Trapezoidal Prism Corridors

• Trapezoidal prism corridors are convex, time-parameterized safe regions in spatio-temporal space used for real-time motion planning in dynamic environments.
• They employ a tilting trapezoidal boundary in the longitudinal-time plane to over-approximate dynamic free-space, enlarging the feasible trajectory volume compared to static cuboidal corridors.
• The method integrates piecewise Bézier curve trajectory optimization under strict safety constraints, achieving improved maneuverability and real-time computational efficiency.

A trapezoidal prism corridor is a convex, time-parameterized region in the spatio-temporal configuration space, designed for real-time, safety-guaranteed motion planning of autonomous vehicles in dynamic environments. Defined in the Frenet-frame coordinates $(S, L, T)$ for longitudinal position, lateral position, and time respectively, the corridor’s cross-section in the $(S, T)$-plane is a trapezoid with constant extent in $L$. It is constructed by over-approximating the free-space carved out by dynamic obstacles into a union of such prisms, within which piecewise Bézier curve trajectory optimization can be performed under strict safety constraints. This construct significantly enlarges the feasible trajectory search space compared to traditional cuboidal corridors and supports more dynamic, non-trivial maneuvers while preserving continuous-time safety guarantees (Deolasee et al., 2022).

1. Formal Structure and Definition

Each trapezoidal prism is defined over a time interval $[T_k, T_{k+1}]$, with bounds for the longitudinal dimension parameterized as line segments whose intercepts and slopes can vary between time endpoints: $$\mathcal{S}_k = \left\{(t, s, l)\;\middle|\; \underline{p}_0^k + h_k\underline{p}_1^k\frac{t-T_k}{h_k} \le s \le \overline{p}_0^k + h_k\overline{p}_1^k\frac{t-T_k}{h_k},\ l_{\mathrm{beg}}\le l\le l_{\mathrm{end}},\ t\in[T_k,T_{k+1}]\right\}$$ Here, $h_k$ is the interval duration, while $\underline{p}_0^k, \overline{p}_0^k$ and $\underline{p}_1^k, \overline{p}_1^k$ are the intercepts and slopes of the lower and upper bounds in $S$. Fixing $l$, the intersection in the $(S, T)$ plane is a 2D trapezoid; stacking these for all $l \in [l_{\mathrm{beg}}, l_{\mathrm{end}}]$ forms the 3D prism. This parameterization allows the corridor to "tilt" in time, thereby tracking moving obstacles or expanding as the available free-space varies.

2. Corridor Construction Around Dynamic Obstacles

Because dynamic obstacles induce a nonconvex and time-varying free-space, the method over-approximates each obstacle as an axis-aligned parallelepiped (with four zero-slope faces). Free-space is sliced along the $L$ axis at each lateral obstacle boundary, resulting in a collection of 3D chunks. Within each chunk, the projection onto the $(S, T)$-plane is processed by a convex corridor generation algorithm: time-sampled lower/upper $S$ limits are grouped into maximal intervals of nearly constant slope, forming the trapezoidal bounds. Each trapezoid inherits a constant $L$ extent and, together, these regions are stacked to build the full sequence of 3D trapezoidal prism safe corridors that cover the reference trajectory.

3. Piecewise Bézier Curve Trajectory Optimization

Within the union of these trapezoidal prisms, planned vehicle trajectories $(s(t), l(t))$ are represented as piecewise nth-order Bézier curves on each interval. Time is reparameterized to $\tau \in [0, 1]$ within each segment, and the curve is expressed as: $$s(t) = \sum_{i=0}^n c_i^{k,s} b_n^i(\tau), \quad l(t) = \sum_{i=0}^n c_i^{k,l} b_n^i(\tau)$$ The global decision variable is the aggregate set of control points for all intervals. The quadratic cost function penalizes deviation from a reference trajectory and its higher derivatives, enforcing both path fidelity and smoothness. Constraints include: boundary conditions (up to second derivative), inter-segment continuity, physical feasibility on velocity/acceleration/jerk (via the hodograph property), and most critically, the inclusion of the Bézier curve (using its convex-hull property and Bernstein-to-monomial basis conversion) entirely within the current trapezoidal prism at each segment.

4. Enlargement of Feasible Solution Space Versus Cuboidal Corridors

In cuboidal corridors (constant boundaries in $S$ over time), the safety constraint demands all Bézier control points in every interval remain between time-invariant lower and upper bounds, thereby drastically restricting the set of valid trajectories, especially in the presence of moving or deforming obstacles. The trapezoidal formulation admits linearly time-varying bounds, so the safe envelope "tilts," always containing the cuboidal corridor and, in practice, often substantially enlarging the admissible volume. Analytic and empirical comparisons show that this extra flexibility enables more aggressive yet safe maneuvers that would otherwise be excluded.

5. Algorithmic Workflow and Computational Performance

The pipeline consists of obstacle over-approximation, lateral slicing, per-slice convexification via time-sample grouping and slope clustering, and prism selection along a warm-start trajectory. These steps are linear in the number of time samples, with quadratic program (QP) formation of cost and constraint matrices scaling as $O((mn)^2)$, where $m$ is the number of time intervals and $n$ the Bézier order. The resultant QP is solved using OSQP with operator splitting, achieving convergence in tens of iterations. The end-to-end process executes within tens of milliseconds on standard laptop hardware, achieving true real-time requirements under typical planning horizons (Deolasee et al., 2022).

6. Empirical Validation and Comparative Results

Simulation in diverse scenarios—including construction zone merging, double lane change, and unprotected left turn—demonstrates superior performance of the trapezoidal approach. For example, in the construction merging scenario over a 7 s horizon, trapezoidal corridors yield lower peak longitudinal acceleration and lateral jerk than cuboidal corridors. Under high initial speeds ($v_s=10.5\, \mathrm{m/s},\ a_s=2\,\mathrm{m/s}^2$), only the trapezoidal corridor planner produces feasible solutions, whereas the cuboidal approach fails unless initial velocity is reduced. In CommonRoad dynamic merging benchmarks, only the trapezoidal-prism planner consistently avoids collision and respects vehicle-dynamic constraints. When the trapezoidal enlargement is unnecessary, both methodologies converge to the same solution, evidencing that the added flexibility never impairs feasibility (Deolasee et al., 2022).

7. Summary and Significance

Trapezoidal prism corridors offer a principled, convex, and computationally tractable method for enlarging the feasible safe region in complex, nonconvex spatio-temporal planning landscapes. By leveraging the geometric tilting of trapezoidal bounds in time and enforcing Bézier-curve trajectory inclusion through convex-hull properties, the method achieves strict continuous-time safety, efficient optimization, and substantial empirical gains in both trajectory quality and planning success rates compared to traditional cuboidal corridor methods (Deolasee et al., 2022).