Customized Potential Field Optimization

• Potential Field Optimization refers to techniques for designing potential fields that steer robotic paths safely and efficiently, defined by harmonic formulations and customizable topological constraints.
• Hybrid optimization combines discrete structural search of workspace obstacles with continuous weight refinement to ensure the navigation path exhibits a desired homotopy signature.
• Empirical results in simulations and real-world deployments confirm that this method achieves globally convergent, collision-free navigation in complex, cluttered environments.

Potential field optimization encompasses a spectrum of analytical, algorithmic, and computational techniques for constructing or selecting potential fields so as to optimize specific physical, geometric, or topological objectives. In robot navigation, harmonic potential fields are widely used owing to their analytical form, global convergence properties, and absence of spurious local minima. However, classical harmonic fields are not customizable with respect to the topological or homotopic properties of the resulting navigation paths. Recent advances, notably "Customize Harmonic Potential Fields via Hybrid Optimization over Homotopic Paths" (Wang et al., 14 Jul 2025), address this limitation by introducing a rigorous framework for generating harmonic potential fields whose induced gradient-descent trajectories exhibit desired homotopic characteristics—even in complex, highly cluttered workspaces.

1. Harmonic Potential Fields: Mathematical Structure and Safety

Harmonic potential fields are constructed as solutions to Laplace’s equation, ensuring analyticity and the absence of non-global critical points except possibly for a set of saddle points of measure zero. The standard formulation in the transformed "point world" domain is

$$\varphi_p(q; w) = w_g \cdot \varphi(q, q_g) - \sum_{i=1}^M w_i \cdot \varphi(q, P_i), \quad \varphi(q, P) = \log(\|q - P\|^2)$$

where $q \in \mathbb{R}^2$ denotes the robot state, $q_g$ is the goal, $P_i$ are (transformed) point obstacles, and $w = [w_g, w_1, ..., w_M]$ are positive weights. This potential can be composed with a logistic map $\sigma(\cdot)$ to ensure boundedness while preserving the key structural properties. The negative gradient field, $-\nabla \varphi_p$, underpins robust, collision-free, globally convergent navigational behavior.

2. Hybrid Optimization: Structural and Parametric Degrees of Freedom

Unlike standard implementations, the cited approach introduces a hybrid optimization paradigm that customizes the harmonic field with respect to user-specified topological constraints on trajectories. This is achieved through:

• Discrete structural search: The workspace is initially represented as a "forest world", where obstacles are grouped into forest-like structures, each being a tree of star-shaped (squircle) obstacles with bounded depth. Structural optimization iterates over possible tree rootings and groupings.
• Continuous parameter refinement: With a forest structure fixed, the problem is mapped via a diffeomorphic transformation to a "point world", and the weights $w$ in the harmonic field are optimized using projected gradient descent to yield gradient-descent paths with target topological signatures.

Formally, the continuous parameter optimization solves

$$\min_{w \in \mathcal{W}} \| D(\tilde{\tau}(t, w)) - D^* \|$$

subject to $\mathcal{W} = \{w > 0 \mid w_g > \sum_i w_i + 1 \}$, where $D(\cdot)$ is the multi-directional D-signature (see §3), $\tilde{\tau}(t, w)$ is the trajectory induced by $-\nabla\varphi_p$, and $D^*$ is the user-specified desired signature.

3. Topological Path Signatures and Homotopy Classes

The central novelty is the explicit characterization, manipulation, and optimization of path homotopy. In the transformed point world, homotopy (and, in computational practice, homology) classes of paths are classified via "D-signatures." This signature encodes both positional and directional information with respect to obstacles:

$$D(\tilde{\tau}) = S(\tilde{\tau}) \odot \bar{d}(\tilde{\tau})$$

where $S(\tilde{\tau}) \in \{+1, -1\}^M$ encodes on which side of each point obstacle the path passes (partitioned via a circumscribed circle and angular sectors) and $\bar{d}(\tilde{\tau})$ records the minimum signed distance to each obstacle along the path. The D-signature is essentially invariant under the diffeomorphic transformation from the original workspace, so matching D-signatures is equivalent (in almost all cases) to matching homotopy classes in the original space. The hybrid algorithm thus allows the designer to explicitly select a desired topological class (e.g., to pass between certain obstacles or avoid certain regions) and then optimizes the harmonic field to realize it.

4. Diffeomorphic Transformations: Simplifying Workspace Topology

Navigation in cluttered, non-convex, or multiply connected environments is rendered tractable by transforming the "forest world" (composed of trees of star-shaped obstacles, possibly with severe overlap) into a point world, where obstacles are isolated. Via a well-defined, invertible diffeomorphism, the geometrically and topologically complex workspace is collapsed onto an analytically manageable domain. Crucially, the transformation preserves the homology and (almost always) homotopy invariants of paths, as demonstrated in Lemma 1 of (Wang et al., 14 Jul 2025). This simplifies both computation of field topology and gradient-based optimization, while ensuring that the final solution, mapped back to the original world, remains safe and topologically correct.

5. Algorithmic Implementation: Optimization Procedure

The hybrid optimization proceeds iteratively:

1. Forest Structural Optimization: Candidate obstacle groupings and tree rootings are generated, and their transformations to the point world are constructed.
2. Continuous Weight Optimization: For each candidate, projected gradient descent with backtracking line search is used to minimize the $\ell_2$ distance to the user-specified D-signature:

$$w_{k+1}' = w_k - \delta_k (D(\tilde{\tau}(t, w_k)) - D^*)^\top \left(\frac{\partial D}{\partial w_k}\right)$$

followed by projection onto the feasible region $\mathcal{W}$.

1. Selection: The configuration and weight set yielding the trajectory whose D-signature most closely matches $D^*$ is selected.

This approach leverages the invariance of topological properties through diffeomorphisms, together with numerical estimation of $\partial D/\partial w$ (as the paths are found by numerical integration rather than in closed form).

6. Empirical Validation: Simulation and Hardware Experiments

Extensive simulations demonstrate the algorithm's efficacy in highly nontrivial environments:

• In polygonal workspaces ($10\,\text{m} \times 10\,\text{m}$) with up to three-level depth trees of squircles as obstacles, the hybrid optimizer systematically enumerates and tests forest structures, ultimately discovering solutions in the prescribed homotopy class.
• In office-like environments ($15\,\text{m} \times 10\,\text{m}$) with complex arrangements of walls, tables, and other obstacles, the system detects and explores over 40 distinct homotopy classes, finally selecting navigation paths according to user criteria.
• The resultant field and path are deployed on differential-drive robots equipped with laser sensors; after real-time SLAM-based mapping and harmonic field computation, the robots navigate reliably and safely, as verified by goal convergence and collision-free operation.

Empirical observations show that major changes in the D-signature correspond to topological transitions (e.g., when the path nears a potential saddle point), confirming the effectiveness of discrete-continuous hybrid search in handling multimodal, highly nonconvex search spaces.

7. Implications and Extensions

The presented framework establishes a route toward fully customizable, analytically robust, and topologically flexible potential field navigation in arbitrary planar environments, without sacrificing classical properties such as global convergence and local-minima-freeness. By translating workspace topology to the point world, and by separating discrete (forest/tree structure) and continuous (weights) components, the method scales to environments of substantial complexity and supports precise control over the topological class of the solution.

Potential applications include:

• Service robotics (e.g., specifying cleaning or delivery paths to include/exclude certain areas)
• Search and rescue (ensuring traversal through or around specific regions)
• Task-specific autonomous navigation (e.g., maintenance robots required to visit subsets of zones in a prescribed sequence)

The construction supports further generalization, such as incorporating task or reward constraints, adapting to dynamic environments (via incremental reoptimization), or extending to higher-dimensional configuration spaces.

Table: Summary of Critical Methodological Steps

Step	Purpose	Technical Mechanism
Forest Structuring	Encode workspace topology	Enumerate candidate trees-of-stars, map via diffeomorphism
Weight Optimization	Customize trajectory	Projected gradient descent on weights in point world
D-signature Matching	Enforce homotopy class	Minimize $\ell_2$ distance to desired D-signature
Field Evaluation	Deploy on robot	Navigate via $-\nabla\varphi_F$ mapped back to workspace

Conclusion

Hybrid optimization of harmonic potential fields over homotopic paths, grounded by diffeomorphic domain transformation and explicit topological control, enables generation of safe, analytically well-conditioned, and path-customizable navigation fields in complex environments (Wang et al., 14 Jul 2025). This methodological advance bridges the gap between global, nonlocal geometric constraints and local, smooth optimization, facilitating the next generation of navigation capabilities in autonomous systems.