Trajectory Velocity Energy (TVE)

• TVE is a framework that defines and optimizes the relationship between trajectories, velocity, and energy-like scalar cost functions across various systems.
• It is applied in domains such as video alignment, robotic manipulator planning, orbital motion, turbulence, and event-driven metrology, each adapting TVE to its metrics.
• TVE integrates methods like energy minimization, artificial potential fields, and PPO-based updates to evaluate trajectories through velocity-dependent evaluations.

Trajectory Velocity Energy (TVE) denotes a class of formulations in which trajectories are analyzed, optimized, or evaluated through scalar quantities that depend on velocity and are coupled to path geometry. The term is defined explicitly in the ODE-native video-alignment framework KVPO, where TVE is the sum of squared velocity residuals along a branch trajectory in flow-matching velocity space (Zhang et al., 14 May 2026). In several other works, the term is not introduced explicitly, but the underlying construction is TVE-like: an energy of velocity along the trajectory is embedded into a robotic artificial potential field, orbital motion is described through coupled trajectory–velocity–energy parameters, turbulent transit is formulated as trajectory optimization under an energy functional, and event-camera metrology reconstructs trajectories in order to estimate velocity and kinetic energy (Uppal et al., 10 Aug 2025, Modestino, 2016, Bollt et al., 17 Jan 2025, Li et al., 8 Jun 2026). The shared structure is not a single universal scalar, but a recurring triad in which trajectory, velocity, and an energy or energy-like functional jointly determine system behavior.

1. Terminological scope and cross-domain meaning

TVE is not used uniformly across the cited literature. In KVPO, it is a named quantity with a precise definition: $$\mathcal{E}_\theta(X^g) = \sum_{b\in B}\sum_{s=1}^S \left\|v_\theta(z_{b,s}^g, t_s, K_{<b}) - \hat{u}_{b,s}^g\right\|_2^2 .$$ This quantity functions as an energy functional over deterministic ODE trajectories in velocity space (Zhang et al., 14 May 2026).

In the robotic manipulator paper, “Trajectory Velocity Energy” is not introduced as an explicit term, but the proposed method is described as realizing precisely an “energy of velocity along the trajectory” embedded into an Artificial Potential Field. The construction combines position-dependent and velocity-dependent terms, and is then coupled to a jerk-minimizing, time-penalizing joint-space trajectory optimizer (Uppal et al., 10 Aug 2025). In the orbital, turbulence, and fragment-reconstruction papers, the TVE connection is likewise interpretive rather than terminological: the papers couple trajectory, velocity, and energy, but do not establish TVE as a standardized field-wide label (Modestino, 2016, Bollt et al., 17 Jan 2025, Li et al., 8 Jun 2026).

Domain	TVE status	Core quantity
ODE-native AR video alignment	Explicit term	$\mathcal{E}_\theta(X^g)$
Robotic manipulator planning	TVE-like reinterpretation	position- and velocity-dependent energy plus jerk-time cost
Orbital motion	TVE connection	fixed $R$, fixed $\beta$, and $v_{orb}(t)$
Turbulent transit	TVE viewpoint	energy minimization over trajectories
Event-driven fragment metrology	TVE pipeline	trajectory $\rightarrow$ velocity $\rightarrow$ kinetic energy

A common misconception is that TVE must denote physical kinetic energy. The cited works do not support that restriction. In KVPO, TVE is a surrogate energy in velocity space; in robotic planning it appears as local position–velocity energies and a global jerk-based functional; in turbulence it is a transit-energy objective; and in fragment metrology it culminates in measured kinetic energy. This suggests that TVE is best understood as a structural relation between trajectories, velocity variables, and scalar costs or energies, rather than as a single invariant formula.

2. Explicit TVE in ODE-native generative modeling

KVPO introduces TVE in the setting of distilled autoregressive video generators trained via flow matching. The generator is a deterministic probability flow ODE,

$$\frac{d x_t}{dt} = v_\theta(x_t, t, K_{<b}),\qquad x_{t=0}=x_T\sim\mathcal{N}(0,I),$$

with training trajectory

$$x_t = t x_0 + (1-t)x_T,\quad x_T \sim \mathcal{N}(0,I),\ t\in[0,1],$$

and ground-truth velocity

$u_t = x_0 - x_T.$

Because the model is a deterministic map rather than an explicit sampling distribution over trajectories, KVPO introduces a surrogate policy in velocity space (Zhang et al., 14 May 2026).

The central construction is the Trajectory Velocity Energy of a branch trajectory $\mathcal{E}_\theta(X^g)$0 under the unperturbed context $\mathcal{E}_\theta(X^g)$1: $\mathcal{E}_\theta(X^g)$2 Here $\mathcal{E}_\theta(X^g)$3 is the cached latent at block $\mathcal{E}_\theta(X^g)$4, solver step $\mathcal{E}_\theta(X^g)$5, and $\mathcal{E}_\theta(X^g)$6 is the rollout velocity target. TVE is therefore the sum of squared residuals along the trajectory, measured in the same velocity space used for flow-matching training. The paper characterizes it as an energy functional over trajectories and uses it to define a Gibbs surrogate policy over branches,

$\mathcal{E}_\theta(X^g)$7

The policy-gradient derivation shows that TVE is not merely a diagnostic metric. Since

$\mathcal{E}_\theta(X^g)$8

the PPO/GRPO update becomes a reward-weighted contrastive flow-matching objective. Above-average branches are pulled toward their rollout velocity targets, while below-average branches are pushed away from them. The paper states that TVE thereby quantifies branch likelihood in flow-matching velocity space and yields a surrogate policy fully consistent with the native ODE formulation (Zhang et al., 14 May 2026).

The significance of this construction is methodological. Existing GRPO-type methods for flows often convert ODEs into SDEs and rely on noise-based exploration. KVPO instead keeps the deterministic probability flow ODE intact, explores semantically through Causal History Routing in the KV cache, and evaluates branches through replayed velocity compatibility. The paper therefore describes TVE as the central statistic connecting replayed ODE velocities, branch probabilities, and PPO-style advantage-weighted updates. Empirically, replacing TVE with a geometric latent $\mathcal{E}_\theta(X^g)$9 surrogate significantly harms performance in long-video alignment, which the authors present as evidence that a velocity-field surrogate is crucial in this regime (Zhang et al., 14 May 2026).

3. Velocity-dependent energy shaping in robotic trajectory planning

In collision-free manipulator planning, the Energy-based Artificial Potential Field (E-APF) framework generalizes classical APF by incorporating both position and velocity into the field construction. Traditional APF uses purely position-dependent scalar potentials,

$R$0

and therefore suffers from local minima and oscillatory motion near obstacles. The E-APF formulation adopts a Lagrangian viewpoint with static potential $R$1 and kinetic-like potential $R$2, defining, for attraction,

$R$3

and

$R$4

For obstacle repulsion it similarly introduces

$R$5

with $R$6 the velocity influence bound (Uppal et al., 10 Aug 2025).

The generalized attractive force derived via the Euler–Lagrange equation is

$R$7

and the repulsive term includes both a classical positional component and a velocity-dependent term,

$R$8

The total E-APF generalized force therefore depends on extended state variables $R$9, not on position alone. The paper identifies three consequences: escape from local minima, momentum-aware behavior, and velocity shaping near obstacles (Uppal et al., 10 Aug 2025).

The same work couples E-APF to a hybrid trajectory optimizer in joint space. The optimizer searches over $\beta$0 and $\beta$1 using

$\beta$2

subject to joint velocity bounds

$\beta$3

and joint acceleration bounds

$\beta$4

The paper states that the integral of squared jerk is a smoothness energy over the trajectory and interprets the full cost as an energy-like functional of the trajectory and its derivatives. In its own summary, the work combines local TVE-like quantities in task space,

$\beta$5

$\beta$6

with a global TVE-like trajectory cost in joint space (Uppal et al., 10 Aug 2025).

The control layer is standard Euler–Lagrange robot dynamics,

$\beta$7

with computed torque control

$\beta$8

yielding

$\beta$9

Simulation on a 7-degree-of-freedom Kinova Gen3 manipulator showed collision-free, smooth, time-efficient, and oscillation-free trajectories, with APF reaching the goal in $v_{orb}(t)$0 s and E-APF in $v_{orb}(t)$1 s; the paper interprets the difference as a smoothness–safety tradeoff rather than as a loss of viability (Uppal et al., 10 Aug 2025).

4. Orbital motion as a trajectory–velocity–energy construction

The paper “Orbital velocity” presents a nonstandard orbital framework in which the trajectory and the orbital velocity are determined from two fundamental scalar parameters: a dynamic distance $v_{orb}(t)$2, which controls potential energy, and an initial momentum parameter

$v_{orb}(t)$3

The geometric distances are

$v_{orb}(t)$4

and the paper derives the central relations

$v_{orb}(t)$5

Potential energy is taken to satisfy

$v_{orb}(t)$6

so fixing $v_{orb}(t)$7 means fixing the potential energy level (Modestino, 2016).

Under constant $v_{orb}(t)$8 and constant $v_{orb}(t)$9, the locus of allowed positions $\rightarrow$0 is an ellipse with one focus at the gravitational center, major axis $\rightarrow$1, and eccentricity $\rightarrow$2. The paper states that, in these conditions, “the elliptical orbit is naturally traced,” thereby identifying

$\rightarrow$3

Velocity along the orbit is then given by

$\rightarrow$4

This is the paper’s central trajectory–velocity–energy relation: the velocity at each point depends on the energy parameter $\rightarrow$5, the momentum parameter $\rightarrow$6, and the local geometry $\rightarrow$7 (Modestino, 2016).

The paper does not explicitly present TVE as a formal label, but it explicitly connects trajectory, velocity, and energy. It interprets $\rightarrow$8 as “the space-time scalar parameter which determines the potential energy,” identifies $\rightarrow$9 with eccentricity and initial momentum, and derives orbital speed from those fixed parameters. The resulting synthesis is that choosing $\rightarrow$0 and $\rightarrow$1 determines the trajectory class, its size and shape, and the velocity distribution along it. The paper applies the framework to solar-system planets and reports close agreement between calculated and measured orbital periods and orbital velocity ranges, with examples including Earth: $\rightarrow$2 and $\rightarrow$3 (Modestino, 2016).

The limitations are equally explicit. The treatment assumes a two-body approximation, Newtonian gravity, plane motion, and closed orbits with constant $\rightarrow$4 and $\rightarrow$5. It does not explicitly treat unbound trajectories, escape velocity, or standard vis-viva reductions. The paper also states that the period relation is “probably to improve with further study, since in the present report the approximation level is not well defined” (Modestino, 2016).

5. Trajectory optimization, velocity scaling, and energy in turbulence

In “Tailwind turbulence,” TVE appears as the minimization of transit energy over possible vehicle trajectories and mean speeds in a turbulent flow. The paper studies point-vehicle models representing rotorcraft or submersibles that interact with their environment through thrust, drag, and weight, move in a plane perpendicular to gravity, and are granted idealized control authority and perfect knowledge of the flow (Bollt et al., 17 Jan 2025).

The dimensional drag law is

$\rightarrow$6

while mechanical power obeys a power law

$\rightarrow$7

After nondimensionalization using the turbulence velocity scale $\rightarrow$8 and correlation length $\rightarrow$9, the mean vehicle speed in the travel direction is

$$\frac{d x_t}{dt} = v_\theta(x_t, t, K_{<b}),\qquad x_{t=0}=x_T\sim\mathcal{N}(0,I),$$0

and two important parameters are

$$\frac{d x_t}{dt} = v_\theta(x_t, t, K_{<b}),\qquad x_{t=0}=x_T\sim\mathcal{N}(0,I),$$1

together with

$$\frac{d x_t}{dt} = v_\theta(x_t, t, K_{<b}),\qquad x_{t=0}=x_T\sim\mathcal{N}(0,I),$$2

The dimensionless energy is defined by

$$\frac{d x_t}{dt} = v_\theta(x_t, t, K_{<b}),\qquad x_{t=0}=x_T\sim\mathcal{N}(0,I),$$3

which the paper interprets as proportional to cost of transport (Bollt et al., 17 Jan 2025).

A central result is the existence of an optimal ratio between vehicle speed and turbulent velocity. In quiescent flow, for linear drag $$\frac{d x_t}{dt} = v_\theta(x_t, t, K_{<b}),\qquad x_{t=0}=x_T\sim\mathcal{N}(0,I),$$4 and rotorcraft $$\frac{d x_t}{dt} = v_\theta(x_t, t, K_{<b}),\qquad x_{t=0}=x_T\sim\mathcal{N}(0,I),$$5,

$$\frac{d x_t}{dt} = v_\theta(x_t, t, K_{<b}),\qquad x_{t=0}=x_T\sim\mathcal{N}(0,I),$$6

In the idealized Tailwind Turbulence bound, for the same vehicle class,

$$\frac{d x_t}{dt} = v_\theta(x_t, t, K_{<b}),\qquad x_{t=0}=x_T\sim\mathcal{N}(0,I),$$7

The asymptotics are explicit: in weak turbulence, $$\frac{d x_t}{dt} = v_\theta(x_t, t, K_{<b}),\qquad x_{t=0}=x_T\sim\mathcal{N}(0,I),$$8; in strong turbulence, $$\frac{d x_t}{dt} = v_\theta(x_t, t, K_{<b}),\qquad x_{t=0}=x_T\sim\mathcal{N}(0,I),$$9, so the optimal dimensional speed approaches the turbulence speed $$x_t = t x_0 + (1-t)x_T,\quad x_T \sim \mathcal{N}(0,I),\ t\in[0,1],$$0 (Bollt et al., 17 Jan 2025).

The Tailwind Turbulence lower bound assumes that the vehicle always sees a tailwind of speed $$x_t = t x_0 + (1-t)x_T,\quad x_T \sim \mathcal{N}(0,I),\ t\in[0,1],$$1 aligned with the direction of travel. For linear drag and $$x_t = t x_0 + (1-t)x_T,\quad x_T \sim \mathcal{N}(0,I),\ t\in[0,1],$$2,

$$x_t = t x_0 + (1-t)x_T,\quad x_T \sim \mathcal{N}(0,I),\ t\in[0,1],$$3

and optimizing over $$x_t = t x_0 + (1-t)x_T,\quad x_T \sim \mathcal{N}(0,I),\ t\in[0,1],$$4 gives a parameter-free lower bound on energy in turbulence. The paper then computes optimal trajectories in Kraichnan’s model of turbulence, parameterizing paths by truncated sine Fourier series and minimizing $$x_t = t x_0 + (1-t)x_T,\quad x_T \sim \mathcal{N}(0,I),\ t\in[0,1],$$5 over both the path coefficients and the mean speed $$x_t = t x_0 + (1-t)x_T,\quad x_T \sim \mathcal{N}(0,I),\ t\in[0,1],$$6. The optimized energies are always less than the quiescent-flow baseline and are slightly larger than but close to the analytical tailwind bound (Bollt et al., 17 Jan 2025).

The physical picture used to explain these results is that favorable trajectories dwell within vortex cells where the local flow acts as a tailwind and cross saddle regions to move between cells. The paper further predicts an optimum turbulence intensity: turbulence can be too weak to provide useful transport or too strong because crosswinds erase the benefit. In a crosswind-augmented tailwind model, it finds a minimum in power near

$$x_t = t x_0 + (1-t)x_T,\quad x_T \sim \mathcal{N}(0,I),\ t\in[0,1],$$7

and numerical results show a minimum in optimized mean power near $$x_t = t x_0 + (1-t)x_T,\quad x_T \sim \mathcal{N}(0,I),\ t\in[0,1],$$8 for the chosen parameters (Bollt et al., 17 Jan 2025). This is a TVE formulation in the strict sense that the trajectory, the operating speed relative to the flow, and the integrated energy cost are optimized jointly.

6. Event-driven reconstruction of trajectories, velocities, and kinetic energies

In fragment-field metrology, TVE is realized not as an optimization criterion but as a measurement chain. The paper “Event-driven dynamic trajectories reconstruction and measurement of mechanical parameters for fragments” seeks to recover 3D position, velocity, and kinetic energy in explosive scenes using event cameras with microsecond-level temporal resolution and high dynamic range (Li et al., 8 Jun 2026).

An event is

$$x_t = t x_0 + (1-t)x_T,\quad x_T \sim \mathcal{N}(0,I),\ t\in[0,1],$$9

where $u_t = x_0 - x_T.$0 are pixel coordinates, $u_t = x_0 - x_T.$1 is the timestamp, and $u_t = x_0 - x_T.$2 is polarity. The method constructs a multi-event-camera vision system and uses three geometric constraints to establish trajectory correspondences: time-correlated epipolar constraint, trifocal tensor line constraint, and local homography constraint. For the binocular event match, candidate pairs satisfy

$u_t = x_0 - x_T.$3

RANSAC is then used to fit straight lines to event point sets, representing 2D fragment trajectories (Li et al., 8 Jun 2026).

The paper converts each geometric criterion into a probability. For the time-correlated epipolar constraint,

$u_t = x_0 - x_T.$4

For trifocal tensor and local homography constraints,

$u_t = x_0 - x_T.$5

These probabilities are fused by an entropy weight method into

$u_t = x_0 - x_T.$6

Low-probability candidates are rejected as mismatches (Li et al., 8 Jun 2026).

Once trajectories are matched across views, 3D reconstruction proceeds through triangulation and nonlinear refinement. For camera $u_t = x_0 - x_T.$7,

$u_t = x_0 - x_T.$8

leading to an overdetermined linear system for $u_t = x_0 - x_T.$9, followed by Levenberg–Marquardt minimization of reprojection error. The paper then computes average 3D velocity from the reconstructed trajectory endpoints and timestamps: $\mathcal{E}_\theta(X^g)$00 With known fragment mass, average kinetic energy is

$\mathcal{E}_\theta(X^g)$01

The reported outputs are directly trajectory–velocity–energy quantities. In simulation, targets moving at 10 m/s were measured at an average of 9.8450 m/s, corresponding to a relative error of 1.55%. In simulated fragment experiments, the mean absolute relative error in speed against an infrared light curtain velocimeter was about 1.0%. In a real fragment experiment, reconstructed fragment speeds ranged from 305.4 to 363.2 m/s, with mean 339.5 m/s, and the total average kinetic energy was approximately 121.7 J (Li et al., 8 Jun 2026). The paper does not use the term TVE, but it explicitly implements the sequence

$\mathcal{E}_\theta(X^g)$02

which is precisely a trajectory–velocity–energy pipeline.

7. Unifying interpretation, limitations, and conceptual boundaries

Across the surveyed literature, TVE does not denote a single closed-form invariant. Instead, it recurs as a pattern with three ingredients. First, a trajectory is represented either as a path in physical space, a sequence of ODE states, or a reconstructed 3D line. Second, velocity is not treated as a mere derivative in the background: it is explicitly parameterized, constrained, replayed, or estimated. Third, a scalar energy or energy-like quantity evaluates the trajectory in a way that depends on velocity. In KVPO, that scalar is a sum of squared velocity residuals in flow-matching space (Zhang et al., 14 May 2026). In E-APF planning, it is a combination of position–velocity energies and a jerk-time functional (Uppal et al., 10 Aug 2025). In orbital motion, it is the coupling of a potential parameter $\mathcal{E}_\theta(X^g)$03, momentum parameter $\mathcal{E}_\theta(X^g)$04, and position-dependent orbital speed (Modestino, 2016). In turbulence, it is a cost-of-transport functional optimized jointly over path and speed (Bollt et al., 17 Jan 2025). In fragment metrology, it is the measured kinetic energy inferred from reconstructed trajectories and velocities (Li et al., 8 Jun 2026).

The literature also sets clear boundaries. TVE is explicit and formalized only in KVPO. Elsewhere, the term is interpretive: the papers present TVE-like quantities or a TVE connection, not a universally adopted nomenclature. Another misconception is that any TVE formulation must be based on physical conservation laws. The examples show otherwise. Some energies are physical, such as $\mathcal{E}_\theta(X^g)$05; some are control-theoretic, such as velocity-dependent APF terms and jerk integrals; and some are surrogate likelihood energies in model space. A plausible implication is that TVE is best regarded as a cross-domain analytical schema rather than a single disciplinary object.

Several limitations recur. The orbital treatment does not explicitly address unbound trajectories or standard effective-potential analyses (Modestino, 2016). The turbulence bounds assume perfect knowledge of the flow and unconstrained control authority (Bollt et al., 17 Jan 2025). The event-driven metrology paper focuses on average velocity over straight-line trajectory segments rather than full acceleration profiles (Li et al., 8 Jun 2026). The robotic manipulator work validates the framework in simulation and identifies future integration with reactive control strategies and physical hardware deployment as subsequent steps (Uppal et al., 10 Aug 2025). KVPO, for its part, ties TVE quality to reward design and to architectures with explicit KV-cache memory, while noting compute and memory costs due to replay (Zhang et al., 14 May 2026).

Taken together, these works show that TVE is most rigorously understood as an organizing principle for systems in which trajectory selection or evaluation is inseparable from velocity-dependent scalar structure. Whether that scalar is called energy, potential, cost, likelihood, or kinetic energy, the governing idea is the same: trajectories are not fully characterized by geometry alone, because their admissibility, optimality, plausibility, or measured consequence is encoded through velocity-dependent energy.