Structure-Preserved Trajectory Expansion

• The paper introduces a polar-coordinate expansion process that preserves kinematic structure by applying coordinated radial scalings and angular offsets to trajectory anchors.
• It improves candidate quality by generating a structured, dense local candidate set for the second-stage refinement in a coarse-to-fine hierarchical diffusion planning pipeline.
• By maintaining smooth and realistic trajectories, the method effectively overcomes the limitations of naive Gaussian perturbations in autonomous driving.

Searching arXiv for the cited paper and closely related work on structure-preserved trajectory expansion in autonomous driving. Structure-Preserved Trajectory Expansion is a trajectory-generation mechanism within HAD, an end-to-end planning framework for autonomous driving that combines a Hierarchical Diffusion Policy with Metric-Decoupled Policy Optimization. Its function is to transform a small number of high-level trajectory anchors into a richer local candidate set for a second-stage planner while preserving kinematic structure, thereby avoiding the oscillations, zigzags, and other unrealistic motions associated with naive Gaussian perturbations. Within HAD, it is the intermediate operation between coarse intention selection and fine local denoising, and it is presented as the mechanism that makes second-stage search local, dense, and structured (Yao et al., 4 Apr 2026).

1. Placement within hierarchical diffusion planning

In HAD, Structure-Preserved Trajectory Expansion occupies the middle stage of a three-part planning pipeline. The first stage, termed Driving Intention Establishment, denoises a sparse set of $M_1$ anchor trajectories and selects the top-$K$ most plausible anchors according to a classification score. The second stage expands each selected anchor into a local candidate grid using polar-coordinate transformations. The third stage, Local Trajectory Refinement, applies a second diffusion decoder for fine-grained denoising and refinement.

This arrangement is explicitly coarse-to-fine. The first decoder identifies plausible global intentions; the expansion stage densifies each intention into a structured neighborhood; the second decoder searches that neighborhood for locally optimal motions. The expansion step is therefore not an output layer but a bridge between global intention and local maneuver refinement. The paper characterizes it as a means of making the second-stage search local, dense, and structured, rather than forcing direct selection from the entire candidate space (Yao et al., 4 Apr 2026).

A plausible implication is that the method addresses a search-allocation problem: global intention uncertainty is handled sparsely, while local maneuver variability is handled densely.

2. Motivation: preserving kinematic structure under local exploration

The method was introduced in response to a specific limitation of diffusion-based trajectory generation in driving. The paper argues that Gaussian noise perturbations around anchors are a poor fit for driving trajectories because pointwise perturbation can destroy kinematic structure, producing oscillatory or zigzag trajectories. These low-quality candidates complicate denoising and inject noise into optimization.

The stated motivation is twofold. First, the expanded candidates should preserve trajectory structure so that candidate motions remain physically plausible. Second, local exploration around a selected intention anchor should be improved so that the refinement stage searches a meaningful neighborhood rather than a noisy one. The paper contrasts this objective directly with naive Gaussian perturbation, which may create zigzag or oscillatory trajectories and degrade both candidate quality and downstream learning.

The key conceptual distinction is between uncoordinated local perturbation and coordinated global deformation. In the paper’s framing, Gaussian perturbation adds local, uncoordinated point noise and therefore breaks smoothness and kinematics. By contrast, the proposed expansion applies global geometric transforms consistently across all timesteps, retaining continuity and realism.

3. Polar-coordinate expansion procedure

The expansion algorithm begins from an input trajectory

$\tau = \{(x_t, y_t)\}_{t=1}^{T}.$

Each point is converted from Cartesian to polar coordinates: $$\rho_t = \sqrt{x_t^2 + y_t^2}, \qquad \theta_t = \operatorname{atan2}(y_t, x_t).$$

The method then uses two coefficient sets: radial scaling coefficients $\{\lambda_u\}$ and angular offset coefficients $\{\delta_v\}$. For each pair $(u,v)$, the trajectory is transformed according to

$$\rho_t^{(u,v)} = \lambda_u \cdot \rho_t, \qquad \theta_t^{(u,v)} = \theta_t + \delta_v.$$

The transformed points are then mapped back to Cartesian coordinates: $$x_t^{(u,v)} = \rho_t^{(u,v)} \cos\theta_t^{(u,v)}, \qquad y_t^{(u,v)} = \rho_t^{(u,v)} \sin\theta_t^{(u,v)}.$$ This yields the expanded trajectory

$$\tau^{(u,v)} = \{(x_t^{(u,v)}, y_t^{(u,v)})\}_{t=1}^{T}.$$

For one anchor, the output is a set of $K$0 expanded trajectories. When the top-$K$1 anchors are expanded, the second-stage candidate set size becomes

$K$2

The appendix presents this as a stepwise algorithm: input a trajectory $K$3, an expansion number $K$4, and the coefficient sets $K$5, $K$6; convert each $K$7 to $K$8; for every radial coefficient and angular coefficient, scale $K$9, shift $\tau = \{(x_t, y_t)\}_{t=1}^{T}.$0, and convert back to Cartesian coordinates; then collect all $\tau = \{(x_t, y_t)\}_{t=1}^{T}.$1 into the expanded set $\tau = \{(x_t, y_t)\}_{t=1}^{T}.$2. The paper describes this as a coarse-to-fine, geometry-aware augmentation (Yao et al., 4 Apr 2026).

4. Structural preservation and downstream use

The preservation claim follows directly from the transformation class. A small radial scaling changes the path’s reach or extent, and a small angular offset rotates it slightly. Because these modifications are applied consistently across all timesteps, the overall shape is preserved much better than under independent per-point noise. This is the sense in which the expansion is “structure-preserved”: the perturbation is coordinated at the trajectory level rather than applied as unstructured point noise.

After expansion, the candidate trajectories are encoded and passed to the Local Trajectory Refinement Decoder: $\tau = \{(x_t, y_t)\}_{t=1}^{T}.$3

$\tau = \{(x_t, y_t)\}_{t=1}^{T}.$4

The expanded trajectories are therefore structured proposals for the second diffusion stage. The refinement stage predicts refined trajectories and multiple metric scores, including distance to the expert trajectory, safety-related metrics, and RL-related metric-decoupled weights. Final aggregation is performed by a softmax over learned scores: $\tau = \{(x_t, y_t)\}_{t=1}^{T}.$5

This formulation makes the role of expansion explicit: it improves the quality of the candidate pool over which the final trajectory is selected. The method is thus not merely a preprocessing augmentation; it is an internal proposal-generation mechanism coupled to second-stage denoising and scoring.

5. Interaction with Metric-Decoupled Policy Optimization

Structure-Preserved Trajectory Expansion is also integral to the RL component of HAD. In Metric-Decoupled Policy Optimization, each expanded local set corresponds to a group of trajectories over which the model computes metric-specific selection probabilities and normalized rewards: $\tau = \{(x_t, y_t)\}_{t=1}^{T}.$6

$\tau = \{(x_t, y_t)\}_{t=1}^{T}.$7

$\tau = \{(x_t, y_t)\}_{t=1}^{T}.$8

In this formulation, the expanded local neighborhood supplies the trajectory groups $\tau = \{(x_t, y_t)\}_{t=1}^{T}.$9 needed for metric-decoupled reward computation. The paper’s interpretation is that, without expansion, RL would operate on a poorer or less diverse set of local motions. The method therefore has a dual role: it supports second-stage diffusion refinement and provides the structured local candidate set required by MDPO.

A plausible implication is that the expansion stage functions as the geometric substrate on which metric-decoupled optimization becomes meaningful, since the normalized rewards are computed within a neighborhood centered on each selected intention anchor.

6. Comparisons, ablations, and terminological context

The paper compares three candidate-generation strategies on NAVSIM v2:

Strategy	Description	EPDMS
Random Noise	Direct Gaussian perturbation of trajectories	84.9
XY Expand	Structured expansion in Cartesian space	85.9
Polar Expand	Proposed structure-preserved expansion	88.6

The comparison is accompanied by a qualitative diagnosis of each method. Random Noise damages kinematic structure and performs worst. XY Expand keeps structure better, but exploration is spatially unbalanced; the paper notes that if one coordinate is small, coverage along that axis becomes limited. Polar Expand provides more uniform local coverage while keeping trajectories realistic (Yao et al., 4 Apr 2026).

Additional ablations emphasize that the method is coupled to hierarchical narrowing. When the first stage selects $$\rho_t = \sqrt{x_t^2 + y_t^2}, \qquad \theta_t = \operatorname{atan2}(y_t, x_t).$$0, performance is 88.1 EPDMS; with $$\rho_t = \sqrt{x_t^2 + y_t^2}, \qquad \theta_t = \operatorname{atan2}(y_t, x_t).$$1, it is 88.5–88.6 EPDMS; with $$\rho_t = \sqrt{x_t^2 + y_t^2}, \qquad \theta_t = \operatorname{atan2}(y_t, x_t).$$2, 86.4 EPDMS; and with $$\rho_t = \sqrt{x_t^2 + y_t^2}, \qquad \theta_t = \operatorname{atan2}(y_t, x_t).$$3, 79.8 EPDMS. This supports the claim that expanding a small set of plausible anchors is preferable to expanding a large unstructured candidate space. Increasing local inference density from $$\rho_t = \sqrt{x_t^2 + y_t^2}, \qquad \theta_t = \operatorname{atan2}(y_t, x_t).$$4 to $$\rho_t = \sqrt{x_t^2 + y_t^2}, \qquad \theta_t = \operatorname{atan2}(y_t, x_t).$$5 improves performance slightly to 88.6 EPDMS, suggesting that denser local search can help, though the paper attributes the main gain to the polar expansion design itself.

A common misunderstanding would be to equate “structure-preserved” here with the broader structure-preserving literature in Hamiltonian simulation. In accelerator physics, for example, structure-preserving methods concern exact symplecticity of transfer maps, expressed by the condition

$$\rho_t = \sqrt{x_t^2 + y_t^2}, \qquad \theta_t = \operatorname{atan2}(y_t, x_t).$$6

and realized through high-order approximate one-turn maps, Lie factorization, Poincaré generating functions, and Cremona maps (Abell et al., 2022). That usage concerns preservation of phase-space geometry under long-term Hamiltonian evolution, not local candidate densification for driving trajectories. A plausible implication is that the two usages share a methodological principle—preserve the relevant structure during approximation—but the preserved structures differ: kinematic path realism in HAD versus symplectic phase-space invariants in accelerator tracking.