Collaborative Successive Hover-and-Fly Structure

• Collaborative successive hover-and-fly (co-SHF) is a multi-agent framework that segments missions into synchronized hover phases for critical tasks and optimized fly segments for rapid transit.
• It reduces complex continuous-time trajectory and scheduling problems to finite-dimensional forms by parameterizing co-hover points, durations, and linearized paths using convex optimization techniques.
• The approach enhances operational efficiency in applications like UAV-USV ISAC and secure communications by ensuring optimal energy usage and coordinated agent performance.

Collaborative successive hover-and-fly (co-SHF) structure is an optimal trajectory and scheduling principle for tightly coupled multi-agent aerial and maritime or multi-UAV systems engaged in wireless communication, inspection, or jamming-aided secure communication missions. Under co-SHF, a set of collaborating vehicles coordinate the sequence and duration of discrete “hover” phases, in which they pause for key sensing, communication, or security objectives, separated by “fly” phases featuring at least one agent traveling at maximum speed. This framework enables provably optimal or near-optimal performance for various resource-constrained, coupled-agent tasks, including integrated sensing and communication (ISAC) and secrecy-maximizing transmission, and allows significant reduction of problem dimensionality compared to continuous-time or time-discretized formulations (Zhang et al., 4 Nov 2025, Wang et al., 28 Jan 2026).

1. Formal Definition and Core Properties

The collaborative successive hover-and-fly structure is characterized by synchronized agent behaviors over a mission interval, where each agent (e.g., UAV, USV, or dual UAVs) traverses a sequence of critical mission waypoints and actively schedules operations at a finite set of “co-hovering” positions. Specifically, for two collaborating UAVs (source, $S$, and jammer, $J$), a co-SHF trajectory consists of:

• A finite sequence of $I \leq K$ co-hovering point pairs $\{(\mathbf{q}_{S,i},\,\mathbf{q}_{J,i})\}_{i=1}^I$, where each agent hovers simultaneously at these positions for a duration $t_i\geq 0$.
• Piecewise-linear “fly” segments between co-hovering pairs, during which at least one agent moves at its maximum speed $V$; the segment duration is thus $\Delta t = \max\{d_S, d_J\}/V$.
• Synchronized scheduling for communication, sensing, or jamming activities, relying exclusively on the co-hovering intervals.
• Under co-SHF, hovering is executed only at synchronized points, and intermediary movement is fully time-optimized, preventing redundant idling or sub-maximal velocity segments.

For air-sea missions (e.g., UAV-USV ISAC inspection), the same paradigm applies: the UAV’s trajectory is segmented into discrete hover-fly episodes, with the USV’s trajectory synchronized through intermediate waypoints, combining energy-aware path planning and adaptive task allocation (Zhang et al., 4 Nov 2025).

2. Mathematical Parameterization

co-SHF fundamentally enables reduction of the infinite-dimensional continuous-time trajectory planning problem to a finite-dimensional form defined by:

• The number $I$ of co-hovering pairs and the coordinates $(\mathbf{q}_{u,i})$ for each agent $u$ at each pair.
• Turning points $\{\mathbf{q}_{u, i, j}\}$ for refining trajectory flexibility between hovers (where $i$ indexes the co-hovering pair and $j$ the segment).
• Hovering durations $\{t_i\}$, which correspond to the time allocation for performance-critical tasks at each co-hover position.
• In flying segments, the position parameterization is given by $$\mathbf{q}_{u,i,j}(z) = \mathbf{q}_{u,i,j} + z(\mathbf{q}_{u,i,j+1} - \mathbf{q}_{u,i,j})$$ for $z\in[0,1]$.
• Physical and mission constraints (speed limits, collision avoidance, total time/costs) are enforced at the finite set of trajectory variables, often requiring only a few auxiliary indicators and first-order relaxations in optimization.

For UAV-USV systems, the co-SHF structure decomposes the mission interval into $E+1$ hover-and-fly stages, whereby each stage is parameterized by hover and path locations $q_e, b^{f}_e,$ and $b^{h}_e$, traveling times $t^f_e, t^h_e$, and associated trajectory, energy, and beamforming optimization variables (Zhang et al., 4 Nov 2025).

3. Problem Reduction and Optimization Methodologies

The co-SHF structure enables explicit reduction from infinite-dimensional, nonconvex trajectory/control problems to tractable finite-dimensional forms:

• The original problem’s controls—continuous-time trajectories $\mathbf{q}_u(t)$ and real-time scheduling $a_k(t)$—are replaced by waypoint sets, segment durations, and discrete scheduling variables ($a_{i,k}\in\{0,1\}$).
• Constraint sets for collision avoidance are recast as minimum-distance infeasibility within each finite segment, with convex lower bounds obtained by Taylor expansion.
• Speed limits and scheduling exclusivity are directly enforced by the segment-based parameterization.

Algorithmically, this enables:

• Successive convex approximation (SCA) for handling nonconvex rate or collision constraints, leveraging concave lower bounds and incremental linearization.
• Semi-definite relaxation (SDR) for beamforming variable control in ISAC hovering intervals.
• Iterative alternating minimization, where each subproblem (e.g., waypoints, beamformers, time allocations) is solved sequentially within each hover-fly episode, leading to robust convergence (Zhang et al., 4 Nov 2025, Wang et al., 28 Jan 2026).

In dual-UAV jamming-aided secure communications, this methodology yields a program whose variables scale as $O(KN)$, rather than the combinatorial complexity of high-resolution time discretization (Wang et al., 28 Jan 2026).

4. Hierarchical Mission Decomposition and Algorithmic Workflow

The co-SHF approach is implemented via hierarchical problem decomposition:

• Stage 1: Hover Point Selection Virtual base station coverage (VBSC)-based clustering algorithms identify rough hover points, grouping targets/ground users into tractable clusters (with at most $Z$ targets per hover).
• Stage 2: Route Sequencing A bi-traveling salesman problem with neighborhood (Bi-TSPN) assigns visiting order, jointly minimizing energetic and operational costs by integrating agent path lengths, environmental currents, and local obstacles.
• Stage 3: Refinement and Time Allocation Hover-point positions and dwell times are refined via linearization and convex optimization, incorporating kinetic and communication/sensing constraints.
• Stage 4: Stage-wise Optimization For each hover-fly segment, detailed agent trajectories and beamforming strategies are optimized via SCA and SDR, alternately updating spatial and temporal allocations until energy or performance improvement converges.

The resulting workflow achieves full mission plans with joint trajectory, resource, and task schedule optimization that fully exploits co-agent coupling (Zhang et al., 4 Nov 2025).

5. Performance, Complexity, and Structural Optimality

The structural optimality of co-SHF is rigorously established in dual-UAV settings by:

• Demonstrating (Proposition 1) that any deviation—such as asynchronous hovering or sub-maximal speed travel—can be strictly improved by redistributing time to synchronized hover events at optimal co-hover pairs.
• Showing (Proposition 2) that no more than $K$ co-hover pairs are necessary, as otherwise per-user throughput can be strictly increased by consolidating hovering at the best-performing point, guaranteeing minimal/unique assignment of hover points per user (Wang et al., 28 Jan 2026).

Tables below compare core performance metrics for co-SHF and baseline methods in representative scenarios:

Method	Energy (kJ)	Min-Secrecy Throughput	Solution Time
co-SHF (UAV-USV)	40.9	—	—
Sequential Access	58.1	—	—
Leader-Follower	50.9	—	—
co-SHF (Dual-UAV)	—	High	Low
TD-SCP (Discretized)	—	Lower	~9× higher

Simulation results confirm that co-SHF-based solutions achieve shorter agent paths, fewer dwelling points, and superior energy efficiency—reducing both computational and operational resources—while enabling finer trade-off between collaborative communication and sensing or secrecy objectives (Zhang et al., 4 Nov 2025, Wang et al., 28 Jan 2026).

6. Domain Applications and Generalizations

The co-SHF structure is directly applicable to:

• UAV-USV maritime ISAC missions, facilitating synchronized inspection with real-time communication under environmental and physical constraints.
• Dual-UAV secure communication networks, maximizing user secrecy throughput via real-time jamming and adaptive hovering.
• Any multi-agent mobile mission in which agent coordination, limited dwell opportunities, and high-dimensional control present coupled optimization challenges.

A plausible implication is that the co-SHF structural reduction generalizes to a wider class of multi-agent mission planning contexts, wherever finite-duration coupled idling/actions and time-optimal travel yield performance and complexity gains.

7. Adaptivity, Limitations, and Future Considerations

The adaptivity of co-SHF hinges on:

• Its ability to dynamically cluster, schedule, and coordinate agent resource allocation in response to target distribution, environmental dynamics (e.g., water currents), and rate/SNR requirements.
• Its compatibility with nonconvex mission spaces and agent heterogeneity, through modular variable decomposition and alternating convexification.

However, the co-SHF approach assumes perfect synchronization and communication between agents over the time horizon, as well as accurate a priori knowledge of target locations and environmental conditions. Future advances may focus on robustification to uncertainty, distributed agent control with limited feedback, and real-time extensions for dynamically evolving operational contexts.

The collaborative successive hover-and-fly structure represents a foundational paradigm for tractable, high-performance multi-agent mission design, with demonstrated utility and provable optimality in both communication and ISAC settings (Zhang et al., 4 Nov 2025, Wang et al., 28 Jan 2026).