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J. Eur. Opt. Society-Rapid Publ. 22, 49( 2026)
interpretable swarm configurations by directly manipulating the 6-DOF state variables s i.
3.2 Trajectory generation and automatic annotation
Building upon the 6-DOF state representation, we generate image sequences by interpolating UAV poses along parameterized trajectories, and derive ground-truth annotations automatically via geometric projection. This subsection describes the trajectory interpolation scheme, the swarm configuration space, the camera model, and the annotation procedure in turn.
For each UAVi, wedefine initial and terminal states as:
i ¼ p ðÞ 0 i; r ð0Þ i; s ðTÞ i ¼ p ðT i s ðÞ 0
Þ r ðTÞ i
; ð3Þ
Fig. 1. Overview of the synthetic UAV-swarm dataset generation pipeline.
this frame. For UAV i 2 { 1,..., N }, the state is parameterized as: titles i ¼ ðp i; r i Þ 2 R 3 SOð3Þ;
where the position vector p i =( x i, y i, z i) specifies the 3D location within the observation volume W R 3, and the orientation r i =(/ i, h i, w i) denotes the Euler angles( roll, pitch, yaw) representing the UAV attitude with respect to the world frame.
The six degrees of freedom can be naturally decomposed into translational and rotational components. The components( x i, y i, z i) correspond to translations of UAV i along the three orthogonal axes of the world frame, as illustrated in Figure 2. The angles(/ i, h i, w i) represent rotations around these axes, describing roll, pitch and yaw motions of the UAV body frame, as depicted in Figure 3. Together, these quantities provide a complete and minimal description of the pose of each UAV in space.
For convenience, the full state of a UAV swarm with N agents is written as
� S ¼ fs 1; s 2;...; s N g 2 R 3 N
SOð3Þ
: ð2Þ
ð1Þ
This representation explicitly separates the per-UAV state into independent pose variables, enabling finegrained control over both individual trajectories and collective formation structures. In contrast to simplified models that only consider 2D positions or planar motion, the 6-DOF formulation allows us to synthesize complex aerial maneuvers, including out-of-plane rotations, coordinated banking motions, and depth-varying formations, which are essential for faithfully emulating realistic UAV swarm behaviors in three-dimensional space.
Moreover, the use of Euler-angle parameterization(/ i, h i, w i) iswellalignedwiththeflight dynamics conventions adopted in most simulation engines and autopilot systems. This facilitates the integration of physically plausible motion primitives and makes it straightforward to map between high-level formation commands and low-level pose specifications in our data generation pipeline. As a result, we can systematically sample diverse yet physically where the superscripts denote the time indices at the beginning and end of the sequence. Given a total of T interpolation steps, the intermediate state at step t 2 { 0, 1,..., T } is computed as follows. The position component is linearly interpolated in Euclidean space:
p ðÞ t i ¼ p ðÞ 0 i þ t T p i; p i ¼ p ðTÞ i � p ðÞ 0 i: ð4Þ
For the orientation component, we employ spherical linear interpolation( SLERP) on unit quaternions to ensure smooth and consistent rotational motion. Let q ðÞ 0 i; q ðTÞ i 2 S 3 denote the unit quaternion representations of the initial and terminal orientations, respectively. The interpolated orientation at step t is given by
q ðÞ t i ¼ SLERP q ð0Þ i; q ðTÞ i; a t
¼ sin ðð
1 � a tÞXÞ q ðÞ 0 i þ sin ð a tXÞ sin X sin X q ðTÞ i ð5Þ where a t = t / T is
the interpolation parameter and X ¼ arccos q ð0Þ i q ðTÞ i is the angular distance between the two orientations. The quaternion
result is then converted back to Euler angles r it ðÞ
¼ / ðÞ t i h ðÞ t i w ðÞ t i for compatibility with the 6-DOF state representation.
This formulation ensures temporal coherence across consecutive frames, mathematically consistent rotational interpolation on SO( 3), and full reproducibility given identical boundary conditions.
For a swarm of NUAVs, the collective configuration at step t is defined as n o
S ðÞ t ¼ s ðÞ t
1; s ðÞ t 2;...; s ðÞ t: ð6Þ
The swarm density is controlled by adjusting the observation volume W and the number of instances N. LetV( W) denote the volume of the observation region; the average spatial density is given by
q ¼
N N
VðWÞ: ð7Þ
By varying q, we systematically generate scenes ranging from sparse formations to dense clusters. The apparent