Gauss-Legendre Rectangle for Collision Risk

• GLR is a two-stage integration method that combines Gauss-Legendre cubature over the ego vehicle’s rectangular footprint with a non-homogeneous Poisson process for temporal aggregation.
• It approximates instantaneous collision probabilities using a 5-point spatial evaluation that preserves vehicle geometry and accounts for trajectory uncertainty.
• The method achieves real-time performance (~1 ms per scenario) with significant accuracy improvements over Monte Carlo and other baseline approaches.

Searching arXiv for the specified GLR paper and closely related Legendre/Gauss–Legendre work to ground the article in current literature. The Gauss–Legendre Rectangle (GLR) is a two-stage numerical integration scheme for continuous-time probabilistic collision risk estimation in motion planning, introduced for autonomous racing overtakes in which safety margins are minimal. In the formulation of "Probabilistic Collision Risk Estimation through Gauss-Legendre Cubature and Non-Homogeneous Poisson Processes," GLR combines Gauss–Legendre cubature over the ego vehicle’s rectangular footprint with a non-homogeneous Poisson process over time, yielding collision probabilities that preserve rectangular vehicle geometry, account for trajectory uncertainty, and remain fast enough for approximately 1000 Hz operation (Weiss et al., 24 Jul 2025).

1. Problem setting and defining quantity

GLR addresses probabilistic collision risk estimation over a continuous time horizon for autonomous racing maneuvers, especially overtakes. The ego vehicle follows a deterministic planned trajectory

$$\mathcal{T}_{ego}:[0,T_F]\to\mathbb{R}^2,\qquad \mathcal{T}_{ego}(t)=\mathbf{x}_{ego}(t),$$

while the opponent is represented by an uncertain future trajectory distribution $p(\mathcal{T})$. For each time $t$, this induces a time-indexed position distribution

$$p_t(\mathbf{x}) = p\bigl(\mathbf{x} \mid p(\mathcal{T}), t\bigr)$$

with mean $\boldsymbol{\mu}_t$ and covariance $\boldsymbol{\Sigma}_t$.

Both vehicles are modeled by rectangular footprints aligned with their headings, with width $W_{car}$ and length $L_{car}$. A collision occurs if at any $t\in[0,T_F]$ the ego rectangle $R_{ego}(t)$ intersects the target rectangle $p(\mathcal{T})$0. If $p(\mathcal{T})$1 denotes the subset of trajectories in $p(\mathcal{T})$2 that collide with $p(\mathcal{T})$3, then the desired quantity is

$p(\mathcal{T})$4

Direct integration over trajectory space is described as intractable. GLR therefore reformulates the task as a two-stage integration problem. First, it computes an instantaneous collision probability $p(\mathcal{T})$5 at each time by Gauss–Legendre cubature over the ego rectangle. Second, it interprets collision events over time through a non-homogeneous Poisson process (NHPP), with hazard $p(\mathcal{T})$6 derived from $p(\mathcal{T})$7, and integrates $p(\mathcal{T})$8 over $p(\mathcal{T})$9 by Gauss–Legendre quadrature. This construction preserves continuous time and rectangular geometry rather than replacing vehicles by simplified circular surrogates.

2. Spatial stage: Gauss–Legendre cubature over the ego rectangle

At fixed time $t$0, the instantaneous collision probability is

$t$1

Exact evaluation for two uncertain rectangles is treated as difficult, so GLR introduces a 5-point approximation of the target rectangle: four corners and one centroid. For each $t$2, a density $t$3 is assigned to the location of that point. All five densities share the covariance $t$4, their means are rigid offsets from $t$5 aligned with target heading, and the centroid density is exactly the original target distribution,

$t$6

Let $t$7 denote the event that point $t$8 lies in $t$9. GLR approximates no-collision at time $$p_t(\mathbf{x}) = p\bigl(\mathbf{x} \mid p(\mathcal{T}), t\bigr)$$0 by assuming independence among these five point events: $$p_t(\mathbf{x}) = p\bigl(\mathbf{x} \mid p(\mathcal{T}), t\bigr)$$1 and therefore

$$p_t(\mathbf{x}) = p\bigl(\mathbf{x} \mid p(\mathcal{T}), t\bigr)$$2

The integral

$$p_t(\mathbf{x}) = p\bigl(\mathbf{x} \mid p(\mathcal{T}), t\bigr)$$3

is the core spatial object. The term “Rectangle” in GLR refers to the integration region $$p_t(\mathbf{x}) = p\bigl(\mathbf{x} \mid p(\mathcal{T}), t\bigr)$$4, which is a vehicle-aligned rectangle. The term “Gauss–Legendre” refers to a tensor-product Gauss–Legendre quadrature rule used as a two-dimensional cubature. For a rectangle

$$p_t(\mathbf{x}) = p\bigl(\mathbf{x} \mid p(\mathcal{T}), t\bigr)$$5

and function $$p_t(\mathbf{x}) = p\bigl(\mathbf{x} \mid p(\mathcal{T}), t\bigr)$$6,

$$p_t(\mathbf{x}) = p\bigl(\mathbf{x} \mid p(\mathcal{T}), t\bigr)$$7

where $$p_t(\mathbf{x}) = p\bigl(\mathbf{x} \mid p(\mathcal{T}), t\bigr)$$8 are the one-dimensional Gauss–Legendre nodes and weights on $$p_t(\mathbf{x}) = p\bigl(\mathbf{x} \mid p(\mathcal{T}), t\bigr)$$9, and $\boldsymbol{\mu}_t$0 are their affine images.

In the ego-vehicle frame this becomes

$\boldsymbol{\mu}_t$1

The paper fixes $\boldsymbol{\mu}_t$2 m and $\boldsymbol{\mu}_t$3 m in the experiments. The Stage-1 cubature order is $\boldsymbol{\mu}_t$4, and the spatial evaluation points in the ego frame can be precomputed. A common ambiguity in the name is resolved explicitly in the formulation: GLR is not Gauss–Legendre quadrature on an arbitrary rectangle in the abstract, but Gauss–Legendre cubature over the ego vehicle’s rectangular footprint.

3. Temporal stage: NHPP hazard model and Gauss–Legendre quadrature in time

The temporal stage converts the function $\boldsymbol{\mu}_t$5 into a collision probability over the full maneuver horizon. The counting process $\boldsymbol{\mu}_t$6, interpreted as the number of collision events that have occurred up to time $\boldsymbol{\mu}_t$7, is modeled as a non-homogeneous Poisson process with intensity $\boldsymbol{\mu}_t$8. Standard NHPP identities give

$\boldsymbol{\mu}_t$9

hence

$\boldsymbol{\Sigma}_t$0

The event $\boldsymbol{\Sigma}_t$1 is interpreted as at least one collision over the maneuver horizon, so

$\boldsymbol{\Sigma}_t$2

Because direct computation of the survival function is treated as intractable, GLR introduces a working hazard function

$\boldsymbol{\Sigma}_t$3

The stated boundary behavior is that if $\boldsymbol{\Sigma}_t$4, then $\boldsymbol{\Sigma}_t$5, and if $\boldsymbol{\Sigma}_t$6, then $\boldsymbol{\Sigma}_t$7.

The time integral is then approximated by one-dimensional Gauss–Legendre quadrature over $\boldsymbol{\Sigma}_t$8: $\boldsymbol{\Sigma}_t$9 with $W_{car}$0 obtained from Gauss–Legendre nodes on $W_{car}$1. The Stage-2 quadrature order is $W_{car}$2 in the experiments. At each time node $W_{car}$3, the method constructs $W_{car}$4, computes $W_{car}$5 by the spatial stage, forms $W_{car}$6, and then evaluates the NHPP expression

$W_{car}$7

This two-stage decomposition is the distinctive feature of GLR: inner two-dimensional Gauss–Legendre cubature in space and outer one-dimensional Gauss–Legendre quadrature in time.

4. Vehicle geometry, uncertainty model, and implementation details

The experimental configuration uses Probabilistic Bézier Curves (PBC) for the target trajectory distribution $W_{car}$8. Each Bézier control point is a Gaussian random variable, the recorded target trajectory is used as the mean of the PBC model, and the control-point covariances are selected so that the marginal $W_{car}$9 is approximately Gaussian at each time, with variance growing over time. The stated construction minimizes average KL divergence to a target Gaussian whose covariance expands from $L_{car}$0 at $L_{car}$1 up to $L_{car}$2 at $L_{car}$3. This yields a closed-form Gaussian

$L_{car}$4

from which the corner and centroid densities $L_{car}$5 are obtained by rigid shifts.

The algorithm itself is explicitly described as agnostic to this uncertainty model. Any source of $L_{car}$6 can be used, including learned predictors or mixture models, provided the densities $L_{car}$7 can be evaluated at the quadrature nodes.

For racing-scale experiments, the method uses $L_{car}$8, $L_{car}$9, and prediction horizon $t\in[0,T_F]$0 with 100 Hz ground-truth sampling. The implementation points identified for 1000 Hz operation are the following. All one-dimensional Gauss–Legendre nodes and weights and all two-dimensional cubature nodes are precomputed in a local vehicle frame. Probability densities are evaluated with vectorized or GPU-accelerated kernels across all spatial nodes and all time nodes. Stage-1 computations at different time nodes are independent and thus parallelizable. The five densities $t\in[0,T_F]$1 share covariance and differ only in mean offsets, which reduces evaluation cost.

The measured runtime is approximately $t\in[0,T_F]$2 ms per scenario, corresponding to 1000 Hz, versus $t\in[0,T_F]$3 ms for dense Monte Carlo ground truth, corresponding to 18 Hz. This runtime profile is central to the method’s intended use in high-speed overtaking, where risk must be updated under tight control deadlines.

5. Empirical performance and comparison with baselines

GLR is evaluated on 446 overtaking scenarios in a high-fidelity Formula One racing simulation and compared against Max Circle, Risk Density, Discounted BIUB, Velocity-Scaled Particle Filter (VS-PF), Mutual Independence (MI), and Quasi-Monte Carlo Gauss–Legendre (QMLGL), an ablation in which Stage 1 is replaced by dense Monte Carlo at the same Gauss–Legendre time nodes (Weiss et al., 24 Jul 2025).

Method	MAE ± $t\in[0,T_F]$4	Time / Rate
GLR	$t\in[0,T_F]$5	$t\in[0,T_F]$6 ms / 1000 Hz
QMLGL	$t\in[0,T_F]$7	$t\in[0,T_F]$8 ms / 125 Hz
VS-PF	$t\in[0,T_F]$9	$R_{ego}(t)$0 ms / 833 Hz
Risk Density	$R_{ego}(t)$1	$R_{ego}(t)$2 ms / 2000 Hz
Discounted BIUB	$R_{ego}(t)$3	$R_{ego}(t)$4 ms / 2000 Hz
Mutual Independence	$R_{ego}(t)$5	$R_{ego}(t)$6 ms / 2500 Hz
Max Circle	$R_{ego}(t)$7	$R_{ego}(t)$8 ms / 1666 Hz

The reported aggregate result is an average error reduction of 77% over five state-of-the-art baselines. Against the best external baseline, VS-PF with MAE $R_{ego}(t)$9, GLR improves to MAE $p(\mathcal{T})$00, which is described as surpassing the next-best method by about 52.6% while maintaining 1000 Hz. QMLGL is slightly more accurate than GLR but approximately eight times slower, which is used to argue that the cubature approximation in the spatial stage remains close to Monte Carlo accuracy.

The comparative interpretation given in the source is that bounding-circle and Boole-inequality-based methods are either overly conservative or structurally biased. GLR is described as less conservative because it uses the full rectangular footprint instead of inflated circles, avoids Boole’s inequality aggregation, and uses an NHPP hazard model rather than independent-timestep assumptions or hard upper bounds. A plausible implication is that GLR’s gain is not attributable to quadrature alone, but to the joint effect of geometry preservation, continuous-time aggregation, and a specific hazard construction.

6. Assumptions, scope, and relation to other Gauss–Legendre “rectangle” constructions

The stated assumptions and limitations are specific. The 5-point approximation of the target rectangle and the independence assumption among the five point events introduce approximation error. The hazard definition

$p(\mathcal{T})$01

is described as heuristic, though well-motivated. The NHPP model assumes independent increments in the counting process. The method is formulated for 2D planar motion; extension to 3D would require three-dimensional cubature and more complex geometry. For multiple opponents, the paper suggests a conditional-independence extension

$p(\mathcal{T})$02

followed by the same NHPP integration.

The framework is nonetheless presented as general: any shape approximable by rectangles can be tiled or bounded by rectangles, and any prediction model that yields evaluable $p(\mathcal{T})$03 can feed Stage 1. The cited broader application classes include warehouse robots, UAVs in cluttered spaces, and highway driving with nontrivial shapes. This suggests that GLR should be understood not as a racing-specific heuristic, but as a geometry-aware continuous-time risk estimator whose racing evaluation serves as a demanding benchmark.

A recurring misconception is terminological. In other arXiv work, Gauss–Legendre constructions on rectangles refer to tensor-product quadrature or Legendre bases on rectangular domains rather than collision-risk estimation. "Gelfand Triplets, Continuous and Discrete Bases and Legendre Polynomials" constructs generalized Legendre polynomial bases on intervals, rectangles, and $p(\mathcal{T})$04-rectangles, providing the orthonormal basis structure from which tensor-product Gauss–Legendre quadrature on rectangles is derived (Celeghini et al., 2023). "Gauss-Legendre Features for Gaussian Process Regression" uses a tensor-product Gauss–Legendre rule on a truncated rectangle $p(\mathcal{T})$05 in frequency space to build low-rank kernel approximations (Shustin et al., 2021). These usages share the rectangle as an integration domain, but they are not the same object as GLR in autonomous racing. In the GLR algorithm proper, the defining rectangle is the ego vehicle footprint, and the defining task is continuous-time probabilistic collision risk.