Force Direction Informed Trees (FDIT*)

• Force Direction Informed Trees (FDIT*) is a sampling-based motion planning algorithm that uses physical force dynamics to guide exploration in high-dimensional configuration spaces.
• It integrates Coulomb’s law and adaptive elliptical k-nearest neighbor queries to leverage invalid vertex repulsion and valid vertex attraction for efficient pathfinding.
• Empirical benchmarks show that FDIT* achieves faster convergence and lower solution costs in complex, narrow, and cluttered environments compared to earlier methods.

Force Direction Informed Trees (FDIT*) are a class of sampling-based motion planning algorithms designed to optimize exploration and pathfinding in high-dimensional configuration spaces. FDIT* distinguishes itself by integrating physical force concepts—particularly Coulomb’s law—and by leveraging information from invalid (collision) vertices to direct exploration toward more feasible, lower-cost paths. It extends earlier frameworks such as Effort Informed Trees (EIT*) and Direction Informed Trees (DIT*) through force-directional neighbor selection and adaptive, elliptical search regions, yielding improved performance in both simulated and real-world environments (Zhang et al., 27 Aug 2025).

1. Foundational Principles of FDIT*

FDIT* is predicated on two core principles: the physical modeling of sampler interactions using concepts from Coulomb’s law and the explicit utilization of obstacle (invalid) vertex information. Unlike standard motion planners which disregard vertices falling in obstacles, FDIT* assigns "charges" to such samples, allowing them to exert repulsive influence and bias the search away from congested areas. Simultaneously, valid vertices exert attractive forces, guiding paths toward the goal region.

The algorithm replaces the traditional spherical k-nearest neighbor (KNN) query with an adaptive elliptical KNN method. This elliptical region adapts dynamically, weighing dimensions in accordance with the net force vector induced by both valid and invalid vertices. Thus, FDIT* ensures that the sampling distribution and tree growth reflect the instantaneous force landscape in the configuration space.

2. Mathematical Framework and Algorithmic Mechanics

The key mathematical construct in FDIT* is the extension of Coulomb’s law to high-dimensional configuration spaces:

$\vec{F} := k_e \frac{q_1q_2}{r^{n-1}} \hat{r}$

where $\vec{F}$ is the force vector, $k_e$ is a constant, $q_1$ and $q_2$ are charges associated with vertices (positive for valid, negative for invalid), $r$ is the Euclidean distance, $n$ is the space dimension, and $\hat{r}$ is the unit vector.

Elliptical KNN queries are computed using a weighted metric:

$$d_\mathrm{elliptical}(x, y) = \sqrt{ \sum_{i=1}^n \left( \frac{x_i - y_i}{v_i} \right)^2 }$$

where the scaling factors $v_i$ are dynamically set based on the magnitude of force contributions per dimension. Charges from invalid vertices—found in obstacles—are aggregated and used to update the force direction and search region per iteration. Algorithmically, each sampling and neighbor selection step considers the aggregated Coulomb force, and tree expansion is biased toward regions of lower repulsion and higher attraction.

3. Role of Invalid Vertices and Directional Exploration

FDIT* uniquely leverages information from invalid vertices. Rather than discarding these collision-prone samples, the algorithm assigns them significant repulsive "charge." This charge:

• Alters the resultant net force vector $\vec{F}_\mathcal{D}$ which guides search direction
• Expands or contracts the axes of elliptical KNN regions to steer exploration away from high-density obstacles
• Allows dynamic adaptation to narrow passages and confined corridors by increasing search focus in "search-worthy" areas

By fusing invalid vertex data with physical dynamics, FDIT* enables a more effective avoidance of infeasible or high-cost solution paths and improves convergence rates compared to methods using uninformed or purely geometric neighbor selection (Zhang et al., 27 Aug 2025).

4. Comparative Perspectives: EIT* and DIT*

Effort Informed Trees (EIT*) employ heuristics based on computational effort, while Direction Informed Trees (DIT*) introduce goal-directedness via vector alignment and direction cost heuristics (Zhang et al., 26 Aug 2025). FDIT* surpasses these by integrating force-based heuristics and expanding on DIT*’s directional cost framework.

• In EIT*, neighbor selection is purely geometric or effort-informed.
• In DIT*, direction cost is computed from vector cosine similarity and sigmoid-scaled penalties.
• In FDIT*, directional heuristics incorporate both attractive and repulsive physical force fields, with mathematically formal neighbor selection and pruning based on net force direction and magnitude.

Empirical evaluations consistently show higher search efficiency and faster initial solutions for FDIT*, especially as configuration dimensionality increases (e.g., up to ℝ¹⁶), and in environments with narrow passages (Zhang et al., 27 Aug 2025).

5. Experimental Validation and Performance Benchmarks

FDIT* has been benchmarked against state-of-the-art planners including RRT, BIT*, AIT*, ABIT*, and EIT*, in both synthetic and real-world settings. In simulations featuring "Dividing Wall-gaps" and "Random Rectangles" environments, FDIT* demonstrates:

• Up to a 34.65% reduction in initial solution cost versus EIT* and other planners
• Fast convergence even in complex, high-dimensional spaces (ℝ⁴ to ℝ¹⁶)
• Superior adaptability in narrow or cluttered regions due to force-directed neighbor selection

On the DARKO robot for container-shelf arrangement tasks with strict (≤5mm) collision tolerances, FDIT* achieves an 80% success rate with reduced planning times and solution costs.

6. Synergies with Force-Informed Action Learning

Research in dexterous manipulation, particularly DexForce (Chen et al., 17 Jan 2025), provides insight into the necessity of force direction and magnitude in manipulation policy learning. DexForce demonstrates that integrating force-informed actions—where targets are augmented by measured contact forces ($x_f = x_o + K_f f$)—is essential for task completion.

A plausible implication is that FDIT*’s framework, centered on dynamic force-based exploration, is well aligned with force-informed policy learning approaches:

• In manipulation domains where both pathfinding and action generation depend on precise force application, FDIT*’s tree-based search may benefit from incorporating force-informed demonstrations as priors or from using force data to guide tree branching and pruning.
• This synergy could lead to planners that not only avoid obstacles efficiently but also respect physical interaction constraints, enhancing both safety and performance in contact-rich environments.

7. Future Research and Methodological Extensions

Several future directions are identified:

• Variable charging schemes: Rather than equal charges for vertices, adaptive charging based on obstacle properties, proximity, or local topology may further enhance planning efficiency.
• Integration with subset optimization techniques to address scalability in higher dimensions or with increasingly complex constraints.
• Applications in bio-inspired robotics, where physical interaction models reflect biological force control strategies.
• Further combining force-directional planning with multimodal sensory feedback, for instance, integrating force, vision, and proprioception data for task generalization and robustness.
• Considerations for human acceptability and ergonomic factors in collaborative path planning.

FDIT* stands as a notable advancement in motion planning, fusing physical force models and invalid vertex information for improved high-dimensional search. Its successful deployment in both simulated and real scenarios confirms its efficacy and marks it as an influential framework for future robotics and automation research.