Dynamics-Aware Distance to Goal

• Dynamics-aware distance to goal is a metric that defines true reachability by accounting for system dynamics, control constraints, and obstacles.
• It employs mathematical formulations like temporal, action, and discounted-cost distances alongside supervised and self-supervised learning methods to approximate reachability.
• This approach improves planning efficiency and robust decision making in motion planning, goal-conditioned reinforcement learning, and model-based control.

A dynamics-aware distance to goal is a metric—learned or computed—that quantifies how difficult it is for an agent, under system dynamics, to reach a given goal from a current state. Unlike geometric distances, such metrics explicitly respect the system’s transition structure, control constraints, environment obstacles, and the agent’s intrinsic dynamics, yielding a measure that meaningfully reflects attainability. This concept is foundational in motion planning, goal-conditioned reinforcement learning, and embodied decision making, where naïve metrics such as Euclidean distance often fail to reflect true task difficulty or cost. Dynamics-aware distances have been developed in both classical robotic planning and modern machine learning, supporting sample efficiency, curriculum learning, robust high-dimensional control, and improved offline RL performance.

1. Formal Definitions and Mathematical Foundations

Dynamics-aware metrics formalize the notion of “distance” as the true or approximate cost, number of actions, or minimum time required to traverse the system’s state space from a source state $s$ to a target $g$, under system dynamics. Key mathematical definitions include:

• Discrete Minimum-Step Distance (Temporal Distance):

$$d^*(s, g) = \begin{cases} 0, & s = g \ 1 + \min_{a \in A} \ \mathbb{E}_{s'\sim p(\cdot|s,a)}[d^*(s', g)], & s \neq g \end{cases}$$

This Bellman recursion defines the minimal expected steps to reach $g$ from $s$ under the optimal policy in a goal-reaching Markov decision process (MDP) (Park et al., 8 Oct 2025).

• Action (Commute-Time) Distance:

$$d^\pi(s, g) = \tfrac{1}{2} m(g\,|\,s) + \tfrac{1}{2} m(s\,|\,g)$$

where $m(g|s)$ is the first-passage time from $s$ to $g$ under a specified policy $\pi$. This is a symmetric, triangle-inequality respecting metric (Venkattaramanujam et al., 2019).

• Expected Discounted-Cost Distance:

$g$0

Here, $g$1 is a step cost, $g$2 is a discount factor, and the expectation is over trajectories $g$3 conditioned on visiting $g$4 then $g$5 (Prakash et al., 2021).

• Latent Temporal Distance (Editor’s term):

In high-dimensional systems, autoencoders and neural networks are used to map states into a latent space and learn a function $g$6 approximating the minimal number of steps under the dynamics from $g$7 to $g$8 (Lee et al., 19 May 2025).

• Dual Goal Representations:

Encode each goal $g$9 by the vector $$d^*(s, g) = \begin{cases} 0, & s = g \ 1 + \min_{a \in A} \ \mathbb{E}_{s'\sim p(\cdot|s,a)}[d^*(s', g)], & s \neq g \end{cases}$$0, i.e., the set of temporal distances from all states to the goal (Park et al., 8 Oct 2025).

2. Learning and Estimation Methodologies

Given that true dynamics-aware distances are rarely available in closed-form except for simple or discretized systems, practical usage relies on learning approximations from data:

• Supervised Regression (Motion Planning):

In sampling-based multi-goal planners, offline datasets of start-goal pairs are processed with a motion planner to collect actual path lengths and planning runtimes. Regression models (e.g., XGBoost) are then trained to predict both path length $$d^*(s, g) = \begin{cases} 0, & s = g \ 1 + \min_{a \in A} \ \mathbb{E}_{s'\sim p(\cdot|s,a)}[d^*(s', g)], & s \neq g \end{cases}$$1 and runtime $$d^*(s, g) = \begin{cases} 0, & s = g \ 1 + \min_{a \in A} \ \mathbb{E}_{s'\sim p(\cdot|s,a)}[d^*(s', g)], & s \neq g \end{cases}$$2 as surrogates for real dynamics-aware distance (Lu et al., 26 Mar 2025).

• Self-Supervised Embedding Learning:

Neural embeddings $$d^*(s, g) = \begin{cases} 0, & s = g \ 1 + \min_{a \in A} \ \mathbb{E}_{s'\sim p(\cdot|s,a)}[d^*(s', g)], & s \neq g \end{cases}$$3 are trained such that $$d^*(s, g) = \begin{cases} 0, & s = g \ 1 + \min_{a \in A} \ \mathbb{E}_{s'\sim p(\cdot|s,a)}[d^*(s', g)], & s \neq g \end{cases}$$4 matches the empirically measured minimal action count (first-passage time) between state pairs, using on- or off-policy rollouts (Venkattaramanujam et al., 2019).

• Online and Offline Temporal Difference Learning:

Parameterized Q-networks are optimized to satisfy a Bellman equation where the value approximates $$d^*(s, g) = \begin{cases} 0, & s = g \ 1 + \min_{a \in A} \ \mathbb{E}_{s'\sim p(\cdot|s,a)}[d^*(s', g)], & s \neq g \end{cases}$$5, enabling the recovery of temporal distances via $$d^*(s, g) = \begin{cases} 0, & s = g \ 1 + \min_{a \in A} \ \mathbb{E}_{s'\sim p(\cdot|s,a)}[d^*(s', g)], & s \neq g \end{cases}$$6 (Tian et al., 2020).

• Discretized Classification via Binning:

Possible (step-based) distances are binned, and neural networks are trained as classifiers to predict distance intervals, using cross-entropy between model output and true bin labels (Prakash et al., 2021).

• Latent-Space Distance Regression:

Autoencoder-based pipelines map states to latent vectors; regression heads are trained to output minimal step-count (within-trajectory) distances between latent code pairs; triangle inequality can be partially enforced via triplet losses (Lee et al., 19 May 2025).

• Dual Head Representations:

Separate neural mappings for states and goals compute inner products $$d^*(s, g) = \begin{cases} 0, & s = g \ 1 + \min_{a \in A} \ \mathbb{E}_{s'\sim p(\cdot|s,a)}[d^*(s', g)], & s \neq g \end{cases}$$7, trained to match the optimal cost-to-go under temporal-difference and expectile losses (Park et al., 8 Oct 2025).

3. Integration into Planning, RL, and Decision Systems

Dynamics-aware goal distances serve key roles across planning and RL architectures:

• Sampling-Based Motion Planning:

Learned distances replace Euclidean or roadmap metrics in TSP-based tour ordering and local steering in multi-goal planners, with cost matrices $$d^*(s, g) = \begin{cases} 0, & s = g \ 1 + \min_{a \in A} \ \mathbb{E}_{s'\sim p(\cdot|s,a)}[d^*(s', g)], & s \neq g \end{cases}$$8 enabling priority expansion toward easier goal sets (Lu et al., 26 Mar 2025).

• Model-Based RL Planning:

In visual goal reaching, model-based planners use dynamics-aware Q-function estimates to score long-horizon plans generated via a learned video model, preferring actions that minimize the learned distance to goal (Tian et al., 2020).

• Offline Model-Based Augmentation:

Temporal distance heads guide where to focus model-based transition synthesis in the latent space for better data coverage in long-horizon goal-reaching scenarios; the distance determines which start-goal pairs are used to generate rollouts (Lee et al., 19 May 2025).

• Goal-Conditioned RL:

The learned distance functions assign sparse rewards as $$d^*(s, g) = \begin{cases} 0, & s = g \ 1 + \min_{a \in A} \ \mathbb{E}_{s'\sim p(\cdot|s,a)}[d^*(s', g)], & s \neq g \end{cases}$$9 when the agent nears the goal (distance below $g$0), and $g$1 otherwise, or are used as an intrinsic shaping term to drive policy learning (Venkattaramanujam et al., 2019, Tian et al., 2020).

• Automatic Curriculum Generation:

In multi-goal curriculum RL, the current state’s learned distances to past states are used to sample new goals of appropriate difficulty, facilitating progression from easily reached goals to challenging ones (Prakash et al., 2021).

• Representation Learning for Policy Input:

Downstream policies are conditioned not on raw goal coordinates or images, but on learned vectors encoding the temporal distance profile of the goal, yielding robustness and invariance to unrelated state noise (Park et al., 8 Oct 2025).

4. Empirical Findings and Performance Characteristics

Evaluation across domains reveals characteristic strengths and trade-offs of dynamics-aware goal metrics:

Study	Application Domain	Main Empirical Findings
(Lu et al., 26 Mar 2025)	Multi-goal motion planning	ML-based planners run $g$2–$g$3 faster, with 35–66% longer paths than strictly optimal, compared to non-ML baselines. Steers expansion efficiently in obstacle-rich spaces.
(Venkattaramanujam et al., 2019)	Goal-conditioned RL	Learned distances match or exceed hand-crafted $g$4 in both low- and high-dim (pixel) domains; robust to domain shifts when trained off-policy.
(Prakash et al., 2021)	Curriculum learning in RL	Automatically generated goal curricula based on DDF yield $g$5–$g$6 increased sample efficiency (steps to 80% success) over random goal sampling.
(Lee et al., 19 May 2025)	Offline model-based long-horizon RL	TempDATA achieves up to 20% return improvements on D4RL AntMaze; omitting temporal-distance supervision degrades performance.
(Park et al., 8 Oct 2025)	Dual-representation offline GCRL	Dual goal representations yield the best average success rates in both state- and pixel-based offline task suites; improved robustness to goal noise.
(Tian et al., 2020)	Visual goal-reaching (model-based RL)	Q-based distances combined with video models outperform VAE-distance and temporal regression baselines; essential for complex object manipulation with distractors.

All approaches observe that naïve Euclidean or latent-$g$7 distances yield poor planning and RL policies in the presence of obstacles, bottlenecks, or intricate dynamics, whereas dynamics-aware metrics drive substantially improved reachability and efficiency.

5. Representational and Theoretical Properties

Dynamics-aware distances exhibit several distinctive theoretical and representational traits:

• Dependence on System Dynamics: Such distances are defined exclusively by the MDP transition probabilities $g$8 and cost/reward functions, being invariant to how states are encoded (e.g., pixel, feature, graph, etc.) (Park et al., 8 Oct 2025).
• Robustness to Exogenous Noise: In settings with observation noise or irrelevant rendering artifacts, dual or embedding-based dynamics-aware metrics filter out such exogenous structure, remaining sufficient for optimal control (Park et al., 8 Oct 2025).
• Sufficiency for Optimal Policy Recovery: Policies that base their choices on minimizing expected dynamics-aware distance to the goal (via direct computation or dual representations) are guaranteed to recover the true optimal goal-conditioned policy under mild regularity (Park et al., 8 Oct 2025).
• Metric Properties: When computed via (symmetrized) first-passsage or commute times, the resulting metrics are symmetric, non-negative, and respect the triangle inequality (Venkattaramanujam et al., 2019).

6. Limitations, Open Questions, and Future Directions

While powerful, dynamics-aware goal distances have several limitations:

• Data and Computation Constraints: Accurate estimation often requires extensive exploration or batch data; early poor coverage biases regression/classification models (Prakash et al., 2021, Lee et al., 19 May 2025).
• Generalization Limits: ML-based surrogate models (e.g., $g$9, $s$0 in (Lu et al., 26 Mar 2025)) are tied to a fixed environment geometry and do not generalize out-of-distribution, necessitating retraining if the system or obstacle layout changes (Lu et al., 26 Mar 2025).
• Resolution Granularity: Discretized classifiers lose fine-grained distance information; regression is preferred for continuous domains but may be less stable (Prakash et al., 2021).
• Lack of Theoretical Admissibility: Most learned surrogates are not guaranteed to be admissible heuristics, so optimality is not preserved; however, they do steer expansion toward tractable regions (Lu et al., 26 Mar 2025).
• Structural Representations: Most approaches focus on scalar (step-count or cost) distances; predicting higher-order trajectory/primitive structure remains underexplored (Lu et al., 26 Mar 2025).

Future work is directed at continuous, uncertainty-aware distance regressors, generalization to variable environments, principled trajectory-structure learning, and adaptive curricula that exploit model uncertainty or zone-of-proximal-development (Prakash et al., 2021, Lu et al., 26 Mar 2025, Lee et al., 19 May 2025).

7. Applications Across Domains

The dynamics-aware distance to goal paradigm has been instantiated in diverse applications:

• Sampling-based robotics and TSP-based multi-goal planners in obstacle-dense environments (Lu et al., 26 Mar 2025)
• Goal-conditioned reinforcement learning in continuous, high-dimensional, and pixel-based control domains (Venkattaramanujam et al., 2019, Prakash et al., 2021, Lee et al., 19 May 2025, Park et al., 8 Oct 2025)
• Model-based visual planning and manipulation for robotic arms and real-world object interaction (Tian et al., 2020)
• Physical reasoning and simulation-based task evaluation using rollout-similarity metrics (Ahmed et al., 2021)

The common principle is to improve both efficiency and efficacy of agent behavior by replacing geometric or arbitrary proximity notions with metrics congruent with the system’s true control and transition structure.

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In summary, dynamics-aware distances to goal transform the landscape of planning and control by grounding attainability and efficiency in the physics and logic of environment transitions. They operationalize a notion of “what is truly close” under constraints, serving as a critical ingredient for scalable, robust, goal-directed learning in robotics, RL, vision, and beyond (Lu et al., 26 Mar 2025, Venkattaramanujam et al., 2019, Prakash et al., 2021, Lee et al., 19 May 2025, Park et al., 8 Oct 2025, Tian et al., 2020, Ahmed et al., 2021).