Clarity: A Normalized Measure for Information

• Clarity is a normalized differential entropy measure defined on (0,1) that captures the epistemic state by mapping uncertainty to certainty in continuous random variables.
• Its dynamics in robotics model both information gain from sensing and decay due to process noise, enabling gradient-based planning and optimization in stochastic environments.
• Integration with Stein Variational Trajectory Optimization and gatekeeper protocols ensures robust, safe trajectory planning with real-time, closed-loop control of information-driven exploration.

Clarity is a technical term that appears in diverse forms across computational sciences, robotics, information theory, automated planning, and interpretability research. In contemporary robotics and information-driven planning, clarity refers most prominently to a normalized, continuous measure of information quality, bounded in (0,1), derived from differential entropy. This measure models the epistemic state of a robot or agent as it gathers, loses, and manages information in a stochastic environment, and is deeply embedded in modern informative trajectory optimization frameworks. Clarity also arises as an interpretability metric for learned representations and as a unifying principle in heterogeneous data comparison. The following sections provide a comprehensive, technically rigorous account of the concept as formalized and deployed in the literature.

1. Formal Definition: Clarity as Normalized Differential Entropy

Clarity, denoted $q[Z]$ for a random variable $Z$, is defined to transform differential entropy into a normalized measure on $(0,1)$:

$$q[Z] = \left(1 + \frac{\exp(2h[Z])}{(2\pi e)^n}\right)^{-1}$$

where $h[Z] = -\int_S \rho(z)\log\rho(z)\,dz$ is the differential entropy of an $n$-dimensional continuous random variable $Z$ with density $\rho$ over domain $S \subset \mathbb{R}^n$ (Naveed et al., 13 Nov 2025, Agrawal et al., 2023, Agrawal et al., 2024).

Key properties:

• $q[Z] \rightarrow 0$ as $Z$0 (maximal uncertainty).
• $Z$1 as $Z$2 ("perfect" knowledge, i.e., zero variance).
• $Z$3 is invariant to translations of $Z$4.

For the scalar Gaussian case $Z$5, $Z$6, which provides a tight and dimensionless mapping of variance to the $Z$7 interval.

2. Spatiotemporal Clarity Dynamics in Informative Planning

In robotics, the environment is partitioned into $Z$8 spatial cells, each associated with a local latent variable $Z$9 governed by a linear Gaussian process:

$(0,1)$0

Clarity in each cell evolves as:

$(0,1)$1

• The first term quantifies information gain due to measurements; the second models information decay arising from process noise.
• When $(0,1)$2, i.e., the robot does not sense cell $(0,1)$3, clarity decays purely via $(0,1)$4.
• The dynamics ensures $(0,1)$5 asymptotically approaches $(0,1)$6 as dictated by both process ($(0,1)$7) and measurement ($(0,1)$8) noise; perfect clarity is unachievable with nonzero $(0,1)$9 (Naveed et al., 13 Nov 2025).
• A typical $$q[Z] = \left(1 + \frac{\exp(2h[Z])}{(2\pi e)^n}\right)^{-1}$$0 is spatially smooth and differentiable (e.g., a Gaussian kernel), so clarity dynamics are $$q[Z] = \left(1 + \frac{\exp(2h[Z])}{(2\pi e)^n}\right)^{-1}$$1, supporting gradient-based control.

This ODE-based propagation of clarity captures the nonuniform, path-dependent, and temporally decaying nature of information in stochastic environments, fundamentally differentiating clarity from static coverage or entropy-based measures.

3. Integration with Stein Variational Trajectory Optimization

The clarity-aware planning problem seeks a robot trajectory $$q[Z] = \left(1 + \frac{\exp(2h[Z])}{(2\pi e)^n}\right)^{-1}$$2 over $$q[Z] = \left(1 + \frac{\exp(2h[Z])}{(2\pi e)^n}\right)^{-1}$$3 that maximally raises all $$q[Z] = \left(1 + \frac{\exp(2h[Z])}{(2\pi e)^n}\right)^{-1}$$4 above prescribed thresholds $$q[Z] = \left(1 + \frac{\exp(2h[Z])}{(2\pi e)^n}\right)^{-1}$$5, operationalized as:

$$q[Z] = \left(1 + \frac{\exp(2h[Z])}{(2\pi e)^n}\right)^{-1}$$6

st. $$q[Z] = \left(1 + \frac{\exp(2h[Z])}{(2\pi e)^n}\right)^{-1}$$7 (the safe set).

To facilitate gradient-based and particle methods, the hinge is relaxed using a softplus surrogate, and the entire control problem is re-framed as Bayesian inference over the space of dynamically feasible controls, giving an unnormalized posterior:

$$q[Z] = \left(1 + \frac{\exp(2h[Z])}{(2\pi e)^n}\right)^{-1}$$8

where $$q[Z] = \left(1 + \frac{\exp(2h[Z])}{(2\pi e)^n}\right)^{-1}$$9 is the smooth surrogate of the time-integrated clarity deficit.

This distribution is approximated via Stein Variational Gradient Descent (SVGD), with $h[Z] = -\int_S \rho(z)\log\rho(z)\,dz$0 particles updated as:

$h[Z] = -\int_S \rho(z)\log\rho(z)\,dz$1

$h[Z] = -\int_S \rho(z)\log\rho(z)\,dz$2

where the log-posterior gradient is decomposable due to the smooth differentiability of $h[Z] = -\int_S \rho(z)\log\rho(z)\,dz$3. This approach enables backpropagation through the entire ODE trajectory, allowing continuous improvement of control policies with respect to clarity-centric objectives (Naveed et al., 13 Nov 2025).

4. Provable Safety via Gatekeeper-Based Trajectory Filtering

While SVGD trajectories are shaped by clarity objectives, trajectory safety with respect to obstacles requires explicit guarantees. The gatekeeper protocol operates at every replan step:

• For each SVGD particle and switch time $h[Z] = -\int_S \rho(z)\log\rho(z)\,dz$4, construct a candidate trajectory concatenating the nominal path up to $h[Z] = -\int_S \rho(z)\log\rho(z)\,dz$5 with a pre-computed, certified-safe backup (under a backup controller $h[Z] = -\int_S \rho(z)\log\rho(z)\,dz$6).
• Safety is validated if the nominal path stays within the admissible set $h[Z] = -\int_S \rho(z)\log\rho(z)\,dz$7 until $h[Z] = -\int_S \rho(z)\log\rho(z)\,dz$8, and the backup remains in a certified-safe subset $h[Z] = -\int_S \rho(z)\log\rho(z)\,dz$9 for its duration, finally reaching a region from which recursive safe operation is guaranteed.
• Among all safe candidates $n$0, select the one minimizing $n$1.
• If no safe candidate exists, execute the previously committed, safe trajectory.

This receding-horizon gatekeeper filtering strategy yields zero collision violations across 100 randomized trials in obstacle-rich environments, in contrast to ≈3% for soft-penalty SVGD alone. The approach is computationally efficient (adds ≈40 ms per step for safety checks) and proven to ensure $n$2 at all times (Theorem 1) (Naveed et al., 13 Nov 2025).

5. Experimental Demonstrations and Quantitative Benefits

Extensive hardware and simulation experiments validate the approach across heterogeneous environments:

• Environments vary in spatial process-noise fields, clarity targets, and the presence of obstacles.
• Case studies include zero-process-noise regimes (robots cease revisiting non-decaying regions once full clarity is reached), quadrants with selective decay (robots prioritize maintenance in decaying regions), and obstacle-rich scenarios (gatekeeper routes avoid all hazards).
• Comparisons show that the Stein variational planner with gating consistently achieves clarity deficits as low or lower than previous methods (lawnmower, ergodic, unconstrained SVGD) while providing strict safety guarantees.
• Gatekeeper adds minimal computational burden and is compatible with real-time hardware-in-the-loop operation (e.g., 6 s quadrotor horizon reevaluated every 100 ms at 70 ms/particle-update + 40 ms/gating overhead).
• No safety violations were observed for the gatekeeper-protected planner; 2.94% of trials resulted in violations without gating (Naveed et al., 13 Nov 2025).

6. Theoretical and Practical Significance

Clarity as a normalized, ODE-propagated information metric delivers several key theoretical and application benefits:

• It provides a bounded, unitless, physically meaningful state variable for informative planning, directly interpretable as a degree of certainty.
• Its ODE-based coupling with robot dynamics facilitates backpropagation of clarity gradients for end-to-end policy optimization.
• The explicit clarity dynamics capture both information gain (sensing) and unavoidable decay (process noise), supporting persistent, adaptive monitoring in stochastic fields.
• Gatekeeper-based control aligns informative exploration with provable, hard safety, a requirement in high-reliability autonomous systems.
• The approach is modular and extensible: clarity dynamics naturally interleave with Gaussian-process models, feedback coverage controllers, and ergodic exploration paradigms (Naveed et al., 13 Nov 2025, Agrawal et al., 2024, Naveed et al., 2023).

Clarity, thus, serves as both a rigorous quantitative abstraction of epistemic state and a practical vehicle for real-time, safe, information-driven autonomous exploration.