Collision Per Kilometer (CPK) Metrics

• Scenario Collision Per Kilometer (CPK) is a metric that normalizes collision events or risk-weighted probabilities per km in autonomous driving simulations.
• It combines empirical event-count data with probabilistic risk assessments to capture collision severity and provide actionable insights.
• CPK is used to compare hazardous driving behaviors and evaluate automated driving system performance across diverse traffic scenarios.

Scenario Collision Per Kilometer (CPK) is a key metric in autonomous driving safety evaluation, quantifying the normalized frequency of collision events or risk-weighted collision probabilities per distance driven within scenario-based simulation frameworks. Leveraged in both empirical simulation (e.g., CARLA-based studies) and probabilistic risk estimation methodologies (notably, fractional-collision risk analytics), CPK enables rigorous comparison of collision propensity across heterogeneous traffic situations, automated driving system (ADS) deployments, and behavioral agent configurations. The metric's design facilitates high granularity in incident attribution (such as injury vs. property-damage), expressivity for counterfactual risk analysis, and awareness of statistical uncertainties inherent in simulation-based studies (Roy-Singh et al., 9 Jun 2025, Mo et al., 2023).

1. Formal Definitions and Mathematical Characterization

CPK is defined in two primary modalities: event-count normalization and probabilistic risk aggregation.

Empirical CPK (event normalization):

Given a simulation scenario involving multiple agents and a defined operational period, denote $N_{\mathrm{collisions}}$ as the total number of recorded collision events and $L_{\mathrm{driven}}$ as the cumulative driven distance in kilometers (across all relevant vehicles):

$$\mathrm{CPK} = \frac{N_{\mathrm{collisions}}}{L_{\mathrm{driven}}}$$

This form is fundamental in studies using omniscient logging of collisions—either across all vehicles or restricted to ego/hazard agents—enabling direct empirical risk quantification in modular scenario-based simulation frameworks such as the CARLA platform (Mo et al., 2023).

Fractional CPK (risk-weighted normalization):

In probabilistic frameworks, each simulation scenario $i$ produces fractional collision probabilities, $p_i^{\rm inj}$ and $p_i^{\rm prop}$, representing the probability of injury-causing or property-damaging collisions (based on severity class mapping). The CPK then aggregates expected risk, normalized by distance:

$$\mathrm{CPK}_{\rm inj} = \frac{\sum_{i=1}^{N} p_i^{\rm inj}}{\sum_{i=1}^{N} d_i} \qquad \mathrm{CPK}_{\rm prop} = \frac{\sum_{i=1}^{N} p_i^{\rm prop}}{\sum_{i=1}^{N} d_i}$$

where $d_i$ is the distance driven in scenario $i$ (Roy-Singh et al., 9 Jun 2025).

2. Computation: Inputs and Workflow

Empirical CPK (as used in HSG-CARLA)

• Collision event detection: Uses built-in “collision sensor” in CARLA attached to each vehicle of interest, logging impact timestamps, agents involved, and impulse vectors.
• Driven distance measurement: At each time step $t_k$ (with update interval $L_{\mathrm{driven}}$0), for each vehicle $L_{\mathrm{driven}}$1, instantaneous speed $L_{\mathrm{driven}}$2 is used to estimate incremental displacement $L_{\mathrm{driven}}$3 in km. Summing over all vehicles and time steps yields $L_{\mathrm{driven}}$4.
• Multi-seed aggregation: Results across multiple random seeds are averaged for statistical robustness (Mo et al., 2023).

Fractional CPK (probabilistic simulation/risk analysis)

• Per-scenario fractional risk: For scenario $L_{\mathrm{driven}}$5, the probability of different crash severities is computed by summing over a probability mass function indexed by bins $L_{\mathrm{driven}}$6 (e.g., brake-jerk, reaction time), where bin weights $L_{\mathrm{driven}}$7 represent joint human-behavioral uncertainties.
• Severity mapping: Each bin is mapped to a severity class ($L_{\mathrm{driven}}$8 for serious injury, $L_{\mathrm{driven}}$9 for less severe injury, $$\mathrm{CPK} = \frac{N_{\mathrm{collisions}}}{L_{\mathrm{driven}}}$$0 for property damage only, or no-collision) via ∆v-based thresholds.
• Bin-wise summation: Injury and property-damage probabilities are estimated as:

$$\mathrm{CPK} = \frac{N_{\mathrm{collisions}}}{L_{\mathrm{driven}}}$$1

where $$\mathrm{CPK} = \frac{N_{\mathrm{collisions}}}{L_{\mathrm{driven}}}$$2 and $$\mathrm{CPK} = \frac{N_{\mathrm{collisions}}}{L_{\mathrm{driven}}}$$3 are indicator functions for the relevant severities.

• Normalization: Aggregate scenario-level expected collisions normalized by $$\mathrm{CPK} = \frac{N_{\mathrm{collisions}}}{L_{\mathrm{driven}}}$$4 yield CPK variants (Roy-Singh et al., 9 Jun 2025).

3. Scenario Granularity: Collisions, Severity, and Weighting

Scenario CPK supports fine-grained differentiation by:

• Collision severity: Separate computation for injury-causing ($$\mathrm{CPK} = \frac{N_{\mathrm{collisions}}}{L_{\mathrm{driven}}}$$5) and property-damage-only ($$\mathrm{CPK} = \frac{N_{\mathrm{collisions}}}{L_{\mathrm{driven}}}$$6) events.
• Total and weighted CPK: Aggregation can combine injury and property damage using scalar weights ($$\mathrm{CPK} = \frac{N_{\mathrm{collisions}}}{L_{\mathrm{driven}}}$$7), e.g.,

$$\mathrm{CPK} = \frac{N_{\mathrm{collisions}}}{L_{\mathrm{driven}}}$$8

Using larger $$\mathrm{CPK} = \frac{N_{\mathrm{collisions}}}{L_{\mathrm{driven}}}$$9 prioritizes injury-risk in downstream analysis (Roy-Singh et al., 9 Jun 2025).

• Scenario meta-weighting: Applied to reflect real-world geographic, usage, or exposure distributions by upweighting or downweighting specific scenarios through scenario weights $i$0:

$i$1

4. Uncertainty Propagation and Confidence Estimation

Statistical uncertainty in CPK arises from both scenario stochasticity (Monte Carlo or behavioral sampling) and, in rare cases, distance measurement inaccuracy. The variance of CPK is propagated using the delta method:

$i$2

where $i$3, typically with $i$4 in deterministic simulation. Per-scenario variances $i$5 are either estimated as $i$6 for $i$7 Monte Carlo samples or as sample variance among replicate runs. Confidence intervals are constructed as $i$8. Scenario-level bootstrapping is used as an alternative for nonparametric interval estimation (Roy-Singh et al., 9 Jun 2025).

5. Application Examples and Behavioral Risk Profiling

In the HUMANOID Scenario Generation (HSG) study in CARLA (Mo et al., 2023), CPK was used to compare risk across five human-driver-derived hazardous agent behaviors:

Behavior	Collisions ($i$9)	CPK (coll/km)
Speeding	18	0.058
Impeding	12	0.038
Occupying & Crimping	29	0.093
Drunk & Drug Driving	20	0.064
Distracted Driving	26	0.083

These results, from Town10 at 150 vehicles (five-minute runs, 20 seeds), demonstrate how scenario-specific risk factors (e.g., lane incursions in occupying/crimping, loss of control in distracted driving) elevate CPK. The driven distance is carefully controlled (e.g., 312.5 km per experimental cell, given 150 vehicles at ~2.08 km per vehicle) (Mo et al., 2023).

6. Large-Scale Implementation and Convergence Monitoring

Large-scale application of CPK in autonomous vehicle risk evaluation requires:

• Parallelization: Batch processing of behavioral-pmf to severity mapping across hardware clusters or GPUs.
• Running accumulation: Numerically stable summing of scenario risk and distance statistics using double precision.
• Convergence diagnostics: Continuous monitoring of running-average CPK versus number of scenarios or distance. Simulation proceeds until the metric stabilizes within a predefined confidence interval.
• Scenario weighting: Adjustment for nonuniform scenario sampling or real-world exposure, either by direct scaling or meta-weights (Roy-Singh et al., 9 Jun 2025).

7. Example Calculation and Interpretation

A representative calculation for illustrative purposes:

Given scenarios:

Scenario $p_i^{\rm inj}$0	$p_i^{\rm inj}$1 [km]	$p_i^{\rm inj}$2	$p_i^{\rm inj}$3
1	0.120	0.05	0.10
2	0.200	0.02	0.03
3	0.080	0.10	0.00

Summing across scenarios:

$p_i^{\rm inj}$4

$p_i^{\rm inj}$5

$p_i^{\rm inj}$6

Such calculations can be extended to thousands of scenarios for robust fleet-level risk analysis (Roy-Singh et al., 9 Jun 2025).

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• Fractional Collisions: A Framework for Risk Estimation of Counterfactual Conflicts using Autonomous Driving Behavior Simulations (Roy-Singh et al., 9 Jun 2025)
• Study on the Impacts of Hazardous Behaviors on Autonomous Vehicle Collision Rates Based on Humanoid Scenario Generation in CARLA (Mo et al., 2023)