SPACE2TIME: Adaptive Spacetime Reformulation

• SPACE2TIME is a framework that recasts spatial disturbances as temporal variations to enable adaptive safety filtering using offline‐computed value functions.
• It employs a finite-horizon reach-avoid approach with viscosity-based barrier functions to balance crash avoidance and tracking performance in unpredictable environments.
• The framework extends to applications in analogue transformation acoustics and speculative physics, demonstrating its versatility across diverse domains.

SPACE2TIME most specifically denotes a framework for the safe and adaptive deployment of offline-learned value-function safety filters under unknown, spatially varying disturbances by reparameterizing spatial variations in disturbance as temporal variations (Tonkens et al., 23 Sep 2025). In the current literature, the same label or closely related “space-to-time” idea also appears in analogue transformation acoustics, in a speculative proposal that the flow of time accompanies the creation of new space, and in interpretive discussions of two-time physics (García-Meca et al., 2013, García-Meca et al., 2014, Muller et al., 2016, Frederico et al., 2010). The recurring motif is not a single unified formalism, but a family of constructions in which spatial structure is recast into temporal structure for control, wave engineering, or foundational interpretation.

1. Safety-filtering motivation and problem statement

In autonomous systems, SPACE2TIME was introduced to address a specific deployment problem for value function-based safety filters. These filters minimally modify a nominal controller to ensure safety, and recent work has used offline learned value functions to scale such filters to high-dimensional systems. The central difficulty is that these methods assume detailed priors on all possible sources of model mismatch, in the form of disturbances in the environment, whereas real deployments expose robots to unknown, spatially varying disturbances such as payload-drone interaction, turbulent airflow, and urban-canyon wind fields (Tonkens et al., 23 Sep 2025).

The formulation is motivated by settings in which a robot moves through a disturbance field that cannot be pre-specified exactly. The paper states four unsatisfactory alternatives. Relying on a single “worst-case” disturbance bound renders the safety filter grossly conservative. Having an oracle that knows the true disturbance field is impossible. Naively switching among a small ensemble of pre-trained value functions, each tied to a constant disturbance bound, offers no guarantee of safety when the disturbance changes between measurement updates. Online re-computation of a reachability value function is only possible in very low dimensions. Within this framing, SPACE2TIME is a deployment strategy for preserving safety without assuming exact prior knowledge of the disturbance field.

A common misunderstanding would be to interpret the method as a new nominal controller or as online Hamilton–Jacobi reachability recomputation. The formulation indicates instead that the nominal controller remains external, while the safety mechanism is an online minimal-correction layer driven by an offline-computed value function.

2. Disturbance recasting from space to time

The key insight is that a spatially varying disturbance field $d = w(x)$ induces, along a trajectory, a time-varying disturbance $d(t) = w(x(t))$. Assuming a known Lipschitz constant $L_d$ for the field $w(\cdot)$ and an upper bound $M_{\tilde f}$ on the system’s flow $\|\tilde f(x,u,d)\|$, the construction obtains $|\dot x| \le M_{\tilde f}$ and therefore

$$|\dot d| = |\nabla_x w(x)\cdot \tilde f(x,u,d)| \le L_d \cdot M_{\tilde f} =: \dot d_{\max}.$$

This converts spatial uncertainty into a bounded rate of temporal disturbance change (Tonkens et al., 23 Sep 2025).

Offline, the original system

$$\dot x = f(x) + g(x)u + d, \qquad d \in \Delta_{\max} = \{d: |d| \le d_{\max}\},$$

is re-parameterized into an augmented, time-varying disturbance system

$$z := [x,\delta]^\top, \qquad \dot z = [f(x)+g(x)u+\eta;\,0],$$

with

$d(t) = w(x(t))$0

where $d(t) = w(x(t))$1 is treated as a constant parameter. The disturbance bound therefore shrinks as one “rewinds” time toward $d(t) = w(x(t))$2 and grows linearly in $d(t) = w(x(t))$3 to the full worst case.

This recasting is the defining “space to time” operation in the robotics usage of the term. It does not assume access to the full field $d(t) = w(x(t))$4; rather, it assumes a bound on how quickly the disturbance can change along trajectories. A plausible implication is that the method separates what must be known offline—$d(t) = w(x(t))$5, $d(t) = w(x(t))$6, and $d(t) = w(x(t))$7—from what may remain unknown online, namely the detailed spatial disturbance profile.

3. Reach-avoid value functions and viscosity-based barrier filtering

Under the augmented dynamics, the framework solves a finite-horizon reach-avoid problem with failure set $d(t) = w(x(t))$8 and a control-invariant target set $d(t) = w(x(t))$9. The reach-avoid value function is

$L_d$0

In the viscosity sense, it satisfies the Hamilton–Jacobi–Isaacs partial differential inequality

$L_d$1

with

$L_d$2

Its zero sublevel sets

$L_d$3

encode the states from which the system can both avoid $L_d$4 and steer into $L_d$5 safely, even under the time-varying disturbance bound $L_d$6 (Tonkens et al., 23 Sep 2025).

The learned $L_d$7 is then used as a viscosity-based control barrier function, $L_d$8. Given a nominal controller $L_d$9, safety is enforced by solving, at each time step, the quadratic program

$w(\cdot)$0

subject to

$w(\cdot)$1

where $w(\cdot)$2 is an extended-class-$w(\cdot)$3 function. The correction is therefore minimal in the explicit sense that the nominal input is modified only as much as required to keep $w(\cdot)$4.

The significance of this construction is operational rather than merely formal. The precomputed reachability object remains usable online even though the actual disturbance is spatially varying and not known a priori. This suggests a mechanism for adaptive conservatism: the filter is neither forced into a global worst-case bound nor deprived of formal reachability semantics.

4. Online estimation, deployment pipeline, and quadcopter results

At run time, the true spatial field $w(\cdot)$5 is unknown, but the current disturbance $w(\cdot)$6 can be measured at a slower rate and its directional time derivative approximated as

$w(\cdot)$7

The method buffers the last $w(\cdot)$8 measurements of $w(\cdot)$9 and sets $M_{\tilde f}$0. It then computes the “time-to-return”

$M_{\tilde f}$1

forms $M_{\tilde f}$2, queries the offline-learned $M_{\tilde f}$3 at $M_{\tilde f}$4 via a PINN network or grid interpolation to obtain $M_{\tilde f}$5, $M_{\tilde f}$6, and $M_{\tilde f}$7, and finally solves the minimal-correction quadratic program to produce $M_{\tilde f}$8 (Tonkens et al., 23 Sep 2025).

Algorithmically, the deployment procedure is: measure $M_{\tilde f}$9, estimate $\|\tilde f(x,u,d)\|$0 and $\|\tilde f(x,u,d)\|$1 every $\|\tilde f(x,u,d)\|$2; update the buffer of the last $\|\tilde f(x,u,d)\|$3 directional derivatives and set $\|\tilde f(x,u,d)\|$4; compute $\|\tilde f(x,u,d)\|$5 clipped to $\|\tilde f(x,u,d)\|$6; evaluate $\|\tilde f(x,u,d)\|$7, $\|\tilde f(x,u,d)\|$8, and $\|\tilde f(x,u,d)\|$9; solve the quadratic program; and apply $|\dot x| \le M_{\tilde f}$0 at the high-rate control loop. The online algorithm is therefore estimation-and-query based, not online PDE solving.

The reported validation uses both simulation and hardware.

Setting	Method	Reported outcome
4D planar simulation	HJR Naive	96% crash rate
4D planar simulation	HJR Worst-Case	0% crash, very poor tracking
4D planar simulation	DeepReach Naive	90% crash rate
4D planar simulation	DR Ours (SPACE2TIME)	34% crash rate; mean goal distance $|\dot x| \le M_{\tilde f}$1; mean trajectory length $|\dot x| \le M_{\tilde f}$2 steps
4D planar simulation	HJR Ours (SPACE2TIME)	2% crash rate; mean goal distance $|\dot x| \le M_{\tilde f}$3; trajectory length $|\dot x| \le M_{\tilde f}$4 steps
6D Crazyflie hardware	DR Naive	20% success $|\dot x| \le M_{\tilde f}$5 trials crashed)
6D Crazyflie hardware	DR Ours (SPACE2TIME)	100% success $|\dot x| \le M_{\tilde f}$6 safe flights)

In simulation, the environment is a quadcopter flying in an urban-canyon-style wind field that increases exponentially as altitude decreases. In hardware, the platform is a 6D Crazyflie model in a motion-capture arena with spoofed velocity disturbances mimicking urban canyon wind. The paper states that these results demonstrate that SPACE2TIME preserves safety under unknown, spatially varying disturbances with minimal conservatism, while significantly outperforming naïve ensemble or worst-case approaches. A careful reading suggests that the strongest empirical result is not absolute optimality, but the best reported trade-off between crash avoidance and tracking performance among the listed baselines.

5. Analogue transformation acoustics and engineered space–time mappings

In acoustics, a distinct SPACE2TIME usage appears in analogue transformation acoustics (ATA). There the issue is not safe control under disturbances, but the implementation of general spacetime coordinate transformations in an acoustic medium. “Spacetime transformation acoustics” states that the standard transformation-acoustics route is not suitable for transformations that mix space and time, whereas ATA rewrites the velocity-potential equation as a covariant Klein–Gordon equation in an analogue Lorentzian metric and thereby exploits full diffeomorphism invariance (García-Meca et al., 2013).

The paper gives an explicit spacetime map,

$|\dot x| \le M_{\tilde f}$7

which mixes $|\dot x| \le M_{\tilde f}$8 and $|\dot x| \le M_{\tilde f}$9 and violates all STA invariance conditions. ATA nevertheless derives realizable acoustic parameters for a real medium,

$$|\dot d| = |\nabla_x w(x)\cdot \tilde f(x,u,d)| \le L_d \cdot M_{\tilde f} =: \dot d_{\max}.$$0

and

$$|\dot d| = |\nabla_x w(x)\cdot \tilde f(x,u,d)| \le L_d \cdot M_{\tilde f} =: \dot d_{\max}.$$1

With a suitable choice of $$|\dot d| = |\nabla_x w(x)\cdot \tilde f(x,u,d)| \le L_d \cdot M_{\tilde f} =: \dot d_{\max}.$$2, this acts as an acoustic frequency converter. The paper states that ATA realizes the “SPACE2TIME” concept, namely a device whose spatial distribution of parameters implements a nontrivial mapping between lab-time and lab-space coordinates.

A related development, “Analogue transformation acoustics and the compression of spacetime,” applies the same ATA machinery to a compressor that simultaneously squeezes space and time in $$|\dot d| = |\nabla_x w(x)\cdot \tilde f(x,u,d)| \le L_d \cdot M_{\tilde f} =: \dot d_{\max}.$$3 dimensions (García-Meca et al., 2014). The mapping is defined through

$$|\dot d| = |\nabla_x w(x)\cdot \tilde f(x,u,d)| \le L_d \cdot M_{\tilde f} =: \dot d_{\max}.$$4

with a piecewise squeezing function $$|\dot d| = |\nabla_x w(x)\cdot \tilde f(x,u,d)| \le L_d \cdot M_{\tilde f} =: \dot d_{\max}.$$5. The transformed medium parameters $$|\dot d| = |\nabla_x w(x)\cdot \tilde f(x,u,d)| \le L_d \cdot M_{\tilde f} =: \dot d_{\max}.$$6, $$|\dot d| = |\nabla_x w(x)\cdot \tilde f(x,u,d)| \le L_d \cdot M_{\tilde f} =: \dot d_{\max}.$$7, and $$|\dot d| = |\nabla_x w(x)\cdot \tilde f(x,u,d)| \le L_d \cdot M_{\tilde f} =: \dot d_{\max}.$$8 are read off algebraically from the transformed metric. The paper further states the scaling law that, in $$|\dot d| = |\nabla_x w(x)\cdot \tilde f(x,u,d)| \le L_d \cdot M_{\tilde f} =: \dot d_{\max}.$$9, space is squeezed by $$\dot x = f(x) + g(x)u + d, \qquad d \in \Delta_{\max} = \{d: |d| \le d_{\max}\},$$0, time by $$\dot x = f(x) + g(x)u + d, \qquad d \in \Delta_{\max} = \{d: |d| \le d_{\max}\},$$1, and event density increases by $$\dot x = f(x) + g(x)u + d, \qquad d \in \Delta_{\max} = \{d: |d| \le d_{\max}\},$$2.

These acoustics papers use “space to time” in a literal wave-engineering sense: spatially and temporally varying material parameters are designed so that wave propagation behaves as though a mixed space–time coordinate transformation had been applied. This is mathematically unrelated to the safety-filtering formulation, even though both rely on recasting a spatially indexed structure as a temporally indexed one.

6. Speculative and foundational usages, and the scope of the term

The phrase also appears in speculative and interpretive settings. “Now, and the Flow of Time” proposes that Hubble expansion takes place in four dimensions rather than in three and identifies the flow of time with the continuous creation of new moments that accompany the creation of new space (Muller et al., 2016). The paper introduces a modified-vacuum ansatz in which a correction to $$\dot x = f(x) + g(x)u + d, \qquad d \in \Delta_{\max} = \{d: |d| \le d_{\max}\},$$3 is proportional to $$\dot x = f(x) + g(x)u + d, \qquad d \in \Delta_{\max} = \{d: |d| \le d_{\max}\},$$4, and it proposes a proper-time increment

$$\dot x = f(x) + g(x)u + d, \qquad d \in \Delta_{\max} = \{d: |d| \le d_{\max}\},$$5

Applied to black-hole mergers, the paper reports a predicted delay of order $$\dot x = f(x) + g(x)u + d, \qquad d \in \Delta_{\max} = \{d: |d| \le d_{\max}\},$$6, with $$\dot x = f(x) + g(x)u + d, \qquad d \in \Delta_{\max} = \{d: |d| \le d_{\max}\},$$7 and $$\dot x = f(x) + g(x)u + d, \qquad d \in \Delta_{\max} = \{d: |d| \le d_{\max}\},$$8 for GW150914. The same source explicitly notes both experimental and theoretical problems. Its evidentiary status is therefore qualitatively different from the control and acoustics uses.

A different interpretive “space $$\dot x = f(x) + g(x)u + d, \qquad d \in \Delta_{\max} = \{d: |d| \le d_{\max}\},$$9 time” theme appears in two-time physics. “The Transition Amplitude for 2T Physics” studies a particle in a space with two times and $$z := [x,\delta]^\top, \qquad \dot z = [f(x)+g(x)u+\eta;\,0],$$0 space dimensions with local $$z := [x,\delta]^\top, \qquad \dot z = [f(x)+g(x)u+\eta;\,0],$$1 symmetry and rigid $$z := [x,\delta]^\top, \qquad \dot z = [f(x)+g(x)u+\eta;\,0],$$2 symmetry, and shows that by constraining initial and final points to suitable hypersurfaces one reproduces lower-dimensional systems such as the massless particle, AdS scalar, nonrelativistic particle, and inverted harmonic oscillator (Frederico et al., 2010). The paper explicitly describes a “space $$z := [x,\delta]^\top, \qquad \dot z = [f(x)+g(x)u+\eta;\,0],$$3 time” interpretation in which different hypersurface choices pick different linear combinations of the original two time-coordinates to serve as the one physical evolution parameter of the reduced system.

As a broader geometric background, “Space and Time models” derives a general spacetime line element from a Helmholtz space model and a chosen time model, writing

$$z := [x,\delta]^\top, \qquad \dot z = [f(x)+g(x)u+\eta;\,0],$$4

with $$z := [x,\delta]^\top, \qquad \dot z = [f(x)+g(x)u+\eta;\,0],$$5 of constant curvature $$z := [x,\delta]^\top, \qquad \dot z = [f(x)+g(x)u+\eta;\,0],$$6, lapse $$z := [x,\delta]^\top, \qquad \dot z = [f(x)+g(x)u+\eta;\,0],$$7, and shift $$z := [x,\delta]^\top, \qquad \dot z = [f(x)+g(x)u+\eta;\,0],$$8 (Bel, 2011). This paper does not present a framework called SPACE2TIME, but it supplies a mathematically precise example of how a spatial model and a time model are combined into a spacetime structure.

Usage	Paper	Core meaning
Adaptive safety filtering	(Tonkens et al., 23 Sep 2025)	Reparameterize spatial disturbance variation as temporal variation
Analogue transformation acoustics	(García-Meca et al., 2013, García-Meca et al., 2014)	Implement space–time coordinate transformations in acoustic media
Cosmological proposal	(Muller et al., 2016)	Creation of time accompanies creation of new space
Two-time physics interpretation	(Frederico et al., 2010)	Hypersurface choice selects an effective physical time

Taken together, these sources suggest that “SPACE2TIME” is a terminologically heterogeneous label rather than a single research program. In robotics, it denotes a concrete disturbance-recasting framework with simulation and hardware validation. In acoustics, it names or exemplifies engineered space–time transformations via analogue metrics. In cosmology and two-time physics, it marks speculative or interpretive claims about how temporal structure may emerge from higher-dimensional or expanding spatial structure. The common phrase therefore masks substantial differences in mathematical apparatus, empirical support, and intended domain of application.