STAF: SubTeaming & Adaptive Formation

• STAF is a hierarchical multi-robot framework that dynamically splits teams into subteams, adapts formations, and merges after navigating narrow spaces.
• It integrates high-level deep graph cut for subteaming, intermediate graph learning for adaptive formation, and low-level policy learning for collision-free navigation.
• Experimental results demonstrate that STAF outperforms rigid formation methods by achieving higher success rates, shorter travel times, and better formation integrity.

Searching arXiv for the specified STAF and related drone-formation papers. SubTeaming and Adaptive Formation (STAF) denotes a coordinated multi-robot navigation capability in which a team of robots dynamically splits into smaller subteams, adaptively controls the subteams to navigate through narrow or complex environments while preserving formations, and then recovers the full-team formation after passage (Deng et al., 19 Sep 2025). In the reported formulation, STAF is implemented as a unified hierarchical learning framework with three coupled levels: high-level deep graph cut for team splitting, intermediate-level graph learning for coordinated navigation among subteams, and low-level policy learning for individual robot control. A related line of work in drone swarms is closely aligned in spirit because it also treats internal organization as a decision variable adapted to environmental conditions, although it does not allow the swarm to split or merge during flight (Alkouz et al., 2020).

1. Problem setting and failure mode of rigid formation control

STAF addresses coordinated multi-robot navigation in which a team must move to goal positions while preserving a mission-specific formation such as circle formation, wedge formation, or line formation (Deng et al., 19 Sep 2025). The team is modeled as an undirected graph

$G=\{V,E\},$

with node set

$V=\{v_1,v_2,\dots,v_n\},$

where each node is

$v_i=\{p_i,g_i,q_i\}.$

Here, $p_i=[p_i^x,p_i^y]$ is the robot position, $g_i=[g_i^x,g_i^y]$ is the goal position, and $q_i=[q_i^x,q_i^y]$ is the velocity in $x/y$. The edge set is represented by the adjacency matrix

$E=\{a_{i,j}\}^{n\times n},$

with $a_{i,j}=1$ if robots $i$ and $V=\{v_1,v_2,\dots,v_n\},$0 are within a radius and $V=\{v_1,v_2,\dots,v_n\},$1 otherwise. The per-robot state is

$V=\{v_1,v_2,\dots,v_n\},$2

where $V=\{v_1,v_2,\dots,v_n\},$3 is the distance to the closest obstacle, and the action is

$V=\{v_1,v_2,\dots,v_n\},$4

The motivating constraint is that rigid formation preservation can become infeasible in narrow or complex environments such as corridors or bridges. The representative example is a team of ten robots in a circular formation encountering a corridor too narrow for the whole team to pass together. Under that condition, the team must be able to split into subteams, adapt the formation within each subteam, and regroup after passing.

The framework is positioned against prior methods that are described as too rigid for such settings, including leader-follower approaches, virtual structure methods, and rigid optimization-based formation control. Prior subteaming work is also characterized as focusing only on division or merging, rather than on how to control subteams during navigation or how to adapt formation shape during transit. The central significance of STAF lies in combining division, adaptive control, and recovery in a single navigation pipeline.

2. Hierarchical organization and graph-cut subteaming

STAF is a three-level hierarchical framework composed of high-level deep graph cut, intermediate-level graph learning, and low-level policy learning (Deng et al., 19 Sep 2025).

Level	Mechanism	Function
High-level	deep graph cut	subteaming
Intermediate-level	graph learning	subteam coordination and adaptive formation
Low-level	policy learning	individual robot control

At the high level, STAF first computes graph embeddings

$V=\{v_1,v_2,\dots,v_n\},$5

where $V=\{v_1,v_2,\dots,v_n\},$6 is a graph attention network. The node projection is

$V=\{v_1,v_2,\dots,v_n\},$7

Attention from node $V=\{v_1,v_2,\dots,v_n\},$8 to node $V=\{v_1,v_2,\dots,v_n\},$9 is defined as

$v_i=\{p_i,g_i,q_i\}.$0

where $v_i=\{p_i,g_i,q_i\}.$1 is the rectified linear unit, $v_i=\{p_i,g_i,q_i\}.$2 is the neighborhood of node $v_i=\{p_i,g_i,q_i\}.$3, $v_i=\{p_i,g_i,q_i\}.$4 denotes concatenation, and $v_i=\{p_i,g_i,q_i\}.$5 is a weight matrix. The final node embedding is

$v_i=\{p_i,g_i,q_i\}.$6

A multi-head mechanism is then used to obtain richer embeddings.

Using $v_i=\{p_i,g_i,q_i\}.$7, the framework formulates team division as a graph cut problem that partitions the team into $v_i=\{p_i,g_i,q_i\}.$8 subgraphs or subteams. The classifier network

$v_i=\{p_i,g_i,q_i\}.$9

consists of two fully connected linear layers followed by SoftMax and outputs

$p_i=[p_i^x,p_i^y]$0

where $p_i=[p_i^x,p_i^y]$1 is the probability that robot $p_i=[p_i^x,p_i^y]$2 belongs to subteam $p_i=[p_i^x,p_i^y]$3, with $p_i=[p_i^x,p_i^y]$4.

The high-level objective is unsupervised and combines three terms: subteam adjacency, subteam balance, and subteam-goal distance. The adjacency term encourages robots in the same subteam to remain connected or adjacent; the balance term encourages equal or near-equal subteam sizes, with target size $p_i=[p_i^x,p_i^y]$5; and the goal-distance term encourages each subteam’s center to align with its corresponding goal center. The text explicitly states that $p_i=[p_i^x,p_i^y]$6 measures the probability that a pair of robots belong to different subteams, while $p_i=[p_i^x,p_i^y]$7 encodes robot adjacency. The subteam center position of subteam $p_i=[p_i^x,p_i^y]$8 is

$p_i=[p_i^x,p_i^y]$9

and the corresponding goal center is

$g_i=[g_i^x,g_i^y]$0

Functionally, this component chooses which robots form each subteam before navigation through a bottleneck. A plausible implication is that STAF operationalizes team division as a structured graph partition rather than as a heuristic split based only on spatial proximity.

3. Adaptive formation control within subteams

The intermediate level coordinates robots within a team or subteam and enables formation adaptation (Deng et al., 19 Sep 2025). Given the graph $g_i=[g_i^x,g_i^y]$1 and robot state $g_i=[g_i^x,g_i^y]$2, a network $g_i=[g_i^x,g_i^y]$3 computes a team-state embedding

$g_i=[g_i^x,g_i^y]$4

The robot state is first projected as

$g_i=[g_i^x,g_i^y]$5

and message passing then aggregates teammates’ state embeddings:

$g_i=[g_i^x,g_i^y]$6

This embedding captures the robot’s own state together with relative spatial relationships to teammates.

Rigid formation is replaced by an adaptive spring-damper formation model. For a robot pair $g_i=[g_i^x,g_i^y]$7, the spring component is based on

$g_i=[g_i^x,g_i^y]$8

where $g_i=[g_i^x,g_i^y]$9 is the expected distance in the original formation and

$q_i=[q_i^x,q_i^y]$0

is the actual distance. The damper component uses the relative velocity term

$q_i=[q_i^x,q_i^y]$1

The combined formation adaptation reward is

$q_i=[q_i^x,q_i^y]$2

where $q_i=[q_i^x,q_i^y]$3 balances spring versus damper.

The stated role of this reward is threefold: it promotes keeping robots close enough to fit through narrow spaces, preserving enough spacing to avoid collisions, and suppressing oscillation and overshoot. This is the specific mechanism by which STAF replaces exact rigid shape preservation with adaptive formation control. The same formation type is preserved within each subteam, but the formation shape may adapt locally.

4. Low-level policy learning, execution flow, and recovery

At the low level, STAF outputs velocity commands for each robot (Deng et al., 19 Sep 2025). Given the state $q_i=[q_i^x,q_i^y]$4 and the embedding $q_i=[q_i^x,q_i^y]$5, the action is produced by

$q_i=[q_i^x,q_i^y]$6

where $q_i=[q_i^x,q_i^y]$7 is a network of two linear layers followed by ReLU. This is part of a policy

$q_i=[q_i^x,q_i^y]$8

parameterized by $q_i=[q_i^x,q_i^y]$9 and trained with reinforcement learning.

The low-level reward encourages moving toward the goal, avoiding obstacles, and avoiding other robots. The description specifies a reward based on distance to goal and a penalty when too close to obstacles or other robots. Collision avoidance is therefore handled through reward penalties rather than through a separate explicit planner.

The overall pipeline consists of five steps. First, the input team graph is constructed, with nodes containing robot state attributes and edges encoding local adjacency. Second, the high-level subteaming stage applies GAT embedding $x/y$0 and classifier $x/y$1 to output $x/y$2, an assignment of robots to subteams. Third, for each team or subteam, $x/y$3 computes team-state embeddings $x/y$4, while the spring-damper reward encourages adaptive formation. Fourth, $x/y$5 produces the action $x/y$6, and the RL policy $x/y$7 learns collision-free navigation. Fifth, after subteams traverse narrow regions, goals are updated and robots merge back into the full formation.

The execution structure is explicitly hierarchical in deployment as well as in learning: the high level is centralized, while the intermediate and low levels are decentralized. Training uses an alternating training algorithm that iteratively trains the high level and then jointly trains the intermediate-low levels. The paper identifies this alternating scheme itself as a limitation.

5. Experimental settings, metrics, and reported performance

STAF is evaluated in three settings: Gazebo simulation in ROS1, Unity-based 3D simulator in ROS1, and physical robot teams running ROS2 (Deng et al., 19 Sep 2025). The scenario types include indoor narrow spaces, outdoor unstructured terrain, narrow corridors and bridges, and different formation types: circle, wedge, and line. The platforms are simulated robots in Gazebo, differential-drive Warthog robots in Unity3D, and differential-drive Limo robots with caterpillar tracks, onboard Intel NCU i7, Wi-Fi communication, and ROS2 in the real world. Real experiments use SLAM for state estimation and mapping, and in Unity3D the linear velocity action is converted into wheel velocities for differential-drive robots.

The quantitative metrics are Successful Rate (SR), Travel Time (TT), and Contextual Formation Integrity (CFI). SR is the proportion of robots in the full team that reach the goal without collisions. TT is the total time for the team to reach the goal. CFI measures real-time adherence to the target formation using a shape threshold $x/y$8, lies in $x/y$9, and uses smaller $E=\{a_{i,j}\}^{n\times n},$0 for stricter formation requirements.

The baselines are Leader-Follower (LF) and Decentralized GNN (DGNN). LF uses one robot as leader while others follow and attempt to maintain formation; DGNN performs hierarchical learning for individual robot velocity control but does not consider team-level formations. In Gazebo, DGNN performs worst on formation integrity because it lacks formation control. LF performs better than DGNN but has only a 40% success rate due to rigid formation constraints. STAF-full achieves 100% success rate in all tested formations. The reported TT values for STAF-full are 102.10 sec for Circle, 69.30 sec for Wedge, and 111.50 sec for Line.

At the subteam level in Gazebo, both subteams achieve 100% success rate across circle, wedge, and line formations. Under $E=\{a_{i,j}\}^{n\times n},$1, the reported CFI values are: for STAF-sub1, Circle 81.56, Wedge 77.22, and Line 91.13; for STAF-sub2, Circle 87.72, Wedge 80.99, and Line 91.78. Qualitatively, LF gets stuck in the corridor, whereas STAF splits the team, one subteam moves first, then the second, and the subteams regroup after passing.

The ablation study shows that removing any of the three graph-cut terms hurts behavior. Using only the balance term yields equal split but is not enough for proper navigation alone. Using only adjacency assigns all robots to one subteam. Using only goal distance aligns subteams toward goals but ignores compactness and balance. More generally, removing the balance term leads to unbalanced division, removing the adjacency term leads to non-compact subteams, and omitting goal distance leads to inefficient navigation.

Generalization is reported for different team sizes of 4, 6, 7, 8 robots and different numbers of subteams of 2, 3, 4. The paper reports about 87% formation integrity for 4 robots under $E=\{a_{i,j}\}^{n\times n},$2, and at least 80% for 8 robots. In Unity3D and on physical robots, STAF successfully handles long outdoor paths, curved trajectories, differential-drive dynamics, and a 9-robot line formation splitting into 3 subteams when needed. Real-world demonstrations further show that 6–8 robots can split and adapt formation in indoor narrow spaces, and that outdoor snowy or uneven terrain is handled despite wheel slippage and action uncertainty.

6. Relation to formation-guided swarm adaptation and scope of the concept

A closely related but narrower formulation appears in "Formation-based Selection of Drone Swarm Services" (Alkouz et al., 2020). That work considers swarm delivery in a skyway network, models each swarm trip over one skyway segment as a Swarm-based Drone-as-a-Service,

$E=\{a_{i,j}\}^{n\times n},$3

and asks which swarm service segments should be selected and which flight formation should be used so that delivery is feasible and energy-efficient. The swarm remains intact throughout the journey: drones stay together, and they do not split or merge mid-flight.

The central decision variable is formation. Five formation types are considered: Column, Front, Echelon, Vee, and Diamond. Two decision modes are defined. In the Fixed approach, one formation is chosen before the mission begins using average wind conditions over the whole service set. In the Adaptive approach, the best formation is reselected at every skyway segment based on the segment’s wind speed and wind direction. The energy model uses

$E=\{a_{i,j}\}^{n\times n},$4

for power and

$E=\{a_{i,j}\}^{n\times n},$5

for segment energy consumption, with feasibility expressed as

$E=\{a_{i,j}\}^{n\times n},$6

Wind speeds are considered in Beaufort numbers 0 through 6, and wind direction is simplified to Front, Right, and Left.

The reported results show that formation choice depends on the aerodynamic model and the wind. Under drag only at Beaufort 5, Column gives the lowest average drag for front wind, whereas Front gives the lowest average drag for right or left wind. When upwash and downwash are included, the best energy-saving formations change: Vee performs best under front wind, and Diamond performs best under right and left wind. The reported best average energy consumption values are 1337.69 W for Vee under front wind, 1365.35 W for Diamond under right wind, and 1354.63 W for Diamond under left wind. Across both distance-based and wind-based evaluations, the Adaptive selection algorithm consistently outperforms the Fixed algorithm in energy consumption, though Adaptive takes longer to execute as network size increases.

This relation is important for delimiting the scope of STAF. Both frameworks treat group structure as a decision variable and both adapt configuration to environment conditions. However, the drone-delivery framework does not model arbitrary subteam membership changes, does not split or merge the swarm, and optimizes primarily for energy consumption under battery constraints. STAF, by contrast, explicitly introduces subteaming, adaptive formation control within each subteam, and team recovery after passing through constrained regions (Deng et al., 19 Sep 2025). This suggests that the 2020 drone work is best understood as a formation-adaptive precursor in spirit, whereas the 2025 STAF framework is a fuller treatment of split, coordinated transit, and regrouping.

7. Limitations, boundaries, and research directions

The reported STAF formulation is explicit about its limitations (Deng et al., 19 Sep 2025). First, high-level division is centralized; future work is suggested in the direction of decentralized team division via gossip or max-consensus. Second, alternating training is described as suboptimal; future work is proposed for end-to-end training with error propagation from low to intermediate and high levels. Third, the number of subteams $E=\{a_{i,j}\}^{n\times n},$7 is manually set as a hyperparameter; future work is suggested to dynamically determine the minimum number of subteams based on corridor width, inferred from maps or real-time sensing.

These constraints also define what STAF is not. It is not an unconstrained coalition-formation framework with arbitrary dynamic membership changes, because the number of subteams is manually chosen and the same formation type is preserved within each subteam. It is not purely a rigid formation controller, because formation shape adapts locally through the spring-damper reward. It is also not merely a split-and-merge heuristic, because the framework couples graph-cut subteaming, graph-based subteam coordination, and RL-based robot control in a unified hierarchy.

Taken together, the available evidence identifies STAF as a specific hierarchical approach to coordinated multi-robot navigation in which graph partitioning, adaptive intra-subteam formation control, and robot-level policy learning are integrated to handle bottlenecks that defeat rigid formation control. The empirical record reported for simulation and physical robots indicates that its distinctive contribution is the combined capability for subteaming, adaptive formation control, and team recovery under challenging navigation constraints (Deng et al., 19 Sep 2025).