Curvature-Aware Longitudinal Control

• Curvature-aware longitudinal control is an anticipative speed regulation approach that uses planned path geometry to adjust acceleration along curved trajectories.
• It integrates virtual border construction, ray-tracing, and predictive dynamics to enforce safety constraints and optimize tracking accuracy.
• Empirical results show reduced lateral acceleration and tracking errors, validating its effectiveness in enhancing autonomous vehicle performance on non-linear roads.

A curvature-aware longitudinal control strategy is an anticipative speed regulation approach for autonomous and connected vehicles, in which upcoming geometric constraints—encoded by the curvature profile of the planned path—directly influence the commanded acceleration (or speed) along the trajectory. Unlike purely reactive controllers, these strategies incorporate path geometry to proactively modulate vehicle speed, ensuring safety, comfort, and tracking accuracy, especially on curvilinear roadways. Modern formulations couple geometric modeling (e.g., virtual track borders, perception-driven curvature estimation), predictive dynamics, and barrier-based or optimization-centric safety constraints.

1. Foundational Principles and Problem Setting

Curvature-aware longitudinal control explicitly utilizes the planned or perceived curvature $\kappa(s)$ of the reference path, where $s$ denotes arc length, to anticipate the need to decelerate or accelerate. The fundamental rationale is that lateral acceleration $a_y = v^2 \kappa$ increases with both speed $v$ and curvature, yielding intrinsic physical limits (friction, comfort bounds) that constrain maximum safe velocity in turns. By embedding these constraints directly in the longitudinal control law, vehicles can decelerate before entering high-curvature segments and regain speed on straighter sections (Lombard et al., 2 Feb 2026, Trisovic et al., 4 Aug 2025, Chen et al., 14 Feb 2025, Li et al., 2023).

Multiple recent works address curvature-awareness in diverse contexts:

• Anticipative adaptation of acceleration based on virtual borders and ray-tracing, synthesizing a "virtual obstacle distance" that tightens in sharp turns (Lombard et al., 2 Feb 2026).
• Speed scheduling via perception-driven parametric curvature estimation, including conservative margins due to sensor uncertainty (Trisovic et al., 4 Aug 2025).
• Platooning and following control in the Frenet frame, compensating for curvature-induced discrepancies between arc-length and physical velocity (Chen et al., 14 Feb 2025).
• Integration of curvature and its higher derivatives with longitudinal jerk in optimal trajectory planning (Li et al., 2023).

2. Virtual Border and Ray-Tracing Approaches

A leading method for curvature anticipation involves constructing offset path borders and performing forward intersection (ray-tracing) to translate path curvature into an effectively "physical" obstacle for speed control (Lombard et al., 2 Feb 2026):

Let the reference path be a smooth planar curve $C : s \mapsto (x(s), y(s))$ with associated unit normal $n(s)$. Virtual left/right borders are defined as: $$B_{\mathrm{left}}(s) = C(s) + w(s) n(s), \quad B_{\mathrm{right}}(s) = C(s) - w(s) n(s)$$ with $w(s)$ the half-track width (plus margin).

At each timestep, the vehicle pose $P_v = (x_v, y_v)$ and heading $\theta_v$ define a ray $s$0, $s$1, $s$2. The minimum $s$3 (i.e., forward distance $s$4) for which $s$5 intersects the union of borders yields the anticipatory "curvature obstacle."

For constant $s$6 (circular arc), the closed-form expression is: $s$7 where $s$8 is the local radius and $s$9 the angle between heading and radius. Otherwise, the intersection is solved numerically along a local horizon (Lombard et al., 2 Feb 2026).

The step-by-step control loop is as follows:

1. Measure vehicle state $a_y = v^2 \kappa$0.
2. Project to current $a_y = v^2 \kappa$1 on reference.
3. Form virtual borders over $a_y = v^2 \kappa$2.
4. Ray-cast and solve for first intersection $a_y = v^2 \kappa$3.
5. Compute free-flow and obstacle-based accelerations.
6. Apply $a_y = v^2 \kappa$4 as the current acceleration command.

3. Formulation of Longitudinal Control Laws

Most curvature-aware approaches employ a minimum-of-two (or more) law combining a "desired" speed tendency with a curvature-constrained limitation. A prototypical example used in (Lombard et al., 2 Feb 2026):

• IDM Free-Flow ($a_y = v^2 \kappa$5):

$a_y = v^2 \kappa$6

with $a_y = v^2 \kappa$7 current speed, $a_y = v^2 \kappa$8 cruise setpoint, $a_y = v^2 \kappa$9 desired acceleration, and $v$0 an exponent (e.g., $v$1).

• Obstacle-Anticipation ($v$2):

$v$3

where $v$4 is comfortable deceleration, $v$5 emergency deceleration, and $v$6 effective reaction time. The more conservative acceleration (smaller value) is selected:

$v$7

This law tightly links speed reduction to anticipated distance to vehicle or curvature-imposed boundaries (Lombard et al., 2 Feb 2026).

Alternative formulations schedule the speed setpoint via a curvature-adaptive maximum,

$v$8

embedding the maximum lateral acceleration $v$9 and allowing for bounded estimation uncertainty via $C : s \mapsto (x(s), y(s))$0 (Trisovic et al., 4 Aug 2025).

Platooning controllers in the Frenet frame combine nominal relative distance laws with curvature-coupled compensation: $C : s \mapsto (x(s), y(s))$1 and map "virtual" inter-vehicle spacing and velocity to physical acceleration, including curvature and curvature-rate terms (Chen et al., 14 Feb 2025).

Optimization-based approaches (constrained ILQR) utilize cost function weights on both lateral acceleration $C : s \mapsto (x(s), y(s))$2 and lateral jerk $C : s \mapsto (x(s), y(s))$3, directly penalizing aggressive maneuvers in higher-curvature regions and coupling longitudinal jerk $C : s \mapsto (x(s), y(s))$4 into the speed profile design (Li et al., 2023).

4. Integration of Curvature, Derivatives, and Uncertainty

Modern strategies do not limit curvature-awareness to instantaneous values; several methods incorporate curvature derivatives ($C : s \mapsto (x(s), y(s))$5, $C : s \mapsto (x(s), y(s))$6) and even explicit uncertainty quantification:

• Higher-order curvature and longitudinal jerk are included in the state and control, enabling finer anticipation of transitions and smooth limiting of speed ahead of sharp curvature changes (Li et al., 2023).
• Perception-driven uncertainty, as in (Trisovic et al., 4 Aug 2025), not only estimates $C : s \mapsto (x(s), y(s))$7 but also its covariance $C : s \mapsto (x(s), y(s))$8, inflating speed constraints based on worst-case or sampled curvature (via robust scheduling: $C : s \mapsto (x(s), y(s))$9).
• Barrier-type safety feedbacks: In platooning (Chen et al., 14 Feb 2025), constructive barrier feedback proportional to the inverse of the relative distance ensures collision avoidance regardless of curvature, and Lyapunov-based proofs provide formal safety guarantees.

5. Implementation Details and Parameterization

Representative parameter settings (extracted from (Lombard et al., 2 Feb 2026, Li et al., 2023)):

Parameter	Value (simulation/real vehicle)	Description
Path width $n(s)$0	$n(s)$1 m (constant)	Virtual border offset
Acceleration $n(s)$2	$n(s)$3 m/s$n(s)$4	Preferred/comfortable acceleration (IDM)
Max deceleration $n(s)$5	$n(s)$6 m/s$n(s)$7	Comfortable deceleration
Emergency decel $n(s)$8	$n(s)$9 m/s$$B_{\mathrm{left}}(s) = C(s) + w(s) n(s), \quad B_{\mathrm{right}}(s) = C(s) - w(s) n(s)$$0	Max. for imminent obstacle
Reaction time $$B_{\mathrm{left}}(s) = C(s) + w(s) n(s), \quad B_{\mathrm{right}}(s) = C(s) - w(s) n(s)$$1	$$B_{\mathrm{left}}(s) = C(s) + w(s) n(s), \quad B_{\mathrm{right}}(s) = C(s) - w(s) n(s)$$2 s	Effective human-like response lag
Speed setpoint $$B_{\mathrm{left}}(s) = C(s) + w(s) n(s), \quad B_{\mathrm{right}}(s) = C(s) - w(s) n(s)$$3	$$B_{\mathrm{left}}(s) = C(s) + w(s) n(s), \quad B_{\mathrm{right}}(s) = C(s) - w(s) n(s)$$4 m/s (fwd), $$B_{\mathrm{left}}(s) = C(s) + w(s) n(s), \quad B_{\mathrm{right}}(s) = C(s) - w(s) n(s)$$5 m/s (rev)	Cruise speed configuration
Acceleration exponent $$B_{\mathrm{left}}(s) = C(s) + w(s) n(s), \quad B_{\mathrm{right}}(s) = C(s) - w(s) n(s)$$6	$$B_{\mathrm{left}}(s) = C(s) + w(s) n(s), \quad B_{\mathrm{right}}(s) = C(s) - w(s) n(s)$$7	Exponent in IDM

Cost weights in ILQR (Li et al., 2023): $$B_{\mathrm{left}}(s) = C(s) + w(s) n(s), \quad B_{\mathrm{right}}(s) = C(s) - w(s) n(s)$$8, $$B_{\mathrm{left}}(s) = C(s) + w(s) n(s), \quad B_{\mathrm{right}}(s) = C(s) - w(s) n(s)$$9, $w(s)$0, $w(s)$1, $w(s)$2, $w(s)$3, $w(s)$4. These choices constrain peak lateral acceleration and jerk, empirically limiting $w(s)$5 m/s$w(s)$6 and $w(s)$7 m/s$w(s)$8.

Loop rates: Control laws are typically executed at 50–100 Hz, with ILQR receding horizon planners running at 10 Hz with $w(s)$920 ms planning latency per cycle.

6. Performance Metrics and Experimental Results

Measured outcomes indicate consistent benefits of curvature-awareness:

• On a scaled Nürburgring track, curvature-aware controllers produce speed profiles with dips aligning to high-curvature segments and rapid recovery on straights. Compared to constant-speed baselines:
  • Peak lateral acceleration reduced by $P_v = (x_v, y_v)$0
  • Maximum lateral tracking error reduced by up to $P_v = (x_v, y_v)$1
• On a real test vehicle, speed profiles adaptively slow to $P_v = (x_v, y_v)$2 m/s in tightest curves ($P_v = (x_v, y_v)$3 m$P_v = (x_v, y_v)$4) and resume $P_v = (x_v, y_v)$5 m/s on straights, consistently avoiding skidding or emergency interventions (Lombard et al., 2 Feb 2026).
• In perception-uncertainty settings (Trisovic et al., 4 Aug 2025), robust scheduling eliminates lateral safety violations and flattens maximum lateral error vs. curvature in randomized trajectory trials.
• Platooning experiments confirm that barrier-augmented laws enforce safety invariance (strictly positive inter-vehicle and road-edge margins), even when baseline controllers permit collision (Chen et al., 14 Feb 2025).
• CILQR-based planners yield RMS lateral deviations $P_v = (x_v, y_v)$6 m, peak lateral acceleration $P_v = (x_v, y_v)$7 m/s$P_v = (x_v, y_v)$8, and RMS longitudinal jerk $P_v = (x_v, y_v)$9 m/s$\theta_v$0, achieving real-time performance (Li et al., 2023).

7. Synthesis and Recommendations

Curvature-aware longitudinal control strategies synthesize geometric path information, anticipative obstacle modeling (via virtual borders or curvature-driven speed limits), and dynamic/optimization-based feedback to realize proactive, safe, and comfortable speed regulation for autonomous vehicles. Their practical implementations integrate:

• Virtual border construction and ray tracing to convert future path curvature into a "virtual obstacle distance" (Lombard et al., 2 Feb 2026).
• Model predictive/optimal control with explicit curvature and jerk penalization (Li et al., 2023).
• Robustification against perception uncertainty and explicit enforcement of safety margins (Trisovic et al., 4 Aug 2025).
• Lyapunov/barrier-based safety guarantees in multi-agent/platooned configurations (Chen et al., 14 Feb 2025).

Empirical validation demonstrates improvements in both safety and tracking. Adopting curvature-aware longitudinal strategies is recommended for any advanced vehicle autonomy stack deployed in environments with non-trivial curvature, tight margins, or scheduling/safety-critical applications.