Heuristic Trajectory Sampling Cluster (HTSC)

• HTSC is a method that clusters whole trajectories using dynamic metrics and fuzzy c-means to identify finite-time coherent sets from sparse data.
• In assistive navigation, HTSC generates a compact cluster of feasible trajectories in the Frenet frame using quintic polynomial interpolation and endpoint smoothing.
• The dual methodology of HTSC acts as a structural prior, reducing redundancy and enabling efficient downstream trajectory optimization in hybrid frameworks.

Searching arXiv for the cited HTSC-related papers to ground the article. Heuristic Trajectory Sampling Cluster (HTSC) appears in the cited literature in two distinct ways. In one usage, it refers generically to clustering trajectories by similarity, and a 2015 cluster-based method operationalizes that principle for extracting finite-time coherent sets from sparse and incomplete trajectory data through spatio-temporal clustering in a space-time feature space. In a later usage, within the Momentum-constrained Hybrid Heuristic Trajectory Optimization Framework (MHHTOF), HTSC denotes the front-end module that generates a compact, feasible, and smooth cluster of candidate trajectories in the Frenet coordinate system for assistive navigation in visually impaired scenarios (Froyland et al., 2015, Zeng et al., 19 Sep 2025, Zeng et al., 16 Apr 2026).

1. Terminological scope and documented usages

In the coherent-set literature, HTSC is described generically: clustering is performed on trajectories according to their whole time-evolution, rather than according to their positions at only the initial or final time. The 2015 method presents itself as a principled, concrete realization of that idea by embedding each trajectory in a product space of all observed states over the time window and then applying fuzzy c-means. In the assistive-navigation literature of 2025–2026, HTSC is defined more specifically as a sampling-based motion primitive generator and as the first-stage candidate-generation component of HTSCMOE within MHHTOF (Froyland et al., 2015, Zeng et al., 19 Sep 2025, Zeng et al., 16 Apr 2026).

Source	Usage of HTSC	Core mechanism
(Froyland et al., 2015)	Generic trajectory-similarity clustering for coherent sets	Dynamic metric in space-time plus fuzzy c-means
(Zeng et al., 19 Sep 2025)	First-stage candidate generator in MHHTOF	Frenet-space generation using third-order interpolation with fifth-order polynomials and MTO constraints
(Zeng et al., 16 Apr 2026)	Front-end of MHHTOF for assistive planning	Polynomial trajectory generation with endpoint smoothing and momentum continuity constraints

This dual usage is not contradictory. Rather, the common element is that trajectories are not treated as isolated points but as structured objects whose temporal evolution is central to clustering or sampling. A plausible implication is that HTSC functions more as a methodological label than as a single fixed algorithm.

2. Generic HTSC as spatio-temporal clustering for finite-time coherent sets

The 2015 formulation targets finite-time coherent sets, defined as regions whose members remain close together over a finite period, using only possibly sparse or incomplete trajectory data. The method treats trajectory clustering as a mathematical optimization problem, requires neither a full dynamical system model nor dense or complete data, and naturally accommodates missing data and short or gappy trajectories (Froyland et al., 2015).

Its central object is a dynamic metric. For a flow map $\Phi(x,t)$ in $\mathbb{R}^d$ and a chosen metric $\rho$, typically Euclidean, the distance between two initial points is

$$\mathbf{D}(x,y) := \int_0^T \rho(\Phi(x,t),\Phi(y,t))^2 \, dt.$$

For discrete trajectory data $\{x_{i,t}\}$ at times $t=0,1,\ldots,T$,

$$\mathbf{D}(x_{i,0},x_{j,0}) = \sum_{t=0}^{T} \rho(x_{i,t},x_{j,t})^2.$$

When $\rho$ is Euclidean, this becomes

$\mathbf{D}(x_{i,0},x_{j,0}) = \|X_i - X_j\|^2,$

with

$$X_i = (x_{i,0},x_{i,1},\ldots,x_{i,T}) \in \mathbb{R}^{d(T+1)}.$$

Each trajectory is therefore a point in space-time. Fuzzy c-means is then applied in this high-dimensional space. With number of clusters $\mathbb{R}^d$0, fuzziness parameter $\mathbb{R}^d$1, memberships $\mathbb{R}^d$2, and centers $\mathbb{R}^d$3, the algorithm iteratively updates centers according to

$\mathbb{R}^d$4

while minimizing

$\mathbb{R}^d$5

This is a soft clustering: each trajectory has a membership value in each cluster, and all points of a trajectory remain assigned to the same cluster at all times.

The method also specifies an explicit missing-data variant. Distances and updates are computed only over observed times; each trajectory is projected into lower-dimensional space where data exists; and center updates at time $\mathbb{R}^d$6 use only data available at $\mathbb{R}^d$7. This permits useful results when there are relatively few trajectories and when there are gaps in observation of the trajectories.

A further diagnostic is the classification entropy

$\mathbb{R}^d$8

which is used to diagnose ambiguous or marginal assignments. High-confidence assignments, such as $\mathbb{R}^d$9, or low entropy, mark coherent regions, while uncertain assignments indicate boundaries or barriers. The resulting workflow consists of gathering sparse or incomplete trajectories, choosing $\rho$0, $\rho$1, and sampling times, constructing the extended vectors $\rho$2, running fuzzy c-means, inspecting memberships and entropy, and interpreting the resulting clusters as approximate coherent sets over $\rho$3.

3. HTSC as Frenet-space candidate generation in assistive navigation

In MHHTOF, HTSC is the foundation of the first stage of the trajectory optimization framework for visually impaired navigation. It is described as a sampling-based motion primitive generator that rapidly produces a cluster of dynamically feasible, smooth, and semantically interpretable candidate trajectories. The cluster is formed in the Frenet frame, with arc-length $\rho$4 along a reference path and lateral offset $\rho$5, which aligns the local planning problem with structured paths such as sidewalks and lanes (Zeng et al., 19 Sep 2025, Zeng et al., 16 Apr 2026).

Each candidate is parameterized by decoupled polynomials,

$\rho$6

These are fifth-order polynomials, or quintic polynomials, and the coefficients are computed from initial and terminal boundary conditions on position, velocity, and acceleration. For the longitudinal component, the coefficients satisfy

$\rho$7

with the remaining coefficients obtained through $\rho$8. Analogous equations are used for the lateral component,

$\rho$9

with $$\mathbf{D}(x,y) := \int_0^T \rho(\Phi(x,t),\Phi(y,t))^2 \, dt.$$0.

Sampling is performed over a grid of feasible terminal states, including different $$\mathbf{D}(x,y) := \int_0^T \rho(\Phi(x,t),\Phi(y,t))^2 \, dt.$$1, $$\mathbf{D}(x,y) := \int_0^T \rho(\Phi(x,t),\Phi(y,t))^2 \, dt.$$2, and possibly duration $$\mathbf{D}(x,y) := \int_0^T \rho(\Phi(x,t),\Phi(y,t))^2 \, dt.$$3. For each sampled terminal candidate state, the method defines initial and terminal position, velocity, and acceleration, solves for polynomial coefficients, applies temporal endpoint smoothing, filters candidates through sparsity conditions, and transforms the surviving Frenet trajectories to Cartesian trajectories for evaluation. The output is a cluster

$$\mathbf{D}(x,y) := \int_0^T \rho(\Phi(x,t),\Phi(y,t))^2 \, dt.$$4

each member being locally feasible in the Frenet frame.

The 2025 description states that HTSC is generated in the Frenet coordinate system using third-order interpolation with fifth-order polynomials and MTO constraints to ensure smoothness and feasibility. The 2026 description emphasizes polynomial trajectory generation with endpoint smoothing and momentum continuity constraints. Taken together, these formulations describe a candidate-generation stage that is simultaneously parametric, constraint-aware, and explicitly oriented toward comfort and safety.

4. Endpoint smoothing, sparsity control, and momentum-constrained optimization

HTSC in the assistive-navigation setting is not limited to polynomial fitting. It also enforces temporal continuity and regulates the density of candidate trajectories. Third-order smoothing is used to enhance temporal continuity and avoid abrupt transitions at segment boundaries. For each dimension $$\mathbf{D}(x,y) := \int_0^T \rho(\Phi(x,t),\Phi(y,t))^2 \, dt.$$5, the interpolated state, velocity, and acceleration are represented as

$$\mathbf{D}(x,y) := \int_0^T \rho(\Phi(x,t),\Phi(y,t))^2 \, dt.$$6

Given endpoint states for position, velocity, and acceleration, the coefficients $$\mathbf{D}(x,y) := \int_0^T \rho(\Phi(x,t),\Phi(y,t))^2 \, dt.$$7 are solved so as to guarantee smooth connections between consecutive segments; this is described as endpoint-aware third-order state interpolation, and the paper explicitly associates it with $$\mathbf{D}(x,y) := \int_0^T \rho(\Phi(x,t),\Phi(y,t))^2 \, dt.$$8 continuity (Zeng et al., 19 Sep 2025).

Cluster sparsity is enforced through a perception-weighted spacing constraint. In the 2026 description, the condition is written as $$\mathbf{D}(x,y) := \int_0^T \rho(\Phi(x,t),\Phi(y,t))^2 \, dt.$$9, where the vector includes differences between a candidate and the cluster center in $\{x_{i,t}\}$0, $\{x_{i,t}\}$1, $\{x_{i,t}\}$2, and $\{x_{i,t}\}$3 at the terminal time, together with the spacing term $\{x_{i,t}\}$4. Here $\{x_{i,t}\}$5 is the nominal minimum terminal spacing, $\{x_{i,t}\}$6 is a perception-weighting factor, and $\{x_{i,t}\}$7 is the full Frenet state vector (Zeng et al., 16 Apr 2026).

The same literature couples HTSC to Momentum-Constrained Trajectory Optimization (MTO). The state is

$\{x_{i,t}\}$8

and the dynamics are given in state-space form by a block-diagonal triple-integrator model,

$\{x_{i,t}\}$9

The trajectory cost is summarized in two related ways. At the HTSC stage, the 2026 account gives

$t=0,1,\ldots,T$0

where $t=0,1,\ldots,T$1 penalizes curvature irregularity, lateral deviation, and boundary proximity, while $t=0,1,\ldots,T$2 penalizes abruptness in longitudinal motion, including velocity and acceleration jerk. In the broader MTO formulation, the 2025 paper writes the long-run objective as

$t=0,1,\ldots,T$3

and specifies a Lagrangian

$t=0,1,\ldots,T$4

expanded to include kinetic energy, guidance and crowd forces, a smoothness term weighted by $t=0,1,\ldots,T$5, and a perception-uncertainty term weighted by $t=0,1,\ldots,T$6. The optimization seeks paths satisfying the Euler-Lagrange condition

$t=0,1,\ldots,T$7

This suggests that, in MHHTOF, HTSC is already a constrained, cost-shaped generator rather than a purely geometric sampler.

5. Position within MHHTOF and reported empirical behavior

Within MHHTOF, HTSC serves as the initial candidate generator. It produces candidate trajectories that are then evaluated, pruned, and refined by subsequent modules. The 2025 paper describes a two-stage hybrid planning system in which HTSC generates feasible and smooth trajectories in the Frenet frame, after which a residual-enhanced actor-critic network with LSTM-based temporal feature modeling adaptively refines trajectory selection in the Cartesian coordinate system. A dual-stage cost modeling mechanism (DCMM) with weight transfer aligns semantic priorities across stages (Zeng et al., 19 Sep 2025).

The 2026 description makes the modular structure even more explicit. HTSC is integrated in the HTSCMOE module as the first stage where candidate trajectories are produced and checked for preliminary feasibility and smoothness before further optimization. These candidates are then further refined by MTO, which applies optimal control and energy-based modeling to suppress undesirable motion artifacts and account for dynamic agent-environment interactions. Final evaluation is then performed via DCMM and DRL adaptation over the candidate cluster using semantic and cost-based multi-objective criteria in the Cartesian frame (Zeng et al., 16 Apr 2026).

The reported experimental outcomes pertain to the overall framework rather than to HTSC in isolation, but they are repeatedly linked to the structure and quality of the candidate cluster. The 2025 abstract states that the proposed LSTM-ResB-PPO attains stable policy performance in approximately half the training iterations required by the PPO baseline, while reducing average cost and cost variance by 30.3% and 53.3%, and lowering ego and obstacle risks by over 77% (Zeng et al., 19 Sep 2025). The 2026 abstract similarly reports convergence in nearly half the iterations of baselines, lower and more stable costs, and stable velocity and acceleration curves with reduced risk in complex dynamic scenarios (Zeng et al., 16 Apr 2026).

At the HTSC level itself, the 2026 description attributes several properties to the cluster: all candidates are guaranteed by construction to be dynamically feasible and comfortable; perception-adaptive sampling and terminal sparsity constraints reduce redundancy; and sampling in the Frenet frame helps respect lane geometry, boundaries, and obstacle proximity. A plausible implication is that HTSC functions as a structural prior over the search space, with downstream learning operating over already filtered motion primitives rather than over an unconstrained action space.

6. Relation to adjacent trajectory-clustering and trajectory-sampling methods

The 2015 coherent-set method situates its generic HTSC interpretation against operator-based approaches. Compared to transfer-operator and Perron-Frobenius methods, it is described as much more data-driven and robust to data limitations, at some cost in interpretive nuance, including the absence of a direct connection to eigenfunctions. It also emphasizes that clustering outcomes are frame-independent or objective under orthogonal, time-dependent coordinate changes, that the method works in any dimension, and that custom metrics can be used, including metrics for spherical data (Froyland et al., 2015).

A different neighboring line of work is void-and-cluster sampling for time-dependent trajectories. That method is a data reduction technique rather than a coherent-set extractor or a motion-primitive generator. It selects a representative subset of trajectories so that samples are well distributed, achieve the blue noise property, can adapt to arbitrary probability densities, and implicitly define an ordering for progressive loading and level-of-detail. For time-dependent trajectories, it continues existing trajectories when possible, starts new trajectories at the largest voids, and can prematurely stop some trajectories to reduce clustering or unevenness (Rapp et al., 2019).

This contrast helps delimit what HTSC is and is not. In the coherent-set usage, HTSC centers on dynamic similarity across whole trajectories and fuzzy partitioning of space-time embeddings. In the assistive-navigation usage, HTSC centers on generating a cluster of candidate trajectories through quintic parameterization, endpoint smoothing, sparsity control, and MTO-related feasibility constraints. By contrast, void-and-cluster sampling addresses representative subset selection with blue noise and error-guided sampling. This suggests that the shared vocabulary of “trajectory,” “sampling,” and “cluster” masks materially different objectives: identification of coherent sets, generation of feasible motion candidates, and reduction of large trajectory datasets.