Trajectory-Expansion Methodology

Updated 4 November 2025

Trajectory-expansion methodology is a systematic approach to generate richer system trajectories from sparse, noisy, or constrained data using principled mathematical techniques.
It leverages tools like block Hankel matrices and persistence of excitation to reconstruct input-output dynamics in both linear and nonlinear systems, enhancing simulation and control tasks.
The methodology underpins applications in urban mobility, robust optimization, and quantum dynamics by filling observational gaps and improving data-driven analysis and planning.

A trajectory-expansion methodology generically refers to frameworks, algorithms, or mathematical principles that construct richer sets of trajectories from limited initial data—whether this data is sparse, noisy, or constrained. In contemporary research, trajectory expansion fills observational gaps, enables data-driven analysis and control, augments simulation and optimization tasks, and facilitates the synthesis of trajectories for complex dynamical systems, often under practical constraints such as optimality, safety, or robustness.

1. Foundational Concepts in Trajectory Expansion

Trajectory expansion builds upon the observation that system behavior can often be captured or reconstructed from limited data by employing principled mathematical procedures. In discrete-time LTI systems, the space of all input-output trajectories is spanned by time-shifts of a single measured trajectory, provided the input is persistently exciting (Berberich et al., 2019). By constructing block Hankel matrices from measured data, one may generate any other valid trajectory via linear combinations, given sufficiently rich excitation.

A more general paradigm is seen in the expansion of OD (origin-destination) data in mobility studies, where sparse records of agent activities (assign, pickup, delivery) are algorithmically expanded into continuous, high-resolution mobility traces via map-based path simulation (Zhang et al., 15 Jul 2025). Trajectory expansion is also critical for data augmentation in sequence modeling (Yoon et al., 9 Jun 2025), trajectory synthesis for optimal estimation (Wilson et al., 2017), safe expansion in unknown environments (Zinage et al., 2020), and robust trajectory planning under uncertainty (Takubo et al., 2022).

2. Data-Driven Trajectory Expansion and Spanning Principles

Central to the data-driven expansion methodology is the spanning principle: a single characterized trajectory—with sufficient excitation—contains all information needed to reconstruct any other trajectory of the system (within observability and controllability constraints). This is formalized via:

The trajectory-spanning theorem: For an LTI system, the Hankel matrices $H_L(u)$ and $H_L(y)$ formed from a measured trajectory can be used to reconstruct any other length- $L$ trajectory by solving

$\begin{bmatrix} H_L(u) \ H_L(y) \end{bmatrix} \alpha = \begin{bmatrix} \bar{u} \ \bar{y} \end{bmatrix}$

for $\alpha$ (Berberich et al., 2019).

Persistence of excitation, defined via

$\text{rank}(H_L(u)) = mL$

is necessary for the Hankel matrix to be a basis.

Nonlinear system extensions (Hammerstein, Wiener, Koopman) utilize basis expansions or kernel methods, lifting trajectories to higher-order feature spaces where similar spanning and expansion applies.

3. Trajectory Expansion from Sparse or Abstract Data

In urban mobility applications, trajectory expansion addresses the scarcity or privacy limitations of high-fidelity GPS data by reconstructing fine-grained trajectories from sparse records. The methodology, as implemented in city-scale delivery rider datasets (Zhang et al., 15 Jul 2025), comprises:

Extraction and sequencing of multi-order “waves” from event records;
Construction of consecutive OD pairs;
Path simulation for each OD pair via a routing API in the relevant transportation mode (e.g., cycling);
Synthesis of continuous trajectories—combining polylines, interpolated timestamps;
Validation against ground-truth metrics, achieving Pearson $r=0.92$ (distance) and $r=0.79$ (duration).

This algorithmic expansion is generalizable to other domains with sparse location/action data and supports studies of urban logistics, traffic flow, and gig economy mobility.

4. Expansion in Simulation, Optimization, and Augmentation Frameworks

Trajectory expansion methodologies are integral to simulation-based optimization (e.g., aerospace trajectory planning (Ruff et al., 2023)), robust evolutionary design (Takubo et al., 2022), and dataset augmentation for sequence modeling (Yoon et al., 9 Jun 2025).

Surrogate neural networks facilitate efficient simulation-based optimization by constructing differentiable models from limited simulation data and guiding input searches to optimize high-cost or nonlinear trajectories, reducing required simulation queries by an order of magnitude (Ruff et al., 2023).
Robust trajectory optimization under uncertainty leverages polynomial chaos expansion (PCE), reformulating stochastic problems so that the system’s response over an ensemble of trajectories (generated by quadrature in uncertain parameter spaces) expands the deterministic solution set, allowing moment-based constraint handling in multi-objective evolutionary algorithms (Takubo et al., 2022).
In data augmentation (ATRADA (Yoon et al., 9 Jun 2025)), transformation of raw trajectories into context vectors via Transformer encoding, dimension reduction (PCA), and latent-space modeling (GMM) enables the expansion of realistic synthetic trajectories from sparse or imbalanced datasets, improving classifier and prediction performance in air traffic applications.

5. Trajectory Expansion in Planning and Control under Safety or Geometric Constraints

In robotic motion planning and control, expansion methodologies are designed to systematically construct collision-free or curvature-bounded trajectories from feasible primitives:

Online generalized shape expansion (3D-OGSE (Zinage et al., 2020)) iteratively creates union/difference regions (spherical sectors) from local sensor data, sampling collision-free paths and optimizing snap-minimized, high-smoothness trajectories that are guaranteed to lie within expanded safe regions. Replanning is triggered as the agent encounters unknown areas, enabling probabilistic completeness and real-time operation.
Trajectory planning in complex nonconvex environments may exploit parameterizations via Cartesian products of balls, projecting the solution into time-varying safe ellipsoidal corridors and solving block-separable convex trust region subproblems for efficient expansion of feasible, smooth trajectories. This decouples problem size from geometric complexity, supporting fast, scalable planning (Jaitly et al., 24 Sep 2025).

Dubins vehicle trajectory expansion strategies systematically enumerate all feasible path types (eight CCC classes), using geometric constructions to concatenate arcs of varying radii and minimize curvature discontinuities, ensuring maximal coverage of the reachable length set between oriented point pairs (Rao et al., 5 Oct 2024).

6. Expansion for Optimal Estimation, Identification, and Quantum Dynamics

Trajectory expansion also plays a pivotal role in optimal experiment design for parameter estimation and in quantum dynamics simulations:

Continuous-time projection-based trajectory synthesis (for Fisher information maximization) seeks trajectories that optimize the minimum eigenvalue of the Fisher information matrix by local descent in the trajectory manifold, substantially reducing estimation errors (Wilson et al., 2017). The expansion occurs in the trajectory function space while adhering to system dynamics and sensitivity constraints.
In strong-field ionization physics (eTraj.jl (Zhu et al., 4 Nov 2024)), expansion methodology separates initial-condition generation (using advanced quantum tunneling models: SFA-SPA, SFA-SPANE, ADK, WFAT) from trajectory evolution (CTMC, QTMC, SCTS), synthesizing high-resolution momentum distributions by propagating large ensembles of electron trajectories with or without quantum phase accumulation.

7. Implications, Limitations, and Applications

Trajectory-expansion methodologies, as advanced in these works, offer scalability, robustness to sparsity, and generalizability to nonlinearities, constraints, and uncertainty. They enable:

Large-scale, privacy-preserving mobility analysis without GPS;
Data-driven control, simulation, and learning without explicit models;
Accelerated, uncertainty-aware optimization in engineering systems;
Augmentation for robust learning tasks in aviation, logistics, and autonomous systems;
Synthesis of experimental and theoretical datasets for quantum and classical dynamics.

Limitations center on assumptions of excitation richness, process noise, and the local versus global optimality accessible with specific expansion algorithms. The general principle, reconstructing richer trajectory sets from sparse or structured primitives via theoretically principled expansion, remains foundational in contemporary research across computational science, engineering, and systems theory.