Velocity-Free Equations: Theory & Applications

• Velocity-free equations are mathematical and physical formulations that remove explicit velocity variables through reformulation, invariances, and emergent dynamics.
• They enable robust control strategies and modeling in areas such as attitude stabilization, fluid kinetics, and pedestrian dynamics by substituting direct velocity feedback with indirect measures.
• Applications span inertial sensor-based control to coordinate transformations in gravity, highlighting both practical benefits and theoretical insights in system stability and regularization.

The term "velocity-free equation" encompasses a diverse range of developments across control theory, fluid dynamics, pedestrian models, differential geometry, and gravitational physics. It generally refers to mathematical or physical formulations in which explicit dependence on velocity—either as a variable, measurement, or feedback quantity—is minimized, eliminated, replaced by indirect methods, or rendered superfluous through coordinate choice, invariances, or functional reformulation. The motivation behind velocity-free approaches is frequently practical (e.g., measurement unavailability, robustness to sensor noise) or theoretical (e.g., removing coordinate artifacts, capturing geometric invariance).

1. Foundational Concepts of Velocity-Free Equations

Velocity-free equations arise when explicit velocity variables are removed or replaced within governing evolution laws. Notable contexts include:

• Control theory: Attitude stabilization and tracking for rigid bodies without relying on direct angular velocity measurements, achieved by employing vector measurements from inertial sensors, auxiliary dynamic systems, and carefully constructed innovation error signals (Tayebi et al., 2012, Wu et al., 2015).
• Fluid and kinetic theory: Reformulations in which the velocity field is encapsulated within a minimization principle, statistical moments, or emergent variables—not as an externally specified function but inferred from energy functionals or stepwise variational principles (Ichinose, 2013).
• Pedestrian and multi-agent system models: Computation of individual motion via interaction-based, collision-free optimization, forgoing acceleration-based dynamics and instead directly constructing velocity from environmental constraints and local geometry (1908.10304).
• Stochastic and deterministic PDEs: Addressing transport and continuity equations with non-smooth velocities by recasting the problem in terms of spatial primitives or renormalized weak solutions, effectively "removing" the explicit action of velocity in the most singular terms (C. et al., 2016, Alberti et al., 2018).
• Gravitational and geometric frameworks: Reinterpreting equations of motion by eliminating velocity-dependent artifacts, often through coordinate transformations (e.g., Painlevé–Gullstrand coordinates in gravitational free fall), seeking formulations where physically meaningful quantities are velocity-free (Berkahn et al., 2017, Engelhardt, 2018, Molinari et al., 2019).

The essential mathematical content, therefore, depends on the particular context: velocity elimination may be achieved through auxiliary systems, integration, coordinate redefinitions, moment relations, or by leveraging structural tensor symmetries.

2. Implementation Strategies and Mathematical Representations

Attitude Control and Tracking (Rigid Body Systems)

In velocity-free attitude stabilization (Tayebi et al., 2012), the evolution law for the rigid body does not require angular velocity as a feedback term. Instead, control torques are synthesized from inertial vector measurements via skew-symmetric injection:

$$z_\gamma = \sum_i \gamma_i S(\hat{b}_i) b_i, \qquad z_\rho = \sum_i \rho_i S(r_i) b_i$$

where $S(\cdot)$ denotes the cross-product mapping, $b_i$ are body-frame measurements, $r_i$ are inertial references, and $\hat{b}_i$ is generated by an auxiliary quaternion system. The control torque is then

$\tau = z_\gamma + z_\rho$

$\beta = -z_\gamma$

$$\dot{\hat{Q}} = \frac{1}{2} \hat{Q} \odot \bar{\beta}$$

No direct velocity feedback is ever employed; stability follows via Lyapunov arguments with quadratic error terms invariant under quaternion sign change, thus eliminating the unwinding phenomenon.

In tracking problems (Wu et al., 2015), observer-based constructions on SO(3) deliver estimated velocity via continuous geometric feedback, where the observer dynamics incorporate the error between estimated and measured orientations, ensuring exponential stability even in the absence of measured rates.

Velocity-Field Theory and Statistical Physics

Field-theoretic approaches (Ichinose, 2013) substitute the phase-space description with stepwise minimization of an energy functional:

$$I_n[u(x); u_{n-1}(x), \sigma_{n-1}(x), \tilde{\rho}_{n-1}(x)] = \int dx \bigg[ \frac{\sigma_{n-1}}{2\tilde{\rho}_{n-1}} (\partial_x u)^2 + V(u) + u \partial_x V^1(x) + \frac{1}{2h} (u - u_{n-1})^2 \bigg]$$

Time evolution is emergent: $t_n = n h$. The recursive minimization provides velocity-field evolution inherently "velocity-free" in the sense that time and motion are encoded not by external velocity specification but by minimization, statistical fluctuations, and geometric averaging (path integrals over configuration space).

Multi-Agent and Pedestrian Models

Collision-free velocity models (1908.10304) compute agent positions via direct geometrical and local interaction constraints:

$\dot{X}_i = V_i \cdot e_i$

with

$$V_i = \min \left\{ V^0_i, \max[0, d_i / T], \ldots \right\}$$

and directional vector $e_i$ constructed by combining desired direction and repulsive influences—often from lateral geometry (ellipses rather than circles) and wall terms. No force-based or acceleration-dependent evolution is required.

Continuity Equations and Loss of Regularity

In transport equations with weak (non-Lipschitz) velocities (Alberti et al., 2018), "velocity-free" refers to a total elimination of regularity propagation: solutions immediately lose all positive Sobolev smoothness, so that any attempt to track regularity through the velocity or flow map is futile. Mixing constructions demonstrate that the inverse flow cannot be Sobolev, which is interpreted as the velocity's regularization effect being rendered "free" of influence.

In stochastic continuity equations (C. et al., 2016), integrating the original equation in space yields the primitive $V(t,x) = \int_{-\infty}^{x} u(t,y) dy$, which satisfies a more regular transport equation, effectively "hiding" the direct (possibly irregular) impact of the velocity field on the solution.

Gravitational Physics and Coordinate Artifacts

Velocity-free equations in general relativity (Berkahn et al., 2017, Engelhardt, 2018) refer to the possibility of removing explicit velocity-dependent terms from equations of motion—commonly present due to coordinate choices. In Schwarzschild coordinates, the radial free-fall acceleration reads

$$\frac{d^2 r}{dt^2} = -\frac{GM}{r^2} (1 - 3 v^2/c^2)$$

A coordinate transformation to Painlevé–Gullstrand coordinates eliminates the velocity-dependent correction, yielding a pure Newtonian form for the coordinate acceleration, with the velocity effect absorbed in the space–time metric. Physical predictions, therefore, depend on whether acceleration is interpreted in coordinate time or proper time.

3. Avoidance of Velocity-Dependence and Unwinding Phenomena

Velocity-free control strategies inherently sidestep the unwinding phenomenon seen in quaternion-based attitude controllers (Tayebi et al., 2012). Control laws structured through innovation terms constructed from measured vectors—and not unit quaternions—are signed-invariant, guaranteeing that initial conditions near the desired orientation do not result in large, spurious attitude excursions.

In the context of gravitational dynamics, removing velocity dependence via coordinate choice (Painlevé–Gullstrand transformation or similar) reveals that many velocity-dependent artifacts are mathematical in origin, not physical, and can be eliminated to yield "velocity-free" acceleration laws.

4. Lyapunov and Stability Foundations for Velocity-Free Formulations

Lyapunov methods dominate rigorous stability analysis in velocity-free frameworks. The composite Lyapunov function in (Tayebi et al., 2012),

$$V(q̃, q, \omega) = 2 q̃^T W_\gamma q̃ + 2 q^T W_\rho q + \frac{1}{2} \omega^T J \omega$$

and its derivative,

$\dot{V} = - z_\gamma^T z_\gamma$

certify asymptotic stability without requiring velocity feedback or explicit attitude reconstruction. Similar observer and controller Lyapunov synthesis in (Wu et al., 2015) shows exponential convergence for tracking and estimation without velocity measurements, robust to ambiguities in attitude representations.

5. Implications, Applications, and Limitations

Control Engineering

Velocity-free attitude control is directly implementable when only inertial vector measurements (from accelerometers, magnetometers, or IMUs) are available. No gyroscope or angular velocity estimator is required, simplifying hardware, lowering cost, and improving robustness in environments with sensor or synchronization failures.

Statistical and Geometric Physics

Velocity-free field theories and Boltzmann equations provide a geometric interface between micro- and macro-dynamics, incorporating fluctuation effects and emergent variables—useful for studying kinetic phenomena and dissipation in open systems with heavy statistical uncertainty (Ichinose, 2013).

Pedestrian and Multi-Agent Systems

Collision-free velocity equations enable realistic simulation of crowd dynamics under high-density, bottleneck, and anisotropic interaction regimes, matching empirical fundamental diagrams and flow-through bottlenecks without the need for force integration or velocity regularity assumptions (1908.10304).

Gravitational Theory and General Relativity

Distinguishing between coordinate and velocity dependence in equations of motion reveals that many apparent velocity corrections in gravitational acceleration are artifacts of coordinate choice; velocity-free equations, obtained through transformation, aid in extracting physical observables (e.g., proper time acceleration) relevant for experiment and theory (Berkahn et al., 2017, Engelhardt, 2018).

Limitations and Controversies

• While velocity-free control methods avoid certain pathologies (unwinding, sensitivity to measurement noise), their performance depends critically on the richness and accuracy of inertial vector measurements; ambiguity or poor sensor configuration can impact stability domains and convergence rates.
• Loss of regularity in fluid transport with non-Lipschitz velocities shows that velocity-free regularization is only meaningful within certain function space frameworks; under strong mixing, all regularity may be lost, challenging classical intuition (Alberti et al., 2018).
• Coordinate-based removal of velocity dependence in gravitational physics does not constitute a physical elimination of velocity effects; understanding which observables are invariant under transformation is critical for interpretation.

6. Extensions and Theoretical Generalizations

Recent work extends the velocity-free paradigm to:

• Observer design on SO(3) manifolds, avoiding singularities inherent in minimal attitude representations, optimizing for exponential separation-type stability in closed-loop systems (Wu et al., 2015).
• Generalized Lagrange and Hamilton equations derived from energy functionals rather than action principles, incorporating non-potential gyroscopic forces naturally within energy-conserving frameworks (Vinokurov, 2015).
• Geometric approaches to defining distribution functions and wave equations via path-integral measures derived from configuration space metrics, facilitating the unification of particle and wave descriptions in statistical mechanics and quantum theory (Ichinose, 2013, Yamaleev, 2017).
• Unique characterizations of privileged flows (shear-, vorticity-, acceleration-free) in cosmological and relativistic space-times, with uniqueness results and exceptions rigorously established (Molinari et al., 2019).

7. Summary Table: Major Themes and Methods

Context	Velocity-Free Mechanism	Key Mathematical Idea
Rigid Body Attitude Control	Inertial vectors + auxiliary system	Skew-symmetric innovation terms, Lyapunov
SO(3) Observer-Based Tracking	Geometric error feedback	Observer/controller separation, exponential stability
Fluid/Kinetic Theory	Field-theoretic energy minimization	Emergent time via variational stepwise flow
Pedestrian Dynamics	Direct geometry-based velocity	Minimum spacing, direction optimization
Stochastic/Deterministic PDEs	Primitive variables, mixing	Renormalization, commutator estimates
Gravitational/Energy Conservation	Coordinate transformation, mass variation	Invariant acceleration, velocity-dependent mass

In conclusion, velocity-free equations represent a family of mathematical and physical frameworks where the explicit inclusion of velocity variables is avoided, indirected, or rendered nonessential. Their effectiveness, behavior, and interpretation depend fundamentally on the synthesis of measurement, symmetry, geometry, and dynamical system theory—each context yielding significant insight into the structure, stability, and evolution of both classical and modern systems.