t\}\\\}and thereby decides the reporting set𝒰t\\mathcal\{U\}\_\{t\}\. Hereη​\(⋅\)\\eta\(\\cdot\)is the declaration\-satisfaction criterion defined above,γ∈\(0,1\]\\gamma\\in\(0,1\]is the discount on future\-action reward,TTis the time horizon, andD¯\\bar\{D\}is the per\-step deadline of coordination communication\. \([17](https://arxiv.org/html/2607.09330#S3.E17)a\) is the per\-step coordination\-delay constraint, which requires that the total delay of all reports of this step on the shared uplink not exceed the deadlineD¯\\bar\{D\}\. This constraint captures the real\-time requirement of a constrained link under heterogeneous deployment\. \([17](https://arxiv.org/html/2607.09330#S3.E17)b\) is the consistency constraint, which captures the coordination relations that physically cannot be violated simultaneously and guarantees that the action set has no explicit conflict\.

Directly solving the above problem faces three structural challenges\. First, heterogeneity\-induced declaration noise\. Since𝒞i,t\\mathcal\{C\}\_\{i,t\}is generated byπi\\pi\_\{i\}throughMiM\_\{i\}, weak LLMs may omit or misstate declarations\. This may cause local violations of \([17](https://arxiv.org/html/2607.09330#S3.E17)b\)\. The coordination mechanism should therefore tolerate individual declaration errors, rather than being bottlenecked by the weakest arm\. Second, cross\-atomic\-task coupling creates cascading conflicts\. Exclusive, grouped, and dependent constraints are coupled through shared actions and states\. The incomplete grouped action may then remove a prerequisite state for a dependent action\. Thus, conflicts propagate over the constraint graph\. Third, partial observability prevents global planning\. Since each arm only observes throughΠi\\Pi\_\{i\}, neither any arm nor𝒟\\mathcal\{D\}has complete access tosts\_\{t\}\. Task ordering must instead emerge from local decisions and the reachable topology\. These challenges motivate a lightweight conflict\-avoidance mechanism that is robust to declaration noise, iteratively convergent, and able to guarantee finite\-step progress without centralized planning\.

## IVProposed Method

### IV\-AUnified Lightweight Orchestrator

To solve the optimization problem formulated in \([17](https://arxiv.org/html/2607.09330#S3.E17)\), we unify the mutual\-exclusion, synchronization, and dependency atomic tasks into a single orchestratorℛ\\mathcal\{R\}of four typed rules that avoids declaration conflicts\.

We first define the mutual\-exclusion rule,TξT\_\{\\xi\}, which handles the scenario in which several admitted actions contend for the same exclusive resource\. Among the actions that contend for the same element ofξ\\xiit keeps only the one with the highest yielding priority and removes the rest\. This enforces the constraint that two actions cannot occupy the same resource at the same time\. It is defined as

Tξ​\(𝒜\)\\displaystyle T\_\{\\xi\}\(\\mathcal\{A\}\)=𝒜∖\{i∈𝒜∣∃\\displaystyle=\\mathcal\{A\}\\setminus\\big\\\{i\\in\\mathcal\{A\}\\mid\\ \\exists\\,\(18\)r∈ξi,t,∃j∈𝒜∖\{i\},r∈ξj,t,ρ\(aj,t\)≻ρ\(ai,t\)\},\\displaystyle r\\in\\xi\_\{i,t\},\\ \\exists\\,j\\in\\mathcal\{A\}\\setminus\\\{i\\\},\\ r\\in\\xi\_\{j,t\},\\ \\rho\(a\_\{j,t\}\)\\succ\\rho\(a\_\{i,t\}\)\\big\\\},whereρ​\(a\)∈\{0,1\}×ℤ≤0×ℤ≤0\\rho\(a\)\\in\\\{0,1\\\}\\times\\mathbb\{Z\}\_\{\\leq 0\}\\times\\mathbb\{Z\}\_\{\\leq 0\}is the yielding priority which is defined as

ρ​\(a\)=\(1​\[p=pe⋆\],−\|pos​\(p\)−pos​\(pe⋆\)\|,−κ​\(i\)\)\.\\rho\(a\)=\\Big\(\\,\\mathds\{1\}\[\\,p=p\_\{e\}^\{\\star\}\\,\],\\ \-\\,\\big\|\\mathrm\{pos\}\(p\)\-\\mathrm\{pos\}\(p\_\{e\}^\{\\star\}\)\\big\|,\\ \-\\,\\kappa\(i\)\\,\\Big\)\.\(19\)The first component𝟙​\[p=pe⋆\]\\mathds\{1\}\[\\,p=p\_\{e\}^\{\\star\}\\,\]is the terminal bit that marks whether the drop slotppis exactly the target resource of objectee\. An action that reaches the final state directly is more worth keeping than one that only relays\. The second component is the negated remaining step count\|pos​\(p\)−pos​\(pe⋆\)\|\\big\|\\mathrm\{pos\}\(p\)\-\\mathrm\{pos\}\(p\_\{e\}^\{\\star\}\)\\big\|, that is, the distance of the drop slotppto the target resourcepe⋆p\_\{e\}^\{\\star\}along the ordinalpos\\mathrm\{pos\}, where a smaller distance is more worth keeping\. In the third component, quantityκ​\(i\)\\kappa\(i\)is the fixed index of the arm, used only to break ties when the first two components are equal\. This priority is built solely from the local features of a single action and needs no inter\-arm negotiation, which keeps𝒟\\mathcal\{D\}lightweight\. For two actions that contend for the same resource, the components are compared from the first to the third, and on the first component where the values differ, the larger value has the higher priority\. Accordingly,TξT\_\{\\xi\}keeps only the highest\-priority contender and removes the others\.

Similarly, we can define the synchronization ruleTζT\_\{\\zeta\}that handles the scenario in which a group of actions that must execute simultaneously is not complete\. The group is kept only when its members name the same group at this step, and all of them still remain in𝒜\\mathcal\{A\}, and otherwise the whole group is removed\. This rules out half\-finished execution\. It is defined as

Tζ\(𝒜\)=𝒜∖\{i∈𝒜∣\\displaystyle T\_\{\\zeta\}\(\\mathcal\{A\}\)=\\mathcal\{A\}\\setminus\\big\\\{i\\in\\mathcal\{A\}\\midζi,t=G≠∅,\\displaystyle\\zeta\_\{i,t\}=G\\neq\\varnothing,\(20\)¬\(∀m∈G:m∈𝒜∧ζm,t=G\)\}\.\\displaystyle\\ \\neg\\big\(\\forall m\\in G:\\ m\\in\\mathcal\{A\}\\wedge\\zeta\_\{m,t\}=G\\big\)\\big\\\}\.
We define the instantaneous\-dependency ruleTαT\_\{\\alpha\}that handles the scenario in which the precondition that an action requires is produced by no one at this step\. For this, we first define the production closure and the holding closure of the admitted set, which are

ψ¯​\(𝒜\)=⋃j∈𝒜\(ψj,t∪χj,t\),χ¯​\(𝒜\)=⋃j∈𝒜χj,t\.\\bar\{\\psi\}\(\\mathcal\{A\}\)=\\bigcup\_\{j\\in\\mathcal\{A\}\}\(\\psi\_\{j,t\}\\cup\\chi\_\{j,t\}\),\\qquad\\bar\{\\chi\}\(\\mathcal\{A\}\)=\\bigcup\_\{j\\in\\mathcal\{A\}\}\\chi\_\{j,t\}\.\(21\)Specifically, all states produced and held by the admitted set at this step\. The rule removes the actions whose precondition is neither in the production closure nor already true inWtW\_\{t\}which is defined as

Tα​\(𝒜\)=𝒜∖\{i∈𝒜∣αi,t⊈ψ¯​\(𝒜\)∪Wt\}\.T\_\{\\alpha\}\(\\mathcal\{A\}\)=\\mathcal\{A\}\\setminus\\big\\\{i\\in\\mathcal\{A\}\\mid\\ \\alpha\_\{i,t\}\\not\\subseteq\\bar\{\\psi\}\(\\mathcal\{A\}\)\\cup W\_\{t\}\\big\\\}\.\(22\)
The persistent\-dependency ruleTωT\_\{\\omega\}handles the scenario in which a state that an action requires to be held at this step is held by no one\. It removes the actions whose required state is neither in the holding closure nor already true, which enforces that the action must execute while some state is being held\. It is defined as

Tω​\(𝒜\)=𝒜∖\{i∈𝒜∣ωi,t⊈χ¯​\(𝒜\)∪Wt\}\.T\_\{\\omega\}\(\\mathcal\{A\}\)=\\mathcal\{A\}\\setminus\\big\\\{i\\in\\mathcal\{A\}\\mid\\ \\omega\_\{i,t\}\\not\\subseteq\\bar\{\\chi\}\(\\mathcal\{A\}\)\\cup W\_\{t\}\\big\\\}\.\(23\)
Building on these four rules, the DT𝒟\\mathcal\{D\}composes them through the unified orchestratorℛ\\mathcal\{R\}and applies the composition to the candidate set repeatedly until it converges to a self\-consistent executable set\. The output action set then realizes the goal of conflict avoidance, that is, it selects from all intended actions of this step a subset that are mutually compatible and can execute together\. This executable set is given by

𝒜t⋆=ℛ​\(\{mi,t\}\)=iter​\(Tξ∘Tζ∘Tα∘Tω\)​\(𝒩tact\),\\mathcal\{A\}\_\{t\}^\{\\star\}=\\mathcal\{R\}\(\\\{m\_\{i,t\}\\\}\)=\\mathrm\{iter\}\\big\(T\_\{\\xi\}\\circ T\_\{\\zeta\}\\circ T\_\{\\alpha\}\\circ T\_\{\\omega\}\\big\)\(\\mathcal\{N\}\_\{t\}^\{\\mathrm\{act\}\}\),\(24\)whereiter​\(⋅\)\\mathrm\{iter\}\(\\cdot\)denotes repeated application of the composite rule in the parentheses until the result no longer changes, and each rule acts on the candidate set𝒜\\mathcal\{A\}and removes the offenders of its type\.

After𝒜t⋆\\mathcal\{A\}\_\{t\}^\{\\star\}is obtained,𝒟\\mathcal\{D\}returns to each vetoed armi∈𝒩tact∖𝒜t⋆i\\in\\mathcal\{N\}\_\{t\}^\{\\mathrm\{act\}\}\\setminus\\mathcal\{A\}\_\{t\}^\{\\star\}a very short downlink instructiondi,t∈\{d0,d1\}d\_\{i,t\}\\in\\\{d\_\{0\},d\_\{1\}\\\}, whered0d\_\{0\}is the yield\-and\-stop instruction andd1d\_\{1\}is the wait instruction\. The actual value is determined by which rule removed the action\. Ifiiyields in a contention by priorityρ\\rhounder the mutual\-exclusion ruleTξT\_\{\\xi\}, thend0d\_\{0\}is returned, because the contention at this step is irrecoverable\. The arm then abandons the action and backs off in place or reselects\. Ifiiis removed because its group is incomplete under synchronizationTζT\_\{\\zeta\}, or because its precondition is not yet satisfied under the dependency rulesTαT\_\{\\alpha\}orTωT\_\{\\omega\}, thend1d\_\{1\}is returned\. In this case the missing condition may be met in a later slot, and the arm keeps its current intention and waits\. Neither instruction prescribes a replacement action for the arm\.𝒟\\mathcal\{D\}only states that this step is infeasible and of which nature, whether to yield or to wait, while the back\-off and the reselection are still carried out by the arm itself\.

Since each of the four rules only removes elements from𝒜\\mathcal\{A\}, so thatT​\(𝒜\)⊆𝒜T\(\\mathcal\{A\}\)\\subseteq\\mathcal\{A\}, their composition is monotonically shrinking and its finite\-step convergence is guaranteed by the following proposition\.

###### Proposition 1\(Finite\-step convergence of the orchestrator\)\.

The compositionΘ=Tξ∘Tζ∘Tα∘Tω\\Theta=T\_\{\\xi\}\\circ T\_\{\\zeta\}\\circ T\_\{\\alpha\}\\circ T\_\{\\omega\}satisfiesΘ​\(𝒜\)⊆𝒜\\Theta\(\\mathcal\{A\}\)\\subseteq\\mathcal\{A\}, so the iteration that starts from𝒩tact\\mathcal\{N\}\_\{t\}^\{\\mathrm\{act\}\}must converge in a finite number of steps to a unique stable executable set, and it is reached in at most\|𝒩tact\|\|\\mathcal\{N\}\_\{t\}^\{\\mathrm\{act\}\}\|rounds\.

###### Proof\.

See Appendix[\-A](https://arxiv.org/html/2607.09330#A0.SS1)\. ∎

The closure and completeness of the three atomic tasks over the coordination space are characterized by the following proposition\.

###### Proposition 2\(Closure and completeness of the three atomic tasks\)\.

In a discrete\-slot scenario where actions are constrained by shared resources and logical states, and where coordination constraints never force any arm to act, so that both arms may remain idle, any coordination constraint between two interacting actions is logically equivalent to a conjunction of mutual\-exclusion and dependency atomic tasks\. Synchronization is precisely the symmetric conjunction of two oppositely directed dependencies\. Therefore, the atomic\-task set mutual exclusion, synchronization, dependency, grounded by the four rulesTξ,Tζ,Tα,TωT\_\{\\xi\},T\_\{\\zeta\},T\_\{\\alpha\},T\_\{\\omega\}, is closed under conjunction,nn\-ary extension, and interval extension, and spans all coordination constraints under this model\. These three relations are pairwise distinct: they require disjoint, simultaneous, and ordered execution, respectively, and none is a special case of another\[[64](https://arxiv.org/html/2607.09330#bib.bib64)\]\.

### IV\-BLearned Communication\-Selection Layer

The coordination layer above already achieves reliable cooperation among the arms without any training\. However, every arm still has to upload its action and constraint declaration to𝒟\\mathcal\{D\}at every step, and these reports accumulate a non\-negligible per\-step latencyDtD\_\{t\}on a heterogeneous and constrained shared uplink\. To further reduce the transmission overhead, this subsection solves the communication\-selection policyπc\\pi^\{c\}of the formulated problem\. Since the latency deadline \([17](https://arxiv.org/html/2607.09330#S3.E17)a\) is coupled step by step in time and its decision observationzi,tz\_\{i,t\}is only a partially observable proxy, it is hard to optimize\{σi,t\}\\\{\\sigma\_\{i,t\}\\\}directly\. To address this we treat it as a sequential decision problem and model it as a constrained partially observable Markov decision process \(C\-POMDP\)\. Its state is the decision observationzi,tz\_\{i,t\}, its action is the report indicatorσi,t\\sigma\_\{i,t\}, and its instantaneous reward is the team\-level shaped signalrtr\_\{t\}\. It maximizes𝔼​\[∑tγt​rt\]\\mathbb\{E\}\[\\sum\_\{t\}\\gamma^\{t\}r\_\{t\}\]under the expectation form𝔼​\[Dt\]≤D¯\\mathbb\{E\}\[D\_\{t\}\]\\leq\\bar\{D\}of \([17](https://arxiv.org/html/2607.09330#S3.E17)a\), whereDt=∑i∈𝒰tδi,tD\_\{t\}=\\sum\_\{i\\in\\mathcal\{U\}\_\{t\}\}\\delta\_\{i,t\}is the total report latency of this step and𝒰t=\{i∣σi,t=1\}\\mathcal\{U\}\_\{t\}=\\\{i\\mid\\sigma\_\{i,t\}=1\\\}is the report set of this step\. This layer is strictly decoupled from the coordination layer\. The coordination layer, namely the lightweight𝒟\\mathcal\{D\}together with the orchestrator, stays training\-free and unchanged\. The policyπc\\pi^\{c\}only decides whether to report, while the reports that arrive are still arbitrated by the same orchestratorℛ\\mathcal\{R\}\.

Then, we define the C\-POMDP of this communication\-selection subproblem element by element\. The agent on each active armiiis the decision unit that decides whether to report\. The observationzi,tz\_\{i,t\}is built only from quantities that armiican read out locally, and it contains neither the information of other arms nor any task semantics\. It is

zi,t=\(bi,t,‖qi−q𝒟‖,hi,Δi,t,τi,t\),z\_\{i,t\}=\\big\(b\_\{i,t\},\\,\\\|q\_\{i\}\-q\_\{\\mathcal\{D\}\}\\\|,\\,h\_\{i\},\\,\\Delta\_\{i,t\},\\,\\tau\_\{i,t\}\\big\),\(25\)wherebi,t∈\{0,1\}b\_\{i,t\}\\in\\\{0,1\\\}is the conflict\-possibility proxy that indicates whether the resource thatiideclares to occupy at this step falls on a shared resource𝒫sh\\mathcal\{P\}^\{\\mathrm\{sh\}\}, that is, whether the action touches a position that another arm may contend for\. This judgment depends only on the static shared\-resource structure𝒫sh\\mathcal\{P\}^\{\\mathrm\{sh\}\}and on the own occupancy declarationξi,t\\xi\_\{i,t\}ofii, and it needs no information of any other arm\. The end\-effector distance to the twin‖qi−q𝒟‖\\\|q\_\{i\}\-q\_\{\\mathcal\{D\}\}\\\|and the channel gainhi=h0​‖qi−q𝒟‖−μh\_\{i\}=h\_\{0\}\\\|q\_\{i\}\-q\_\{\\mathcal\{D\}\}\\\|^\{\-\\mu\}jointly characterize the local link state of armii\. The quantityΔi,t∈\{0,1\}\\Delta\_\{i,t\}\\in\\\{0,1\\\}is the intention novelty, namely whether the action of this step is the same as the previous step\. The quantityτi,t∈ℤ≥0\\tau\_\{i,t\}\\in\\mathbb\{Z\}\_\{\\geq 0\}is the number of consecutive rounds that armiihas not reported since its last report\. The actionσi,t∈\{0,1\}\\sigma\_\{i,t\}\\in\\\{0,1\\\}is the report indicator\. At each step every agent first produces an intended actionai,ta\_\{i,t\}on its own, and thenπc\\pi^\{c\}outputsσi,t\\sigma\_\{i,t\}fromzi,tz\_\{i,t\}\. Let the report set be𝒰t=\{i∣σi,t=1\}\\mathcal\{U\}\_\{t\}=\\\{i\\mid\\sigma\_\{i,t\}=1\\\}\. The orchestrator then computes the consistent set only over𝒰t\\mathcal\{U\}\_\{t\}as𝒜t⋆=ℛ​\(\{mi,t\}i∈𝒰t\)\\mathcal\{A\}\_\{t\}^\{\\star\}=\\mathcal\{R\}\(\\\{m\_\{i,t\}\\\}\_\{i\\in\\mathcal\{U\}\_\{t\}\}\)\. The action of an armi∉𝒰ti\\notin\\mathcal\{U\}\_\{t\}that does not report does not enter conflict avoidance and lands directly through the execution mapΦ\\Phiunder its own intention\. It bypasses the veto, so that its conflict is only guarded by the physical gate\. The communication cost is the total latency of this step on the shared uplink,Dt=∑i∈𝒰tδi,tD\_\{t\}=\\sum\_\{i\\in\\mathcal\{U\}\_\{t\}\}\\delta\_\{i,t\}, which under an equal channel is proportional to the report count\|𝒰t\|\|\\mathcal\{U\}\_\{t\}\|\.

The reward is the team\-level per\-step signal

rt=Rt−λ​Dt−β​ft,r\_\{t\}=R\_\{t\}\-\\lambda\\,D\_\{t\}\-\\beta\\,f\_\{t\},\(26\)whereRt=∑e∈ℰ⋆\(ςe​\(st\+1\)−ςe​\(st\)\)≥0R\_\{t\}=\\sum\_\{e\\in\\mathcal\{E\}^\{\\star\}\}\\big\(\\varsigma\_\{e\}\(s\_\{t\+1\}\)\-\\varsigma\_\{e\}\(s\_\{t\}\)\\big\)\\geq 0is the task\-goal progress of this step, namely the same per\-step reward\. Hereλ≥0\\lambda\\geq 0is the latency dual variable,Dt=∑i∈𝒰tδi,tD\_\{t\}=\\sum\_\{i\\in\\mathcal\{U\}\_\{t\}\}\\delta\_\{i,t\}is the total report latency of this step, andftf\_\{t\}is the number of physical failures caused by unreported actions at this step\. The action of an armi∉𝒰ti\\notin\\mathcal\{U\}\_\{t\}that does not report bypasses the orchestration and lands directly throughΦ\\Phi, and whether it is compatible with the concurrent actions of this step determines success or failure\. Let the active set that is actually executed concurrently at this step be the union of the admitted arms and the unreported arms that land directly,𝒜texe=𝒜t⋆∪\(𝒩tact∖𝒰t\)\\mathcal\{A\}\_\{t\}^\{\\mathrm\{exe\}\}=\\mathcal\{A\}\_\{t\}^\{\\star\}\\cup\(\\mathcal\{N\}\_\{t\}^\{\\mathrm\{act\}\}\\setminus\\mathcal\{U\}\_\{t\}\)\.ft=\|\{i∈𝒩tact∖𝒰t∣η\(i∣𝒜texe,Wt\)=0\}\|f\_\{t\}=\\big\|\\\{\\,i\\in\\mathcal\{N\}\_\{t\}^\{\\mathrm\{act\}\}\\setminus\\mathcal\{U\}\_\{t\}\\ \\mid\\ \\eta\(i\\mid\\mathcal\{A\}\_\{t\}^\{\\mathrm\{exe\}\},W\_\{t\}\)=0\\,\\\}\\big\|is the number of actions that land without orchestration yet violate the consistency criterionη\\etadefined in Section[III](https://arxiv.org/html/2607.09330#S3)\. Such incompatibility shows up physically as a collision or a gate failure and is counted by the execution mapΦ\\Phiat landing\. Hereβ​ft\\beta\\,f\_\{t\}is a reward shaping based on physical execution feedback rather than an injection of the coordination rule\. The task progressRtR\_\{t\}is sparse, since most steps are zero\. The gain of reporting or not reporting is reflected only indirectly through physical execution, so credit is hard to assign\. In contrast,ftf\_\{t\}, as the direct physical consequence of a missed conflict, supplies the sparse reward with a dense and observable signal\. Whenbi,t=1b\_\{i,t\}\{=\}1, not reporting causes a collision,ftf\_\{t\}rises, and the return drops, which drives the policy to report when a conflict is possible\. Whenbi,t=0b\_\{i,t\}\{=\}0, not reporting causes neither a collision nor a communication cost, which drives the policy to stay silent when it is safe\.

The policy adopts a per\-agent Bernoulli parameterization\. A networkgθg\_\{\\theta\}maps the observationzi,tz\_\{i,t\}to a report probabilitypθ​\(zi,t\)=1/\(1\+e−gθ​\(zi,t\)\)∈\[0,1\]p\_\{\\theta\}\(z\_\{i,t\}\)=1/\\big\(1\+e^\{\-g\_\{\\theta\}\(z\_\{i,t\}\)\}\\big\)\\in\[0,1\]which is given by

πθc​\(σi,t∣zi,t\)=pθ​\(zi,t\)σi,t​\(1−pθ​\(zi,t\)\)1−σi,t\.\\pi^\{c\}\_\{\\theta\}\(\\sigma\_\{i,t\}\\mid z\_\{i,t\}\)=p\_\{\\theta\}\(z\_\{i,t\}\)^\{\\,\\sigma\_\{i,t\}\}\\big\(1\-p\_\{\\theta\}\(z\_\{i,t\}\)\\big\)^\{\\,1\-\\sigma\_\{i,t\}\}\.\(27\)Here, the probability of reporting withσi,t=1\\sigma\_\{i,t\}=1ispθ​\(zi,t\)p\_\{\\theta\}\(z\_\{i,t\}\)and the probability of not reporting withσi,t=0\\sigma\_\{i,t\}=0is its complement1−pθ​\(zi,t\)1\-p\_\{\\theta\}\(z\_\{i,t\}\)\. The value is given by a value networkVϕV\_\{\\phi\}that estimates from the observation the expected discounted return of that state,Vϕ​\(zi,t\)≈𝔼πθc​\[∑k≥tγk−t​rk∣zi,t\]V\_\{\\phi\}\(z\_\{i,t\}\)\\approx\\mathbb\{E\}\_\{\\pi^\{c\}\_\{\\theta\}\}\\big\[\\sum\_\{k\\geq t\}\\gamma^\{k\-t\}r\_\{k\}\\mid z\_\{i,t\}\\big\]\. The advantage function is defined as

A^t=G^t−Vϕ​\(zi,t\),G^t=∑k≥tγk−t​rk,\\hat\{A\}\_\{t\}=\\hat\{G\}\_\{t\}\-V\_\{\\phi\}\(z\_\{i,t\}\),\\qquad\\hat\{G\}\_\{t\}=\\sum\_\{k\\geq t\}\\gamma^\{k\-t\}r\_\{k\},\(28\)whereG^t\\hat\{G\}\_\{t\}is the Monte Carlo return fromtton andA^t\\hat\{A\}\_\{t\}measures the gain of the report decision of this step relative to the baselineVϕV\_\{\\phi\}\. Both are used for the policy improvement and the value fitting of the PPO solver below\.

The action of this subproblem is per\-agent binary, and the observation is low\-dimensional, and after the shaping above, the reward is already densified\. We therefore use PPO\-Lagrangian as the solver because its policy gradient naturally couples to the primal–dual ascent of the latency constraint, which is standard modern practice for constrained RL\. Relaxing the per\-step latency constraint \([17](https://arxiv.org/html/2607.09330#S3.E17)a\) with a multiplierλ≥0\\lambda\\geq 0yields the saddle\-point objective as

minλ≥0⁡maxθ⁡𝔼πθc​\[∑tγt​\(Rt−β​ft\)−λ​\(∑tDt−T​D¯\)\],\\min\_\{\\lambda\\geq 0\}\\ \\max\_\{\\theta\}\\ \\mathbb\{E\}\_\{\\pi^\{c\}\_\{\\theta\}\}\\\!\\Big\[\\sum\_\{t\}\\gamma^\{t\}\\big\(R\_\{t\}\-\\beta\\,f\_\{t\}\\big\)\-\\lambda\\big\(\\textstyle\\sum\_\{t\}D\_\{t\}\-T\\bar\{D\}\\big\)\\Big\],\(29\)which is solved by alternating between its inner and outer levels\. The inner level, which fixesλ\\lambdaand improves the policy, ascends with PPO\. It reuses the advantageA^t\\hat\{A\}\_\{t\}and the Monte Carlo return targetG^t\\hat\{G\}\_\{t\}defined above, replacing single\-step bootstrapping with the return\-to\-go so that the sparse terminal reward propagates back stably over a short horizon\. With the probability ratioϱt​\(θ\)=πθc​\(σi,t∣zi,t\)/πθoldc​\(σi,t∣zi,t\)\\varrho\_\{t\}\(\\theta\)=\\pi^\{c\}\_\{\\theta\}\(\\sigma\_\{i,t\}\\mid z\_\{i,t\}\)/\\pi^\{c\}\_\{\\theta\_\{\\mathrm\{old\}\}\}\(\\sigma\_\{i,t\}\\mid z\_\{i,t\}\)it maximizes the clipped surrogate objective

ℒclip​\(θ\)=𝔼t​\[min⁡\(ϱt​\(θ\)​A^t,clip​\(ϱt​\(θ\),1−ϵ,1\+ϵ\)​A^t\)\],\\mathcal\{L\}^\{\\mathrm\{clip\}\}\(\\theta\)=\\mathbb\{E\}\_\{t\}\\big\[\\min\\big\(\\varrho\_\{t\}\(\\theta\)\\,\\hat\{A\}\_\{t\},\\ \\mathrm\{clip\}\(\\varrho\_\{t\}\(\\theta\),1\-\\epsilon,1\+\\epsilon\)\\,\\hat\{A\}\_\{t\}\\big\)\\big\],\(30\)and the value networkVϕV\_\{\\phi\}is fitted byminϕ⁡𝔼t​\[\(Vϕ​\(zi,t\)−G^t\)2\]\\min\_\{\\phi\}\\mathbb\{E\}\_\{t\}\[\(V\_\{\\phi\}\(z\_\{i,t\}\)\-\\hat\{G\}\_\{t\}\)^\{2\}\]\. The outer level, namely the latency dual, performs a PID primal–dual ascent onλ\\lambda\. With the per\-episode latency excessg^=1T​∑tDt−D¯\\hat\{g\}=\\frac\{1\}\{T\}\\sum\_\{t\}D\_\{t\}\-\\bar\{D\}it updatesλ←\[λ\+kP​g^\+kI​∑g^\+kD​Δ​g^\]\+\\lambda\\leftarrow\[\\lambda\+k\_\{P\}\\hat\{g\}\+k\_\{I\}\\sum\\hat\{g\}\+k\_\{D\}\\Delta\\hat\{g\}\]\_\{\+\}\. The proportional term responds quickly, the integral term removes the steady\-state excess, and the derivative term suppresses overshoot\. The PID form is adapted from the transmission scheduling of goal\-oriented semantic communication\[[30](https://arxiv.org/html/2607.09330#bib.bib30)\]\.

## VExperiments

For experimentation, we evaluate on the RoCo/MuJoCo physics simulator\[[11](https://arxiv.org/html/2607.09330#bib.bib11)\]\. The evaluation covers three classes of multi\-arm coordination, which correspond one\-to\-one to the three constraint operators of the orchestrator\. They are mutual exclusion \(e\.g\., confined\-space sorting, three arms relay\-sort objects through a shared panel, exclusionTξT\_\{\\xi\}\), synchronization \(move rope, barrierTζT\_\{\\zeta\}\), and dependency \(cabinet, hold\-then\-fetch, dependencyTωT\_\{\\omega\}\)\. The team heterogeneity is induced solely by the capability gap of the per\-armMiM\_\{i\}\. From strong to weak the models are fixed as DeepSeek\-R1\-Distill\-Qwen\-1\.5B\[[65](https://arxiv.org/html/2607.09330#bib.bib65)\]\>\>Qwen2\.5\-1\.5B\-Instruct\>\>Qwen2\.5\-0\.5B\-Instruct\[[66](https://arxiv.org/html/2607.09330#bib.bib66)\]\. Unless otherwise stated, the main experiments use a homogeneous default team that shares Qwen2\.5\-1\.5B\. For the heterogeneity\-robustness analysis, we additionally form three team configurations with weak, mid, and strong heterogeneity, where a larger capability gap among the per\-arm models indicates stronger heterogeneity\. For comparison, we consider four baselines, grouped into a dialogue\-based group and a centralized group as follows\.

- •RoCo\-NL\[[11](https://arxiv.org/html/2607.09330#bib.bib11)\]is the representative dialogue\-based paradigm, where multiple arms negotiate the division of labor through several rounds of natural\-language \(NL\) dialogue\.
- •AutoGen\[[12](https://arxiv.org/html/2607.09330#bib.bib12)\]is a real\-library implementation of the dialogue\-based paradigm, where a group\-chat manager orchestrates multi\-round multi\-agent dialogue\.
- •Centralized\-LLM \(fused\)\[[21](https://arxiv.org/html/2607.09330#bib.bib21),[63](https://arxiv.org/html/2607.09330#bib.bib63)\]is the representative centralized single\-brain paradigm, where one central LLM aggregates the whole team’s local observations and generates the joint action in one shot, with*fused*denoting a compressed observation input\.
- •Centralized\-Classical\[[42](https://arxiv.org/html/2607.09330#bib.bib42)\]replaces the central LLM with a classical greedy task\-allocation scheduler, representing an LLM\-free centralized ceiling on coordination quality\.

### V\-APerformance Evaluation

TABLE II:Success rate and per\-episode communication \(homogeneous 1\.5B\)\.Table[II](https://arxiv.org/html/2607.09330#S5.T2)reports the success rate \(SR\) and the per\-episode communication overhead of the five methods during collaboration with8080round in total\. Success rate is the fraction of successful episodes over all evaluation episodes, where an episode succeeds if and only if all task\-relevant objects meet the per\-object criterionςe\\varsigma\_\{e\}at termination\. From Table[II](https://arxiv.org/html/2607.09330#S5.T2), we can see that the considered methods have a similar SR band, while the proposed method has the lowest communication, which is7373to90×90\\timeslower than the dialogue methods and still1\.91\.9to2\.6×2\.6\\timeslower than the centralized methods\. This shows that the proposed method matches the baselines’ success rate band at the lowest communication overhead\. This is because the dialogue methods must transmit NL every round and the centralized methods must upload every arm’s full local observation, whereas each arm of ours completes its semantic understanding locally and uploads only a compact constraint tuple compressible into fixed\-width identifiers\.

![Refer to caption](https://arxiv.org/html/2607.09330v1/fig5_scalability.png)Figure 3:Per\-step communication vs\. team sizenn\.Fig\.[3](https://arxiv.org/html/2607.09330#S5.F3)shows how the per\-step communication changes as the team sizennvaries\. From Fig\.[3](https://arxiv.org/html/2607.09330#S5.F3), we can see that RoCo\-NL and AutoGen inflate rapidly withnn, while ours and the two centralized methods grow linearly, and ours stays the lowest throughout\. This shows that the communication advantage of ours keeps widening as the team grows and is not reversed by scale\. This is because the dialogue payload is proportional to the negotiation rounds times the number of participants and is therefore quadratic, whereas each arm of ours sends only one fixed\-length constraint tuple per step so that the total grows linearly withnn\.

![Refer to caption](https://arxiv.org/html/2607.09330v1/fig_latency.png)Figure 4:End\-to\-end per\-decision coordination latency vs\. team sizenn\.Fig\.[4](https://arxiv.org/html/2607.09330#S5.F4)shows how the end\-to\-end per\-decision latency changes asnnvaries\. From Fig\.[4](https://arxiv.org/html/2607.09330#S5.F4), we can see that the latency of the proposed method is independent ofnnand stays a constant1313to1818ms and the lowest throughout, while Centralized\-Classical rises linearly to0\.240\.24s and Centralized\-LLM to0\.550\.55s, and RoCo\-NL and AutoGen rise to the order of seconds \(8\.18\.1and13\.413\.4s\)\. From Fig\.[4](https://arxiv.org/html/2607.09330#S5.F4), we can also see that atn=8n=8the latency of the proposed method is only about1/7401/740of that of the slowest benchmark, so it alone sustains a real\-time\-level latency as the team scales\. This is because each arm of ours reads a bounded local context and scores in parallel, whereas the dialogue methods decode autoregressively, round by round, over a context that accumulates withnnduring LLM inference, and the centralized methods must collect every observation before serial processing\.

![Refer to caption](https://arxiv.org/html/2607.09330v1/fig_compute.png)Figure 5:Per\-episode LLM inference compute vs\. team sizenn\.Fig\.[5](https://arxiv.org/html/2607.09330#S5.F5)shows how the per\-episode LLM inference compute \(the total of prompt and completion tokens\) changes asnnvaries\. From Fig\.[5](https://arxiv.org/html/2607.09330#S5.F5), we can see that the proposed method grows linearly and RoCo\-NL and AutoGen inflate quadratically\. We can also see that atn=8n=8, the dialogue methods require44to66times as much computation as the proposed method\. This shows that the dialogue methods explode not only in communication overhead and latency but also in compute costs\. This is because the total generated tokens of multi\-round dialogue are proportional to the rounds times a growing dialogue context, whereas the total compute of ours is linear innnand parallelizable across arms, so its compute grows while its latency stays flat \(cf\. Fig\.[4](https://arxiv.org/html/2607.09330#S5.F4)\)\.

TABLE III:Success rate under increasing team heterogeneityTable[III](https://arxiv.org/html/2607.09330#S5.T3)reports how the SR varies as the team heterogeneity increases from weak to strong\. From Table[III](https://arxiv.org/html/2607.09330#S5.T3), we can see that the proposed method stays near\-flat throughout, while AutoGen degrades monotonically as the heterogeneity grows and RoCo\-NL stays lower and drops to its lowest under the strongest heterogeneity\. This shows that, as the team becomes more heterogeneous, the proposed method still maintains coordination, whereas the dialogue methods degrade\. This is because our delegates coordinate correctness to the deterministic DT resolver and depend on no single strong brain, requiring each arm only to express a local intent, whereas the coordination quality of the dialogue methods is directly dragged down by the language ability of the weakest participant\.

TABLE IV:Effect of AutoGen dialogue rounds on communication and success rate \(Sort, 30 seeds\)\.Dialogue roundsRRCommunication data size \(msg tok/ep\)SR13770\.43726740\.49139780\.55241,2820\.62161,8950\.633Table[IV](https://arxiv.org/html/2607.09330#S5.T4)reports the communication overhead and SR of AutoGen as the number of dialogue roundsRRincreases from11to66\. As shown in Table[IV](https://arxiv.org/html/2607.09330#S5.T4), the communication overhead increases by55times, whereas the SR improves by only44%44\\%\. This indicates that dialogue\-based methods cannot obtain proportional performance gains by simply increasing the number of negotiation rounds\. The reason is that the gains from deeper negotiation quickly saturate and are disproportionate to its communication cost, as the residual coordination bottleneck shifts to physical execution\. Therefore, additional dialogue rounds cannot overcome the execution\-level performance ceiling, while the proposed method can achieve a comparable SR with a single communication round\.

### V\-BAblation Study

TABLE V:Component ablation by removing one module of LDT\-Coord\.Table[V](https://arxiv.org/html/2607.09330#S5.T5)reports how the four metrics change when one core component is removed, where SR and latency are controlled and near\-invariant \(see the setup\) so that each ablation degrades only its target metric \(in bold\)\. From Table[V](https://arxiv.org/html/2607.09330#S5.T5), we can see that removing the DT constraint resolver raises the invalid execution attempts by about30%30\\%more collision trials per episode, and removing the learned gate raises the reporting needed to reach the same coordination quality by55times reports per step\. This shows that the DT contributes to execution efficiency and the RL gate contributes to communication efficiency, and neither expresses its value through the success rate\. This is due to the fact that DT vetoes mutual\-exclusion conflicts with a one\-bit STOP and blocks collisions before execution, while the learned gate reports only at the conflict\-prone handoff steps and is communication\-efficient\.

### V\-CConvergence and Mechanism: Dual Dynamics of the Learned Gate

![Refer to caption](https://arxiv.org/html/2607.09330v1/fig13_triptych.png)Figure 6:Dual dynamics of the learned communication gateπc\\pi^\{c\}under tight and loose latency deadlines over training episodes: \(a\) dual variableλ\\lambda, \(b\) per\-step report count, \(c\) success rate\.Fig\.[6](https://arxiv.org/html/2607.09330#S5.F6)shows the dual dynamics of the learned communication gateπc\\pi^\{c\}under tight and loose latency deadlines as training proceeds, with the three panels giving the dual variableλ\\lambda, the per\-step report count, and the success rate\. From Fig\.[6](https://arxiv.org/html/2607.09330#S5.F6), we can see thatλ\\lambdarises monotonically with training and drives the per\-step report count down into the latency budget, while the SR stays near the execution ceiling throughout\. This is because the PID dual ascent encodes the communication constraint into the reward, so that a tighter constraint yields a largerλ\\lambdaand a heavier reporting penalty and forces the policy to report only at the most necessary conflict steps, reaching a low\-communication, high\-success tradeoff within the feasible region and satisfying the per\-step latency deadline\.

## VIConclusion

This paper presented LDT\-Coord, a digital\-twin coordination middleware for networks of heterogeneous LLM\-driven embodied agents\. In LDT\-Coord, each agent perceives and decides autonomously and reports only a structured action together with typed temporal constraints, while the digital twin runs a single training\-free orchestrator that unifies mutual\-exclusion, synchronization, and dependency atomic tasks into four orchestration rules and applies them iteratively until convergence\. To further reduce the upload communication overhead, a PPO\-Lagrangian\-based communication\-efficient layer is introduced\. Simulation results show that the proposed coordinate\-without\-training and communicate\-by\-learned\-optimization framework can enable efficient collaboration among embodied agent teams\. For future work, we will investigate collaborative perception among agent teams to extend each agent’s perceptual boundary\.

### \-AProof of Proposition[1](https://arxiv.org/html/2607.09330#Thmproposition1)

###### Proof\.

Each rule only removes elements, so the composite operator satisfies the shrinkage propertyΘ​\(𝒜\)⊆𝒜\\Theta\(\\mathcal\{A\}\)\\subseteq\\mathcal\{A\}\. Let𝒜\(0\)=𝒩tact\\mathcal\{A\}^\{\(0\)\}=\\mathcal\{N\}\_\{t\}^\{\\mathrm\{act\}\}and𝒜\(k\+1\)=Θ​\(𝒜\(k\)\)\\mathcal\{A\}^\{\(k\+1\)\}=\\Theta\(\\mathcal\{A\}^\{\(k\)\}\), so that𝒜\(0\)⊇𝒜\(1\)⊇⋯\\mathcal\{A\}^\{\(0\)\}\\supseteq\\mathcal\{A\}^\{\(1\)\}\\supseteq\\cdotsis a monotonically decreasing chain on the power\-set lattice of the finite set𝒩tact\\mathcal\{N\}\_\{t\}^\{\\mathrm\{act\}\}\. Any nontrivial iteration removes at least one element; otherwise,𝒜\(k\+1\)=𝒜\(k\)\\mathcal\{A\}^\{\(k\+1\)\}=\\mathcal\{A\}^\{\(k\)\}is already a fixed point\. Since the set is finite, the decreasing chain stabilizes after at mostK≤\|𝒩tact\|K\\leq\|\\mathcal\{N\}\_\{t\}^\{\\mathrm\{act\}\}\|steps, and the stable point is the greatest fixed point of the shrinkage operator on this finite power\-set lattice\[[49](https://arxiv.org/html/2607.09330#bib.bib49)\]\. This ends the proof\. ∎

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