ETF: Efficient UAV Transition for Multicasting

• ETF is an algorithmic framework that ensures UAV transitions remain within designated transmission ranges using a two-phase approach to verify direct trajectories and construct replacement paths when necessary.
• The method employs low-complexity geometric checks—using straight-line trajectory verification and midpoint heuristic adjustments—and Dijkstra’s algorithm for forwarder chain selection.
• Simulations demonstrate ETF’s effectiveness with lower multicast delay, higher throughput, and reduced energy per bit in both short- and long-distance UAV multicasting scenarios.

Efficient Transition Formation (ETF) is an algorithmic framework for fast yet resource-efficient UAV transitions in a multicasting environment, designed to maintain high multicasting performance while a mobile UAV moves between an origin $A$ and a destination $B$. In "Resource-Efficient Seamless Transitions For High-Performance Multi-hop UAV Multicasting" (Tu, 6 Jul 2025), ETF is formulated as a two-phase procedure: it first evaluates whether a direct straight-line trajectory (SLT) is seamless, using low-complexity computations or a chain of fast checks with controlled traffic overheads, and only when the SLT is interrupted does it construct a new trajectory with a minimum number of seamless straight lines joined at specially selected locations. The method is developed for multi-hop UAV multicasting with local, piggy-backed knowledge of forwarder locations and referred transmission ranges.

1. Formal problem statement

ETF considers a multicast of rich media among fixed forwarding UAVs and one mobile UAV $m$. The forwarding set is

$\mathcal U=\{u_0,\dots,u_{n_f-1}\},$

where each forwarder $u_i$ has Cartesian coordinates $\mathbf p_i=(x_i,y_i,z_i)$ and a referred coverage radius $r_i$. Its referred transmission range (RTR) is the closed ball

$$S_i=\{\mathbf x\in\mathbb R^3:\|\mathbf x-\mathbf p_i\|\le r_i\}.$$

The mobile UAV moves from $A\in\mathbb R^3$ to $B\in\mathbb R^3$, with $B$0 denoting the forwarders whose RTRs contain $B$1 and $B$2, respectively (Tu, 6 Jul 2025).

The trajectory sought by ETF is piecewise linear: $B$3 The objective is fourfold: the path must remain entirely inside the union of selected RTRs $B$4; it must use as few segments $B$5 as possible; it must minimize the total travel distance $B$6; and it must require only local, piggy-backed knowledge of $B$7.

Two explicit constraints govern feasibility. First, each segment must be fully covered by some forwarder: $B$8 Second, consecutive RTRs must overlap: $B$9

A central premise is that a single SLT $m$0 is provably the shortest possible path. ETF therefore treats direct-line verification as the first stage, and resorting to a multi-segment construction only when seamless straight-line travel is impossible.

2. Seamlessness criteria for direct motion

ETF distinguishes between short-distance and long-distance transitions according to whether the RTRs of $m$1 and $m$2 overlap (Tu, 6 Jul 2025). The short-distance case is defined by

$m$3

whereas the long-distance case corresponds to non-overlapping RTRs.

For the short-distance case, ETF introduces two geometric points on the line segment $m$4: $m$5, the exit point of $m$6 from the sphere $m$7, and $m$8, the entry point of $m$9 into the sphere $\mathcal U=\{u_0,\dots,u_{n_f-1}\},$0. With

$\mathcal U=\{u_0,\dots,u_{n_f-1}\},$1

the SLT is seamless if and only if one of two conditions holds. The first is

$\mathcal U=\{u_0,\dots,u_{n_f-1}\},$2

which corresponds to the absence of a gap between the two spheres along the line. The second is the existence of a third forwarder $\mathcal U=\{u_0,\dots,u_{n_f-1}\},$3 such that both

$\mathcal U=\{u_0,\dots,u_{n_f-1}\},$4

Operationally, $\mathcal U=\{u_0,\dots,u_{n_f-1}\},$5 and $\mathcal U=\{u_0,\dots,u_{n_f-1}\},$6 are obtained by solving the quadratic line-sphere intersection equations, selecting the root near the relevant endpoint.

For the long-distance case, ETF first computes the set of covering forwarders

$\mathcal U=\{u_0,\dots,u_{n_f-1}\},$7

where the distance from $\mathcal U=\{u_0,\dots,u_{n_f-1}\},$8 to the segment $\mathcal U=\{u_0,\dots,u_{n_f-1}\},$9 is given in closed form by Heron’s formula: $u_i$0 If $u_i$1, the SLT is not seamless. Otherwise, ETF applies a progressive checking procedure along $u_i$2: starting from the first exit point $u_i$3 of $u_i$4, it iteratively selects a forwarder that covers the current exit point, moves to that forwarder’s far-exit point, and continues until either the chain reaches $u_i$5 or no further covering forwarder exists.

This two-case structure makes ETF a geometric coverage-verification method rather than a purely graph-theoretic mobility planner. A plausible implication is that seamlessness is treated as a path-continuity property in Euclidean space first, and only secondarily as a connectivity problem over forwarder relations.

3. Construction of replacement trajectories

When the SLT is interrupted, ETF constructs a replacement path whose segments remain seamless and whose number is minimized relative to the identified forwarder chain (Tu, 6 Jul 2025). The replacement strategy differs between the short-distance and long-distance settings.

In the short-distance replacement, ETF defines the midpoint

$u_i$6

and solves

$u_i$7

selecting the root $u_i$8. The turning point is then

$u_i$9

The resulting trajectory is the two-segment path $\mathbf p_i=(x_i,y_i,z_i)$0. The paper states that this path remains strictly inside $\mathbf p_i=(x_i,y_i,z_i)$1, and that the turning point is selected in $\mathbf p_i=(x_i,y_i,z_i)$2 arithmetic.

In the long-distance replacement, ETF first identifies a chain of forwarders connecting $\mathbf p_i=(x_i,y_i,z_i)$3 to $\mathbf p_i=(x_i,y_i,z_i)$4. If $\mathbf p_i=(x_i,y_i,z_i)$5 and $\mathbf p_i=(x_i,y_i,z_i)$6 lie on the same multicast path, that path provides the forwarder chain directly. Otherwise, ETF builds a graph $\mathbf p_i=(x_i,y_i,z_i)$7 whose nodes are the forwarders and whose edges satisfy

$\mathbf p_i=(x_i,y_i,z_i)$8

Each edge is weighted by

$\mathbf p_i=(x_i,y_i,z_i)$9

and Dijkstra’s algorithm is run from $r_i$0 to $r_i$1 to select the minimal-hop chain

$r_i$2

Turning points are then generated recursively. With $r_i$3, ETF defines

$r_i$4

for each consecutive pair of forwarders, and computes

$r_i$5

where $r_i$6 is chosen so that $r_i$7 lies on the boundary of $r_i$8. This ensures that each straight sub-segment lies in exactly one $r_i$9 and meets the next without gap.

The replacement mechanism is therefore midpoint-based and boundary-constrained. The paper explicitly presents it as a fast construction, not as a global optimization over all possible turning-point configurations.

4. Algorithmic organization and complexity

ETF is organized into two phases (Tu, 6 Jul 2025). Phase I performs SLT evaluation and fast checks. Phase II handles interrupted SLTs and constructs the multi-segment replacement trajectory. In pseudocode terms, the algorithm first branches on the overlap condition between $$S_i=\{\mathbf x\in\mathbb R^3:\|\mathbf x-\mathbf p_i\|\le r_i\}.$$0 and $$S_i=\{\mathbf x\in\mathbb R^3:\|\mathbf x-\mathbf p_i\|\le r_i\}.$$1, then either verifies seamless direct travel or invokes a short-distance or long-distance replacement procedure.

The computational complexity reflects this staged design. In the short-distance case, checking the condition of Theorem 1 requires $$S_i=\{\mathbf x\in\mathbb R^3:\|\mathbf x-\mathbf p_i\|\le r_i\}.$$2 time to scan the overlapping neighbors of $$S_i=\{\mathbf x\in\mathbb R^3:\|\mathbf x-\mathbf p_i\|\le r_i\}.$$3 and $$S_i=\{\mathbf x\in\mathbb R^3:\|\mathbf x-\mathbf p_i\|\le r_i\}.$$4. The short-distance replacement requires $$S_i=\{\mathbf x\in\mathbb R^3:\|\mathbf x-\mathbf p_i\|\le r_i\}.$$5 time because it solves a single quadratic equation. In the long-distance case, computing the covering set and executing the progressive chain check each require $$S_i=\{\mathbf x\in\mathbb R^3:\|\mathbf x-\mathbf p_i\|\le r_i\}.$$6, so long-distance checking is $$S_i=\{\mathbf x\in\mathbb R^3:\|\mathbf x-\mathbf p_i\|\le r_i\}.$$7 overall. The long-distance replacement requires Dijkstra’s algorithm on up to $$S_i=\{\mathbf x\in\mathbb R^3:\|\mathbf x-\mathbf p_i\|\le r_i\}.$$8 nodes, giving $$S_i=\{\mathbf x\in\mathbb R^3:\|\mathbf x-\mathbf p_i\|\le r_i\}.$$9 time, or $A\in\mathbb R^3$0 with a Fibonacci heap and $A\in\mathbb R^3$1 edges. The subsequent turning-point construction is $A\in\mathbb R^3$2. The overall worst-case time is therefore

$A\in\mathbb R^3$3

with space complexity

$A\in\mathbb R^3$4

for storing $A\in\mathbb R^3$5 and the adjacency structure.

The information model is deliberately local. ETF requires only piggy-backed location and radius information, and the abstract emphasizes low-complexity computations such as Euclidean distances and a chain of fast checks with controlled traffic overheads. This suggests that resource efficiency in ETF refers not only to path geometry, but also to control-plane frugality.

5. Reported performance in NS2.35

The performance evaluation was conducted in NS2.35 under two main scenarios, with results reported for average multicast delay (AMD), average multicast throughput (AMT), average mobile throughput (AMoT), average additional energy per received bit (AAEB), and additional control overhead (ACO) (Tu, 6 Jul 2025).

Scenario	Configuration	Reported outcomes
Small-group	9 UAVs, 4 receivers, 1 mobile; traffic rates from 512 Kb/s to 2.176 Mb/s	Lowest AMD $A\in\mathbb R^3$6, highest AMT, least AAEB, ACO $A\in\mathbb R^3$7
Large-group	165 UAVs, 3 mobiles; traffic 128 Kb/s to 960 Kb/s	AMT $A\in\mathbb R^3$8 up to 960 Kb/s; ACO constant at $A\in\mathbb R^3$9
Multi-mobility tests	1 to 12 mobiles; random speeds 10–35 m/s	AMoD and AMoT remained essentially constant

In the small-group scenario, the results were reported as the mean of 20 runs. ETF always achieved the lowest AMD and the highest AMT, outperforming LCRT, T-LCRT, EGMP, and ETTA when seamlessly transiting UAV group members. The paper further reports that ETF admitted up to $B\in\mathbb R^3$0 more traffic with acceptable AMD and AMT. In AAEB, ETF consumed the least energy per bit, attributed to maximizing received bits while using near-SLT paths. For ACO in the small group, the reported values were approximately $B\in\mathbb R^3$1 kb/s for EGMP, $B\in\mathbb R^3$2 kb/s for T-LCRT, and $B\in\mathbb R^3$3 kb/s for ETTA and ETF.

In the large-group scenario, ETF maintained high AMT $B\in\mathbb R^3$4 up to 960 Kb/s, whereas ETTA maintained this level only up to 780 Kb/s, described as approximately $B\in\mathbb R^3$5 less input capacity. In tests with increasing mobility multiplicity—from 1 to 12 mobiles at random speeds from 10 to 35 m/s—ETF’s AMoD and AMoT remained essentially constant, while the other methods degraded sharply. For ACO in the large group, the reported values were approximately $B\in\mathbb R^3$6 kb/s for EGMP increasing to $B\in\mathbb R^3$7 kb/s with 6 mobiles and $B\in\mathbb R^3$8 kb/s with 12 mobiles; approximately $B\in\mathbb R^3$9 kb/s for T-LCRT increasing to $B$00 kb/s and $B$01 kb/s; and a constant approximately $B$02 kb/s for ETTA and ETF.

Taken together, these results characterize ETF as a transition mechanism that preserves multicast service quality under both higher offered traffic and increased mobility concurrency, while keeping control overhead low.

6. Assumptions, limitations, and interpretive context

The assumptions underlying ETF are explicit (Tu, 6 Jul 2025). Each UAV’s irregular real-world range is abstracted by a sphere of radius $B$03, the RTR, known via GPS or exchanged piggy-backed. Obstacles are assumed negligible or delegated to a separate avoidance subsystem, because SLTs exclude detours. Coverage is modeled as binary: a point is either inside or outside a sphere.

The limitations are equally explicit. ETF does not optimize globally over all possible turning points; instead, it uses midpoint-based heuristics for fast computation, which may not produce the absolutely shortest path in some geometries. It assumes one UAV moves at a time, so simultaneous group mobility is not treated. It also ignores dynamic channel fading and time-varying $B$04.

These limitations clarify a common potential misconception. ETF should not be interpreted as a globally optimal continuous-space planner over arbitrary 3D environments. Rather, it is a fast seamless-transition procedure constrained by RTR geometry and multicast connectivity. This suggests a deliberate trade-off between exact geometric optimality and low-overhead, tractable execution.

The paper identifies several potential extensions: jointly optimizing turning points, for example via convex optimization; integrating obstacle avoidance so that SLT segments can flex around 3D obstacles; handling simultaneous transitions of multiple UAVs with possible interdependence; extending the coverage model beyond spheres to anisotropic or height-dependent forms; and incorporating channel-diversity or multi-band scheduling to reduce interference during transitions. These directions indicate how ETF could be generalized without altering its core two-phase structure.

7. Position within UAV multicasting methodology

ETF is framed as a seamless-transition algorithm for high-performance multi-hop UAV multicasting, rather than as a general-purpose routing or formation-control framework (Tu, 6 Jul 2025). Its central methodological contribution is the combination of geometric direct-path verification with a fallback construction based on overlapping RTR chains. The first stage exploits the fact that the SLT is the shortest possible path; the second stage seeks a minimum number of seamless straight-line segments when that ideal path is interrupted.

Within this design, seamlessness is the organizing principle. A segment is valid not merely because endpoints are connected in a graph, but because the entire segment remains within a suitable RTR. Likewise, replacement trajectories are defined by joining points selected to control the mobile UAV’s seamless travel distances. This emphasis differentiates ETF from approaches that rely primarily on ad hoc forwarder negotiation or zone-based multicast control.

The performance claims reported in simulation, together with the stated computational bounds and local-information requirements, position ETF as a resource-aware transition mechanism for multicast-rich UAV networks. A plausible implication is that its strongest fit is in settings where the continuity of media delivery during UAV motion is at least as important as the mere existence of a connected route.