Speed-to-Fly Cost Component

• Speed-to-fly cost component is a control construct that balances incremental energy expenditure against time savings to optimize trajectories.
• It is integrated into Model Predictive Control schemes for both aircraft and AUVs, guiding real-time speed decisions under varying conditions.
• Tuning parameters such as current gating and cost indices enable energy-efficient navigation while ensuring mission timing is maintained.

Speed-to-fly (STF) cost components are optimal control constructs designed to balance time- and energy-related costs in trajectory planning for vehicles such as aircraft and Autonomous Underwater Vehicles (AUVs). Originally rooted in the theory of gliding flight, the approach mathematically formalizes the operational trade-off between propulsion expenditure and mission time, generalizes to dynamically-varying control contexts, and now extends to modern energy-constrained vehicles operating in fluid environments. STF cost formulations are central to advanced Model Predictive Control (MPC) schemes for energy-efficient navigation, and they provide an analytic foundation for adaptive real-time speed selection in Flight Management Systems (FMS) and AUV mission planners (Syntakas et al., 31 Jan 2026, Kaptsov et al., 2022, Silva et al., 2024, Silva et al., 2024).

1. Theoretical Foundation and Definition

The STF principle emerges from seeking a speed (or generalized “pace”) at which the marginal increase in energy cost is exactly balanced by the marginal reduction in time-related cost. For powered vehicles, this balance takes the form

$$\text{marginal time cost} = -\frac{d (\text{energy cost})}{dv}$$

where $v$ is the vehicle's speed and "energy cost" reflects either direct fuel/battery usage or, for hybrid terms, a general operational cost functional.

The classical “speed-to-fly” results—originating in gliding theory—dictate optimal airspeed selection under varying wind conditions to minimize time or energy en route to a goal. This paradigm is now generalized to engineered systems, where STF is encoded as a cost term in optimal control problems, typically within receding-horizon MPC or variational frameworks (Syntakas et al., 31 Jan 2026, Kaptsov et al., 2022, Silva et al., 2024, Silva et al., 2024).

2. STF Cost Component in Ocean Current-Aware MPC for AUVs

In oceanic environments, the STF component is incorporated into stage-wise MPC cost functionals to exploit ambient flows:

$$\ell_k^{\mathrm{STF}} = s_k \kappa_{\mathrm{eff}} u_k^\top R_u u_k + s_k w_{\mathrm{glide}} \| \dot{x}_{k,\mathrm{pos}} - v_{c,k} \|^2$$

Here, $u_k$ encapsulates thrust commands, $R_u$ encodes thrust penalties, $s_k$ is a scalar "helpfulness gate" (selectively activating STF only when ocean currents assist progress), $\dot{x}_{k,\mathrm{pos}}$ is predicted ground speed, and $v_{c,k}$ the local ocean current (Syntakas et al., 31 Jan 2026). The first term steepens the propulsion penalty when currents are favorable, biasing control toward minimal active thrust (i.e., passive drift or “gliding”), while the second term penalizes departures from the regime where the AUV’s ground velocity matches the flow, enforcing near-zero water-relative velocity.

The gate $s_k$ is formally defined as: $$s_k = \frac{1}{2}(1 + \hat{e}_k^\top \check{c}_k) \cdot \tanh\left(\frac{\| v_{c,k} \|_{\epsilon_c}}{V_{\mathrm{scale}}}\right)$$ where $\hat{e}_k$ points from the current position to the goal, and $\check{c}_k$ is the normalized current vector. $V_{\mathrm{scale}}$ sets the ocean current strength threshold to activate gating, and $\| \cdot \|_{\epsilon}$ are smooth norms for $C^1$-differentiability.

For $s_k \approx 1$, the MPC solution strongly favors exploiting currents; for $s_k \approx 0$, the STF terms vanish and the standard energy-effort tracking cost is recovered. Differentiability is preserved globally, critical for CasADi + IPOPT-based solvers (Syntakas et al., 31 Jan 2026).

3. STF Cost Formulation in Aircraft Optimal Control

3.1. Cruise and Climb with Cost Index

For aircraft, the STF cost is constructed as a direct-operating-cost minimization that aggregates energy (fuel or battery) cost against time-dependent penalties using a tunable cost index (CI). For steady cruise flight:

$$\mathrm{DOC} = \int_0^{t_f} (C_t + C_e \dot{E}(t)) dt = C_e \int_0^{t_f} \left( \mathrm{CI} + \dot{E}(t) \right) dt$$

where $\mathrm{CI}$ is the ratio $C_t/C_e$ and reflects the airline's willingness to trade energy against time (Kaptsov et al., 2022, Silva et al., 2024, Silva et al., 2024).

The optimal STF speed $v^*$ solves the condition: $$- \mathrm{CI} \frac{\Delta x}{v^{*2}} - \left.\frac{\partial E_f}{\partial v}\right|_{v = v^*} = 0$$ where $E_f(v)$ is the final energy as a function of cruise speed, and $\Delta x$ is the mission segment length. Raising CI moves $v^*$ higher, favoring time savings at the expense of increased energy use.

3.2. Variable Cost Index

Allowing CI to vary dynamically (e.g., under Air Traffic Control instructions) leads to necessary conditions of the form: $$- (\mathrm{CI}_0 - \mathrm{CI}_{\mathrm{in}}) \frac{\Delta x}{v^2} e^{ -\Delta x / (\tau v)} - \mathrm{CI}_{\mathrm{in}} \frac{ \Delta x }{ v^2 } - \frac{ \partial E_f }{ \partial v } = 0$$ The speed update algorithm involves evaluating this stationarity condition in real time as operational requirements (CI) change (Silva et al., 2024, Silva et al., 2024).

4. Parameterization and Tuning

Tuning of STF cost weights directly shapes operational behavior:

• $V_{\mathrm{scale}}$ in AUVs selects the current strength to initiate STF activation; values should align with the prevailing environmental median (Syntakas et al., 31 Jan 2026).
• $\kappa_{\mathrm{eff}}$ scales the thrust penalty; higher values drive more aggressive energy saving but can increase trip time.
• $w_{\mathrm{glide}}$ trades off current-following (gliding) against strict trajectory adherence. Too-small values forfeit potential energy savings; excessively large values risk loss of path fidelity (Syntakas et al., 31 Jan 2026).
• CI parameters in aircraft directly control the time-energy trade-off. Low CI emphasizes fuel/battery saving, high CI emphasizes operational punctuality (Kaptsov et al., 2022, Silva et al., 2024, Silva et al., 2024).

Example parameter settings are tabulated below:

Parameter	Context	Typical Range
$V_{\mathrm{scale}}$	AUV (MPC)	0.05–0.3 m/s
$\kappa_{\mathrm{eff}}$	AUV (MPC)	1.5–3.0
$w_{\mathrm{glide}}$	AUV (MPC)	0.15–0.35
CI	Aircraft (FMS)	0 (energy-only) – 1.0+ (time)

5. Implementation and Computational Aspects

For AUVs, the STF cost terms function as plug-and-play modifications in $C^1$-class MPC frameworks, maintaining solver convergence properties and constraint satisfaction even as the cost landscape dynamically adapts to environmental inputs (Syntakas et al., 31 Jan 2026). Simulations (e.g., with BlueROV2 and Copernicus current fields) establish significant energy savings—up to 38.4% in descent and 12% in level cruise—without compromising arrival time or safety envelope.

Aircraft applications integrate the STF principle in FMS logic. After pilot (or dispatch) selection of CI, the system computes optimal $v^*$ by numerically solving the stationarity condition at each update, referencing pre-computed tables for rapid evaluation. With time-varying CI (e.g., due to ATC direction), $\tau$-filtered trajectories ensure smooth transitions (Kaptsov et al., 2022, Silva et al., 2024, Silva et al., 2024).

6. Comparative Table: STF Formulations Across Domains

Domain	STF Cost Formulation	Key Control/Decision Variable
AUV	$$s_k \kappa_{\mathrm{eff}} u_k^\top R_u u_k + s_k w_{\mathrm{glide}} \|\dot{x}_{k,\mathrm{pos}} - v_{c,k} \|^2$$	$u_k$ (thrust), $\dot{x}_{k,\mathrm{pos}}$
Aircraft	$\int [\mathrm{CI} + \dot{E}] dt$ with $-\mathrm{CI} \Delta x/v^2 - \partial_v E_f = 0$	$v$ (true airspeed/groundspeed)

Across all domains, the STF cost realization enforces a balance between energetic frugality (favoring slow, efficient motion or "gliding") and operational urgency (favoring higher speeds when time is valuable).

7. Applications, Impact, and Extensions

STF cost components have demonstrated substantial energy and cost savings in both simulation and analytical studies for AUVs and advanced aircraft systems (Syntakas et al., 31 Jan 2026, Kaptsov et al., 2022, Silva et al., 2024, Silva et al., 2024). In AUVs, the approach enables exploitation of ocean currents for near-zero water-relative "gliding," reducing primary propulsion demands. In aviation, STF logic provides a theoretical and computational basis for ECON speed management and enables real-time compliance with dynamically shifting operational regimes under ATC control.

A plausible implication is that further integration of STF principles with online environmental forecasting and robust optimization could yield additional performance gains in both energy and schedule adherence, extending benefits to new classes of energy-constrained and autonomous vehicles.