Spatially Dependent CAV Control

• Spatially dependent CAVs are advanced systems that integrate both local and aggregated nonlocal data to optimize control policies for enhanced safety and efficiency.
• The methodologies employ reinforcement learning, model predictive control, and convex optimization to manage heterogeneous traffic and anticipate upstream disturbances.
• Empirical findings demonstrate reduced deceleration waves, lower collision risks, and increased throughput in mixed-autonomy scenarios.

Spatially dependent connected and automated vehicles (CAVs) leverage spatial context—explicit or aggregated information about nearby vehicles and infrastructure—to optimize their control policies for safety, throughput, and energy efficiency. Unlike conventional methods that treat each vehicle as an independent entity or that are limited to local information, spatially dependent approaches fuse nonlocal information flows, encode geometric separation in optimization constraints, and model heterogeneous blocks of mixed human-driven vehicles (HDVs) and CAVs as unified, spatially-structured systems. These strategies have been instantiated across domains such as car-following on highways, flow stabilization in mixed autonomy, intersection management, and coordinated trajectory planning under geometric constraints and uncertainty.

1. Mathematical Formulations of Spatial Dependencies

Spatially dependent CAV frameworks adopt vehicle and traffic models where explicit spatial relationships or fused aggregate measures are central elements of the system state. In car-following tasks, the “CAV–AHDV–CAV” paradigm represents a chain of HDVs between two CAVs as a single fused entity, with state variables defined as weighted aggregates of headways, velocities, and relative speeds:

$$S_{k\to k+1}(t) = (\,\bar{s},\,\bar{v},\,\overline{\Delta v}\,)(t)$$

where, for $m$ HDVs,

$$\bar{s} = \frac{1}{W}\sum_{j=k+1}^{k+m} w_j\,s_j;\quad \bar{v} = \frac{1}{W}\sum_{j=k+1}^{k+m} w_j\,v_j;\quad \overline{\Delta v} = \frac{1}{W}\sum_{j=k+1}^{k+m} w_j\,\Delta v_j;\quad W = \sum_j w_j$$

weights $w_j$ encode spatial or behavioral significance. These fused states appear as inputs to reinforcement learning (RL) policies driving each CAV in the chain, alongside local headway and velocity information. The resulting control law directly exploits nonlocal, spatially aggregated information, enabling each CAV's policy to anticipate and mitigate deceleration waves or traffic oscillations cued by upstream disturbances (Chen et al., 23 Jun 2024).

At the macroscopic level, extensions to traffic flow models such as the anisotropic Aw–Rascle–Zhang (ARZ) system incorporate nonlocal (spatially-averaged) density in the relaxation term for CAVs:

$$\bar\rho(x,t) = \frac{1}{L}\int_{x}^{x+L} \rho(y,t)\,dy$$

with $L$ as the CAV look-ahead distance. Flow stability is analyzed via linear wave perturbations, yielding conditions where increased spatial anticipation (greater $L$) systematically modifies the decay or amplification of traffic waves—summarizing the inherent system-scale impact of spatially dependent vehicle responses (Hui et al., 30 Jul 2024).

2. Collision Avoidance and Convexification in Spatial Trajectory Planning

Optimal control formulations for intersection management, lane-free crossing, and trajectory optimization on curved roads encode geometric and kinematic constraints via spatial constructs. CAVs are treated as moving convex polytopes ($\zeta_i$), and collision avoidance requires separation from both other vehicles and road boundaries. The fundamental constraint,

$\text{dist}(\zeta_i,\zeta_j) \geq d_{\min}$

is nonconvex in the vehicle coordinates but is convexified through Lagrangian duality and conic programming, producing a set of affine and conic inequalities in augmented variable space. For instance, introducing dual variables $\lambda_{ij}$, $s_{ij}$ yields separation conditions that are smooth and tractable for gradient-based nonlinear programming solvers, supporting real-time optimization (Amouzadi et al., 2022).

In lane-free intersection crossing, all spatial dependencies—pairwise and with boundaries—are thus enforced exactly (to polytope approximation) during joint solution, ensuring collision-free, dynamically feasible passage even in dense multi-CAV scenarios (Amouzadi et al., 2022, Amouzadi et al., 2022).

3. Spatial Domain Transformation and MPC for Curved and Uncertain Environments

Infrastructure-assisted trajectory planning on curved roads and at unsignalized intersections operates in spatial coordinates, typically via curvilinear or path-based coordinates. This approach reparameterizes the vehicle's state and control evolution in terms of the path coordinate $s$ (or $p$), lateral offset $n$, heading error $e_\theta$, and local dynamics such as curvature $\kappa(s)$:

$$z(s) = [n(s),\,e_\theta(s),\,v(s)]^\top;\quad z'(s) = f(z(s),\,u(s))$$

Constraints on speed, collision avoidance, and adherence to road boundaries all become simple functions of $s$, streamlining formulation and online implementation in rolling-horizon model predictive control (MPC) (Yi et al., 2021).

Uncertainty in HDV trajectories is integrated as both path and speed uncertainty, with bounding regions for possible positions, velocities, and timings encoded as spatial functions. This robustifies CAV–HDV collision avoidance and ensures safety despite non-deterministic HDV maneuvers, using constraint tightening and robust linearization within the spatial NLP (Zhao et al., 5 Dec 2024).

4. Metrics, Empirical Findings, and Efficiency of Spatial Strategies

Quantitative assessments of spatially dependent CAV control hinge on precise metrics:

• $\sigma_{\rm thw}$: standard deviation of time headway, indicating uniformity of inter-vehicle spacing
• $\rho_{\rm col}$: proxy for collision risk (e.g., frequency of negative headways)
• $Q_{\rm flow}$: throughput, typically as vehicles per hour

For the “CAV–AHDV–CAV” RL framework, results on large-scale datasets (HighD, NGSIM, SPMD, Waymo, Lyft) at moderate CAV penetration (30%) show: $\sigma_{\rm thw}$ reduced by 22%, collision rate $\rho_{\rm col}$ dropping from 0.8% to 0.1%, and throughput $Q_{\rm flow}$ increased by 12% versus standard IDM baselines (Chen et al., 23 Jun 2024).

Macroscopic look-ahead models reveal that the optimal look-ahead length (e.g., $L = 100$ m) minimizes oscillation decay time, as excessive or minimal look-ahead increases convergence time. Market penetration above 20–40% ensures rapid stabilization regardless of spatial CAV placement (uniform or clustered) (Hui et al., 30 Jul 2024).

For intersection crossing, minimum-time, spatially constrained OCPs yield crossing times that remain invariant with the number of CAVs (for moderate $N$), outperforming state-of-the-art reservation and lane-free heuristics by 40–54% in total crossing time reduction (Amouzadi et al., 2022, Amouzadi et al., 2022). Real-time iteration (RTI) schemes for spatial-domain MPC achieve optimality gaps less than 2.3% while reducing computation time by two orders of magnitude compared with full SQP convergence, making online deployment feasible in multi-vehicle unsignalized intersection scenarios (Zhao et al., 5 Dec 2024).

5. Robustness, Anticipation, and Long-Horizon Coordination

Explicit modeling of spatial dependencies equips CAVs with enhanced anticipation of upstream and downstream disturbances. Aggregation of nonlocal information (e.g., via $\bar{s}_{\rm forward},\,\bar{v}_{\rm forward}$ in RL state vectors) enables policies that preemptively react to traffic oscillations, rather than relying solely on immediate predecessor feedback. In macroscopic models, finite look-ahead transforms the system's stability domain, expanding the basin for oscillation absorption and deceleration wave attenuation (Chen et al., 23 Jun 2024, Hui et al., 30 Jul 2024).

Robust spatial-domain MPC in the presence of HDV uncertainty yields collision-free and regulation-compliant trajectories, even in stochastic or adversarial scenarios. Tightened time-gap constraints and receding horizon adaptation ensure that desired safety margins are respected with all minimum time-gaps over 1.1 s in tested mixed-traffic scenarios (Zhao et al., 5 Dec 2024).

6. Applications and Deployment Implications

Spatially dependent CAV control has immediate relevance for:

• Highway flow stabilization: mitigating stop-and-go waves in mixed autonomy, requiring only moderate CAV penetration.
• Intersection management: lane-free, batch-optimized algorithms enable intersection throughput limited by geometry, not by fleet size.
• Trajectory optimization on complex roads: spatial framing accommodates arbitrary curvature, infrastructure data, and real-time re-optimization.
• Mixed traffic environments: CAVs coexisting with uncertain HDVs benefit from robust spatial approaches that preserve safety and efficiency under uncertainty.

These approaches, validated on both microscopic (RL, car-following) and macroscopic (flow, intersection) models as well as in large-scale simulation on diverse real and synthetic traffic datasets, demonstrate how explicit spatial dependency modeling underpins superior and safer CAV control and coordination compared with traditional local or myopic methods (Chen et al., 23 Jun 2024, Hui et al., 30 Jul 2024, Amouzadi et al., 2022, Amouzadi et al., 2022, Yi et al., 2021, Zhao et al., 5 Dec 2024).