Velocity-Powered Transport Cost

• Velocity-powered transport cost is a cost function that depends explicitly on vehicle speed and dynamic variables, capturing energetic and economic trade-offs.
• These models use physics-based elements like inertial forces, aerodynamic drag, and rolling resistance to provide realistic and nuanced optimization for routing and control.
• Their applications span eco-routing, drone delivery optimization, and autonomous trajectory planning, offering actionable insights for efficient transport operations.

A velocity-powered transport cost is any cost function for travel or transport that depends explicitly and nontrivially on the vehicle, agent, or system velocity (possibly with additional dependencies on acceleration, state of charge, or other system variables). Such cost models arise across transportation network problems, vehicle routing with advanced powertrains, optimal control, and generalized mass transport. Unlike classical distance- or time-based costs, velocity-powered cost functions encode physical, energetic, or economic penalties that are sensitive to local or segment-wise speeds, opening more realistic and nuanced optimization landscapes for routing, operation, and control.

1. Physical Basis and Mathematical Formulation

The foundational aspect of velocity-powered transport cost is the explicit dependence of energy or operational cost on traversal speed. In the context of longitudinal vehicle motion, the instantaneous tractive power required is given by

$P_t(v,a,\theta) = F_t(v,a,\theta) \cdot v$

where $F_t$ includes inertial ($m a$), gravitational ($m g \sin \theta$), rolling resistance ($C_{rr} m g \operatorname{sign}(v)$), and aerodynamic drag ($1/2\,\rho\,C_dA\,v^2$) terms (Qiao et al., 2016). Powertrain losses, conversion efficiencies, and hybridization further modulate the fuel or energy flow, producing cost rates per unit time or distance that are strictly nonlinear in $v$.

In drone delivery and hybrid vehicle applications, power consumption models capture the complex trade-off between speed, range, and efficiency. For example, multirotor drones experience U-shaped energy-per-unit-distance curves in $v$, with both high-speed drag and low-speed hover losses penalizing extremes, yielding operational cost curves that are convex in $v$ (Tamke et al., 2021).

Generalizing further, mass transport theory introduces velocity-powered costs as action minimizers over admissible controls (velocities), formalized as

$$c(x, y) = \inf_{u(\cdot)} \int_0^T \mathcal{L}(x(t), u(t))\,dt, \quad x(0)=x,\,x(T)=y$$

with Lagrangians $F_t$0 embodying, e.g., $F_t$1 kinetic, energy, or control costs [(Hindawi et al., 2011); (Elamvazhuthi et al., 4 Apr 2025)].

2. Velocity-Dependent Cost Models in Routing and Trajectory Optimization

In practical routing and planning problems, velocity-powered cost appears in several operational forms:

a) Segment-wise Parameterization via Conversion Factors

In (Qiao et al., 2016), road network links are assigned conversion factors ($F_t$2 miles/gal, miles/kWh) mapped to traffic-inferred average velocities. Each segment cost is then

• Conventional vehicle (CV): $F_t$3
• Hybrid / Electric (HEV, BEV, PHEV): Fast-switching between fuel or electric cost rates based on battery state-of-charge (SOC) and segment velocity class

This model integrates both physical tractive dynamics and empirical (e.g., PSAT) energy mappings, updating SOC and operational mode as the route proceeds, and delivers path costs that adapt to both velocity regime and powertrain architecture.

b) Joint Routing and Speed Optimization

The joint routing and speed optimization problem (JRSP) (Fukasawa et al., 2016) treats per-segment speeds $F_t$4 as decision variables, minimizing

$F_t$5

where $F_t$6 is a strictly convex fuel cost function of speed, e.g., $F_t$7 (truck) or higher-degree polynomial (ship). Here, the speed-dependent cost creates couplings between routing, travel time, and consumption, requiring set-partitioning decompositions and specialized branch-cut-and-price algorithms for tract-able solution (Fukasawa et al., 2016).

c) Drone and Multi-Modal Vehicle Routing

Routing with drone speed selection explicitly models discrete system velocities $F_t$8, with segment or triplet costs evaluated using aerodynamics-informed power and time equations (Tamke et al., 2021). Optimal routing is closely tied to velocity, since higher speeds shorten mission time but may sharply increase energy use and degrade operational feasibility.

d) Trajectory Planning for Autonomous Systems

In trajectory planning, learned velocity fields provide a vectorized representation of "local" velocity-powered cost, encoding, per location and scene, the preferred velocity for smooth, safe, and efficient travel (Xin et al., 2024). Rather than scalar cost maps, velocity fields enable planners to align their trajectories with both heading and magnitude priors, directly penalizing deviations in sampled velocities relative to the field guidance.

3. Optimal Transport and Dynamic Ground-Costs

Velocity-powered costs also appear in generalized optimal transport (OT), where transport between distributions is penalized by kinetic- or control-energy corresponding to system velocities.

a) LQ Cost Functions

For linear-quadratic (LQ) systems, the cost

$F_t$9

yields a quadratic-increment cost over endpoint pairs $m a$0 after Riccati/Hamiltonian analysis. The optimal map is the gradient of a convex potential (up to linear change of coordinates), extending Brenier's theorem to general quadratic action (Hindawi et al., 2011).

b) Nonlinear Dynamics and Norm-Invariant Flows

For more general and physically-motivated dynamics, e.g., the Euler equations of angular velocity for rigid bodies with drift terms, the ground cost is again determined as the minimum control energy across admissible trajectories. For norm-invariant drifts, this is shown to reduce to classical squared-distance kinetic cost; for more general nonlinearities, only bounds or variational characterizations may be available (Elamvazhuthi et al., 4 Apr 2025).

4. Application Domains and Computational Implications

Velocity-powered transport cost models are crucial in several modern applied domains:

• Eco-routing and powertrain-aware navigation: Route guidance for advanced vehicles leverages velocity-adaptive cost to minimize real-world energy and economic expenditure; path selection can differ drastically from shortest-distance or shortest-time routes, with reported average trip cost savings between 6–15% (and maxima up to 60%) for various vehicle types (Qiao et al., 2016).
• Drone operations and multi-modal logistics: Operational policies favor intermediate speeds that balance range and delivery time; selection of per-flight optimums yield empirically demonstrated 10–20% operational savings, with negligible direct energy cost from drones relative to fuel/labor components (Tamke et al., 2021).
• Joint vehicle routing and speed scheduling: Explicit coupling of speed and fuel-use in routing produces complex mixed-integer convex programs, which are solved efficiently via advanced set-partitioning and dynamic pricing methods for realistic network sizes (Fukasawa et al., 2016).
• Autonomous trajectory planning: Vector-valued cost representations such as velocity fields enable efficient sample-based planners to produce lower collision rates and higher route progress than scalar cost maps, with substantial computational gains due to reduced dense map evaluation (Xin et al., 2024).
• Generalized optimal transport: LQ and nonlinear ground costs provide mathematically rigorous underpinnings for control over measure-valued flows, with direct links to Riccati, Pontryagin, and PDE-based potential theory (Hindawi et al., 2011, Elamvazhuthi et al., 4 Apr 2025).

5. Structural Properties and Theoretical Results

A unifying feature of velocity-powered transport cost models is the emergence of convexity or strict convexity in the cost-speed relationship, critical for both the well-posedness of optimization (uniqueness, regularity of mappings) and for computational tractability:

• Convexity in Speed: Guarantees unique minima for per-segment or per-arc speed settings, enables column-generation and dynamic programming subroutines to be efficient and reliable (Fukasawa et al., 2016).
• Piecewise Linearization and Discrete Modes: Many applied systems reduce velocity-powered cost to look-up or table-based cost assignment via conversion factors, facilitating efficient Dijkstra-style solvers even in large networks (Qiao et al., 2016).
• Optimal Transport with Generalized Lagrangians: The existence and uniqueness of optimal transport maps with velocity-powered ground cost derive from smoothness and convexity of the corresponding value function, with direct ties to Monge–Ampère theory (Hindawi et al., 2011).
• Dominance and Pruning in Discrete Speed-sets: Elimination of inefficient velocity settings (dominated speeds) through analytical criteria compresses problem size and accelerates MILP solution, especially in drone or multi-modal routing (Tamke et al., 2021).

6. Limitations, Assumptions, and Ongoing Directions

Assumptions and modeling limitations in velocity-powered transport cost must be noted:

• Granularity of Velocity Assignment: Averaging over segments or driving cycles may mask sub-segment or dynamic effects, especially in highly time-varying systems.
• Empirical vs. Physics-Based Models: Black-box conversion factors carry risk of inaccuracy outside validation regimes; more physically accurate continuous models (e.g., full tractive force integration) yield greater precision at computational expense.
• Learning-based Representations: Data-driven velocity fields may fail to extrapolate or generalize in unexplored regions or rare event scenarios, inheriting biases from demonstration data (Xin et al., 2024).
• Computational Scaling: Jointly optimizing routing, speed, powertrain, and dynamic constraints results in high-dimensional mixed-integer or nonconvex subproblems, challenging for very large networks or continuous-time control (Fukasawa et al., 2016, Tamke et al., 2021).

A plausible implication is that hybrid methods combining physical modeling, empirical calibration, and hierarchical optimization (e.g., segment-level conversion factors guided by continuous-time energetic models) may remain central for realistic, efficient, and robust application across transport domains.

7. Summary Table: Velocity-Powered Transport Cost Models

Application Domain	Core Cost Formulation	Key Reference
Advanced vehicle eco-routing	Conversion-factor per segment $m a$1, network Dijkstra	(Qiao et al., 2016)
Joint routing and speed	$m a$2 convex MICP	(Fukasawa et al., 2016)
Drone delivery optimization	U-shaped $m a$3, discrete $m a$4	(Tamke et al., 2021)
Trajectory planning (AV)	Local velocity field $m a$5	(Xin et al., 2024)
LQ optimal transport	Quadratic action value $m a$6	(Hindawi et al., 2011)
Nonlinear OMT (Euler)	Control energy in nonlinear drift	(Elamvazhuthi et al., 4 Apr 2025)

Each of these models instantiates the fundamental principle that velocity choice, not just route or distance, is critical for accurately capturing the real energetic, economic, and stability costs of transport, with rigorous mathematical structure and practical algorithmic consequences.