Speed-Adherence Score: Autonomous Driving Control

• Speed-Adherence Score is a quantitative metric for desired-speed conditioned autonomous driving that measures how accurately an autonomous policy follows a user-specified speed profile along a route.
• It computes adherence by applying an exponential penalty to the relative speed errors at each trajectory point and aggregates these using distance-based weighting.
• The metric isolates speed adherence from other driving criteria, enabling researchers to evaluate control precision without conflating overall safety, efficiency, or route completion.

Speed-Adherence Score is a quantitative controllability metric for desired-speed conditioned autonomous driving. In its exact published usage, it is introduced in Bench2Drive-Speed to measure how faithfully a closed-loop policy executes a user-specified target speed along a reference route, while remaining compatible with standard autonomous-driving evaluation (Shao et al., 26 Mar 2026). The term is not introduced as a separate named formula in several adjacent literatures that also couple “speed” with some form of adherence, adaptability, or compliance. Instead, those works use domain-specific constructs such as a power-law exponent for batting and bowling adaptability in cricket, a multi-metric speed–contour-adherence trade-off in superpixel segmentation, schedule deviation and punctual rate in bus control, a style prompt adherence similarity score in prompt-based TTS, and training-speed estimators in neural architecture search (Rizvi, 2024, Giraud et al., 2019, Guan et al., 11 Aug 2025, Cho et al., 9 Jan 2026, Ru et al., 2020).

1. Definition and evaluative role

Bench2Drive-Speed defines Speed-Adherence Score as a metric for desired-speed conditioned autonomous driving, motivated by the observation that standard autonomous-driving benchmarks typically evaluate safety, route completion, traffic-rule compliance, comfort, and efficiency, but do not explicitly measure whether a policy follows a user-specified speed command (Shao et al., 26 Mar 2026). The score is therefore positioned as a controllability metric rather than a replacement for conventional driving metrics.

The underlying task is route-conditioned closed-loop driving with explicit user control over desired target speed and, separately, overtake or follow instructions. Within that setting, Speed-Adherence Score quantifies how closely the executed speed matches the target speed during rollout. The paper treats this as a first-class capability because users may wish to customize whether the vehicle drives faster or slower, but that controllability must still coexist with safety, comfort, and route-following objectives (Shao et al., 26 Mar 2026).

This situates the metric in a broader class of adherence measures that evaluate compliance with an externally specified control variable. In the Bench2Drive-Speed formulation, that variable is the speed profile along the route. The score is thus fundamentally a route-level compliance functional over realized motion, not a latent-intent measure and not a general-purpose safety indicator (Shao et al., 26 Mar 2026).

2. Formal mathematical definition

The metric is defined over a closed-loop trajectory on a reference route of length $L$. Given a route $\mathcal{R}$ and an ego trajectory $\{(x_i, y_i, \mathbf{v}_i)\}_{i=1}^N$, the ego position at each step is projected onto the route to obtain an arc-length coordinate

$s_i \in [0,L].$

The actual speed is

$v_i^{\text{actual}} = \|\mathbf{v}_i\|_2,$

and the target speed at that route position is

$v_i^{\text{target}} = v^{\text{target}}(s_i).$

To reduce bias from non-uniform sampling and stationary states, the benchmark uses distance-based weights

$w_i = \|(x_i,y_i)-(x_{i-1},y_{i-1})\|_2.$

The per-step relative speed error is

$$e_i = \frac{|v_i^{\text{actual}} - v_i^{\text{target}}|}{\max(v_i^{\text{target}},\epsilon)}.$$

This normalization by $\max(v_i^{\text{target}},\epsilon)$ is explicitly introduced to avoid division by zero or instability when the target speed is very low or zero. The relative error is then mapped to a bounded per-step score through

$\text{score}_i = \exp(-\alpha e_i),$

where $\mathcal{R}$0 is a penalty-strength hyperparameter. Exact speed matching yields score $\mathcal{R}$1, and larger deviations yield exponentially smaller scores. The overall Speed-Adherence Score is the distance-weighted average

$\mathcal{R}$2

Before reporting, this is a soft score in $\mathcal{R}$3; in the benchmark tables it is presented on a 0–100 scale, with values such as 68.79 and 41.54 (Shao et al., 26 Mar 2026).

The paper supports segment-wise target-speed specification. In the appendix, a route is represented as

$\mathcal{R}$4

where $\mathcal{R}$5 are normalized progress intervals and $\mathcal{R}$6 is the target speed assigned to that segment. During runtime,

$\mathcal{R}$7

This expresses the target-speed query as a function of current route position rather than raw time, which is consistent with the distance-weighted episode-level aggregation (Shao et al., 26 Mar 2026).

3. Inputs, computation protocol, and special handling

Computing the score requires four inputs: the ego trajectory from closed-loop rollout, including positions $\mathcal{R}$8 and velocity vectors $\mathcal{R}$9; the route geometry $\{(x_i, y_i, \mathbf{v}_i)\}_{i=1}^N$0 with arc-length parameterization; the target-speed profile $\{(x_i, y_i, \mathbf{v}_i)\}_{i=1}^N$1; and a projection rule from ego pose to route progress $\{(x_i, y_i, \mathbf{v}_i)\}_{i=1}^N$2 (Shao et al., 26 Mar 2026). Accurate projection is methodologically central because the target speed is queried by route position.

The benchmark uses CARLA-based closed-loop evaluation on 48 routes, with 16 routes per difficulty level: easy, medium, and hard. Speed-Adherence Score is computed per route from the executed trajectory. Distance weighting is explicitly used to reduce the effect of long stationary periods, frame-rate artifacts, and dense sampling in stopped states. Because standing still contributes little under this weighting, the score should be interpreted as adherence over traveled distance rather than a pure time average (Shao et al., 26 Mar 2026).

A notable caveat appears in follow scenarios with a slower lead vehicle. The paper states that when the ego vehicle is constrained by a slower lead vehicle, with

$\{(x_i, y_i, \mathbf{v}_i)\}_{i=1}^N$3

the penalty is softened. The provided description does not include the exact softened formula, only the statement that the penalty is reduced in this case. This makes the metric more realistic in interactive driving, because the inability to reach the commanded speed may be imposed by traffic rather than by policy failure (Shao et al., 26 Mar 2026).

The metric also handles low-speed regimes through the denominator safeguard $\{(x_i, y_i, \mathbf{v}_i)\}_{i=1}^N$4. However, the exact value of $\{(x_i, y_i, \mathbf{v}_i)\}_{i=1}^N$5 is not specified in the supplied text. This suggests that faithful reproduction of the implementation requires either the codebase or supplementary details, particularly for low-speed handling and the follow-scenario softening (Shao et al., 26 Mar 2026).

4. Relation to standard autonomous-driving evaluation

Speed-Adherence Score is designed to be compatible with standard autonomous-driving metrics, not to replace them. Bench2Drive-Speed also reports Driving Score (DS), Success Rate (SR), efficiency, comfort, multi-ability metrics, traffic-rule compliance, and route completion, and the benchmark is built upon Bench2Drive so that standard CARLA/Bench2Drive evaluation is preserved (Shao et al., 26 Mar 2026).

This separation matters because Speed-Adherence Score evaluates instruction following, whereas DS, SR, traffic-rule compliance, and related measures evaluate whether the vehicle drives safely, completes the route, and behaves acceptably under standard autonomy criteria. The benchmark is therefore explicitly structured to study whether a model can follow target speed without degrading the rest of driving quality (Shao et al., 26 Mar 2026).

The reported experiments indicate that speed-conditioned models can improve speed adherence while maintaining comparable Driving Score and Success Rate. The paper further reports that comfort changes only slightly and efficiency may improve. It also distinguishes target-speed following from overtaking: speed adherence can be achieved relatively well, whereas executing overtaking commands remains difficult, especially in hard scenarios where aggressive lane changes can induce safety violations (Shao et al., 26 Mar 2026).

A common misconception would be to treat a high Speed-Adherence Score as evidence of globally good driving. The benchmark explicitly resists that interpretation. A high score means that realized speed closely matches the user-specified target along the route; it does not, by itself, imply safety, legality, comfort, or success in interactive maneuvers (Shao et al., 26 Mar 2026).

5. Empirical behavior and benchmark results

The baseline comparison reported in Bench2Drive-Speed shows that the metric is sensitive to whether target-speed conditioning is present. On the 48 evaluation routes, $\{(x_i, y_i, \mathbf{v}_i)\}_{i=1}^N$6 scores about 41.54, whereas TCP-Speed trained with Expert2.1k scores about 68.79, TCP-Speed with Virtual2.1k scores about 69.23, and another Virtual2.1k variant scores about 68.36 (Shao et al., 26 Mar 2026). The paper interprets this as evidence that adding speed conditioning substantially improves adherence and that re-annotated regular driving data can match expert-parameter supervision.

The difficulty breakdown follows the expected pattern: easy routes have the highest scores, medium and hard routes are lower, but conditioned models still outperform the vanilla baseline across all difficulty levels. This supports the claim that the score is responsive to increasing scenario complexity and interactive traffic difficulty (Shao et al., 26 Mar 2026).

The appendix compares long versus short virtual target-speed settings. The reported result is that virtual-short tends to give slightly better speed adherence, while virtual-long is comparable but less stable. The paper attributes this to extrapolation noise increasing with longer horizons (Shao et al., 26 Mar 2026).

These results also reinforce the metric’s intended semantics. Because the score is defined through route projection, relative speed error, exponential penalization, and distance-weighted aggregation, it is sensitive both to sustained command following and to systematic deviation from the specified speed profile. This suggests that the benchmark is probing controllability at the level of executed longitudinal behavior rather than merely detecting whether the model has received a speed token (Shao et al., 26 Mar 2026).

6. Related constructs in other research domains

Outside Bench2Drive-Speed, several papers study closely related ideas but do not define a literal Speed-Adherence Score. The terminological overlap is therefore partial rather than exact.

Domain	Construct	Relation to Speed-Adherence Score
Desired-speed conditioned autonomous driving	Speed-Adherence Score	Exact named metric (Shao et al., 26 Mar 2026)
Cricket performance modeling	Power-law exponent $\{(x_i, y_i, \mathbf{v}_i)\}_{i=1}^N$7	Practical adherence/adaptability indicator, but not a named Speed-Adherence Score (Rizvi, 2024)
Superpixel segmentation	Runtime + BR/UE/PR-F + ASA + CO	Multi-metric speed–adherence trade-off, not a composite score (Giraud et al., 2019)
Bus control on arterial corridors	Schedule deviation and punctual rate	Closest proxy to adherence under speed control, not a named score (Guan et al., 11 Aug 2025)
Prompt-based TTS	SPAM similarity score	Prompt-adherence metric that includes speed as an acoustic factor (Cho et al., 9 Jan 2026)
Neural architecture search	TSE, TSE-E, TSE-EMA	Training-speed estimators, not adherence scores (Ru et al., 2020)

In the cricket study, the closest construct is the power-law exponent $\{(x_i, y_i, \mathbf{v}_i)\}_{i=1}^N$8 in

$\{(x_i, y_i, \mathbf{v}_i)\}_{i=1}^N$9

where $s_i \in [0,L].$0 is run-scoring rate and $s_i \in [0,L].$1 is innings half-life. The paper explicitly states that $s_i \in [0,L].$2 can function as an adaptability indicator across formats, with high $s_i \in [0,L].$3 associated with better adaptation to faster play and low $s_i \in [0,L].$4 with better suitability for longer formats. The supplied description notes that the text does not introduce a named “Speed-Adherence Score” as a separate formula; rather, the operative score-like quantity is the power-law exponent $s_i \in [0,L].$5 (Rizvi, 2024).

In SCALP, the problem is framed as a trade-off among computational speed, adherence to image contours, and regularity or compactness. The paper does not define a composite metric called “Speed-Adherence Score.” Instead, the closest quantitative framework is a joint reading of runtime, Boundary Recall, Undersegmentation Error, Precision–Recall/$s_i \in [0,L].$6-measure, Achievable Segmentation Accuracy, and Compactness, with SCALP presented as improving contour adherence and regularity without sacrificing speed much (Giraud et al., 2019).

In the bus-control study, the closest equivalent is bus schedule adherence, defined as the deviation between actual arrival time and scheduled arrival time. Speed control is treated as a control action for improving punctuality rather than as a separately scored adherence variable. The relevant outcomes are schedule deviation, punctual rate for arrivals within 30 seconds of schedule, and headway standard deviation (Guan et al., 11 Aug 2025).

In prompt-based TTS, SPAM is a Style Prompt Adherence Metric rather than a speed-adherence score. Its speech-prompt fusion module includes explicit speed, energy, and pitch branches, and the final adherence score is a cosine similarity between speech and prompt embeddings. This makes it adherence-oriented and speed-aware in the acoustic sense, but not a route-level execution metric (Cho et al., 9 Jan 2026).

In neural architecture search, the closest analogue is the Training Speed Estimator family—TSE, TSE-E, and TSE-EMA—which uses accumulated early training loss as a speed-based predictor of eventual generalization performance. The paper explicitly states that it does not use the exact phrase “Speed-Adherence Score” (Ru et al., 2020).

7. Scope, limitations, and interpretive boundaries

The Bench2Drive-Speed metric has several explicit limitations. It depends on route projection, so projection noise can affect the score. Its relative-error normalization can make low-speed regimes delicate, even though $s_i \in [0,L].$7 provides a safeguard. Its follow-scenario softening is described conceptually but not fully formalized in the supplied text. Its distance weighting is deliberate but changes the meaning of the result from a time-average notion of adherence to a traveled-distance notion of adherence (Shao et al., 26 Mar 2026).

The metric also evaluates realized speed rather than latent intent. If a model is constrained by traffic, its score may decrease unless the special follow-case softening applies. Relatedly, route-level scores are computed from the executed trajectory, so early episode termination due to failure affects the score through the realized rollout rather than through a separate adherence-failure flag (Shao et al., 26 Mar 2026).

Across the broader literature, a recurrent boundary is that “speed” and “adherence” often enter evaluation through domain-specific proxies rather than a universal scalar. The autonomous-driving formulation is a direct compliance score against a target-speed profile. The cricket formulation reduces the trade-off to an adaptability exponent. The superpixel formulation keeps speed and contour adherence as separate axes within a multi-metric evaluation. The bus-control formulation evaluates schedule adherence as an arrival-time deviation objective. The TTS formulation scores style-prompt adherence with speed as one acoustic factor. The NAS formulation uses speed as a predictor of eventual performance rather than as a command-following measure (Rizvi, 2024, Giraud et al., 2019, Guan et al., 11 Aug 2025, Cho et al., 9 Jan 2026, Ru et al., 2020).

A plausible implication is that the exact term “Speed-Adherence Score” is presently most precise when reserved for desired-speed conditioned autonomous driving as formalized in Bench2Drive-Speed. In adjacent areas, similar language points to related but non-identical constructs, and those distinctions are necessary for accurate comparison across fields.