Polynomial Navigation Functions

• Polynomial navigation functions are polynomial potential fields that encode 3-D workspace geometry and obstacles using quadratic implicit functions.
• They combine attractive quadratic potentials with repulsive formulations to guarantee a unique, nondegenerate global minimum at the target while avoiding spurious local minima.
• Empirical studies confirm that tuning the gain parameter k localizes gradients, enabling reliable navigation in complex environments including intersecting obstacles.

Polynomial navigation functions are a class of artificial potential functions designed for feedback-based motion planning and collision avoidance in three-dimensional (3-D) workspaces populated with obstacles. These functions encode both workspace geometry and obstacle regions using polynomial (specifically quadratic) implicit functions and generate navigation fields whose gradient flow directs a mobile agent towards a desired goal while provably ensuring obstacle avoidance. When constructed with sufficient gain parameters and appropriate obstacle compositions, these polynomials guarantee the absence of spurious local minima in the free configuration space, including scenarios with pairwise intersecting obstacles, and admit unique, nondegenerate global minima precisely at the goal position (Lang et al., 14 Jan 2026).

1. Workspace and Obstacle Encodings

Polynomial navigation functions are typically implemented in a 3-D Euclidean domain $E^3$. The workspace is modeled as a closed ball of radius $r_0$ centered at the origin, represented by the quadratic (thus polynomial) implicit function

$$\mathcal W = \{\,x\in\mathbb R^3 : \|x\|^2 \le r_0^2\,\}, \qquad \beta_0(x) = r_0^2 - \|x\|^2.$$

Internal obstacles are represented using two parametric quadratic forms:

• Spherical obstacles: Each has center $p_i$ and radius $r_i$, encoded as

$\beta_{i,s}(x) = \|x - p_i\|^2 - r_i^2.$

• Capped cylindrical obstacles: Each consists of a cylindrical core of radius $r_i$ with axis direction $\hat v_i$ and segment endpoints $p_{i,1}, p_{i,2}$, encoded by

$$\beta_{i,c}(x) = \| \hat v_i \times (x - p_{i,1}) \|^2 - r_i^2$$

with additional hemispherical end-caps defined by a continuously differentiable, piecewise polynomial

$$\beta_{i,cc}(x) = \begin{cases} \| \hat v_i \times (x-p_{i,1}) \|^2 - r_i^2, & \text{if } (\hat v_i\cdot(x-p_{i,1}))\,(\hat v_i\cdot(x-p_{i,2})) < 0, \ \|x-p_{i,1}\|^2 - r_i^2, & \text{if } \hat v_i\cdot(x-p_{i,1}) \le 0 \wedge \hat v_i\cdot(x-p_{i,2}) \le 0, \ \|x-p_{i,2}\|^2 - r_i^2, & \text{if } \hat v_i\cdot(x-p_{i,1}) \ge 0 \wedge \hat v_i\cdot(x-p_{i,2}) \ge 0. \end{cases}$$

The free space is defined as

$$\mathcal F = \{x \in \mathcal W: \beta_1(x) > 0, \dots, \beta_m(x) > 0 \},$$

where

$\beta(x) = \prod_{i=0}^m \beta_i(x).$

2. Construction of Polynomial Navigation Functions

Given a target point $p_d$, the attractive potential is modeled as a quadratic form: $\gamma_d(x) = \|x - p_d\|^2.$

The base polynomial navigation function is constructed as

$$\hat\varphi(x) = \frac{(\gamma_d(x))^k}{\beta(x)}, \qquad k \in \mathbb N.$$

This form realizes a combined attractive-repulsive artificial potential, where repulsion from obstacles and workspace boundaries is polynomial in nature and multiplicative across all obstacles. Two normalized variants are frequently used in simulations: $$\varphi(x) = \frac{\gamma_d(x)}{\left(\gamma_d(x)^k + \beta(x)\right)^{1/k}}, \qquad \psi(x) = \frac{\gamma_d(x)}{\gamma_d(x) + \beta(x)^{1/k}}.$$

The gain parameter $k$ regulates the flatness and gradients: large $k$ values localize all non-minimal critical points near obstacle or boundary surfaces, suppressing spurious minima within the workspace interior.

3. Theoretical Guarantees and Analysis

The central theorem establishes that, provided obstacle boundaries are encoded as above, no more than pairwise intersections occur among obstacles (no triple intersections), and the exponent $k$ is chosen suitably large (dependent on minimal clearance $\epsilon$), the constructed polynomial navigation functions have the following rigorous properties:

• Smoothness on $\mathcal F$ and continuity up to $\partial\mathcal F$.
• A unique, nondegenerate global minimum located at $x = p_d$.
• Attainment of a constant maximal value (either $+\infty$ for $\hat\varphi$, or $1$ for normalized forms) precisely on $\partial\mathcal F$.
• All additional critical points (if any) in $\mathcal F$ are nondegenerate saddles.

The proof consists of partitioning $\mathcal F$ into regions near each obstacle, near the workspace boundary, and away from obstacles, determining that potential non-minimal critical points are forced toward boundaries as $k$ increases. Hessian analysis is used to establish that such boundary-adjacent critical points are always saddle points.

For intersecting obstacles, a p-Rvachev smooth composition is employed: $\beta_{ij} = R_p(\beta_i, \beta_j)$ ensuring the merged implicit function $\beta_{ij} = 0$ on the union boundary with gradient nonvanishing throughout, thus avoiding local minima on intersection curves via similar Hessian arguments.

4. Computation of Gradients and Hessians

Letting $\nu(x) = \gamma_d(x)^k$ and $\delta(x) = \beta(x)$, derivatives are given by

$$\nabla\hat\varphi = \frac{1}{\delta^2}\left( \delta\,\nabla\nu - \nu\,\nabla\delta \right), \qquad D^2\hat\varphi|_{\nabla\hat\varphi=0} = \frac{1}{\delta^2}\left(\delta\,D^2\nu - \nu\,D^2\delta \right).$$

For the attractive component,

$$\nabla\gamma_d(x) = 2(x - p_d), \qquad D^2\gamma_d(x) = 2I.$$

The product structure yields

$$\nabla\beta(x) = \sum_{i=0}^m \left( \nabla\beta_i \cdot \prod_{j \ne i} \beta_j \right),$$

with explicit closed-form expressions for both spheres and cylinders. This structure ensures that as any obstacle or boundary is approached ($\beta_i(x) \rightarrow 0^+$), $\hat\varphi(x) \rightarrow +\infty$, guaranteeing that the gradient descent dynamics $\dot{x} = -\nabla\hat\varphi(x)$ forces solutions to remain in free space and converge to the global minimum at $p_d$.

5. Empirical Validation and Numerical Results

Extensive numerical simulations confirm the theoretical properties of polynomial navigation functions:

• General random tests: With $r_0 = 5$ m, $m = 10$ mixed obstacles, 10 targets, and 15 random starting conditions per target, the normalized navigation function $\psi$ (with damping $c=0.6$) resulted in all 150 runs converging successfully for $k=11$, while $k=9$ yielded three parasitic local minima.
• Truss obstacle composite: For a “truss” of 36 assembled obstacles and 100 initial positions near the workspace boundary, using $k=40$, all trajectories converged to the goal safely, with maximal speed $<0.4$ m/s and acceleration $<0.24$ m/s$^2$.
• Obstacle merging employing p-Rvachev composition reduced effective obstacle count (e.g., 36 to 1), allowing substantially lower $k$ (such as $k=10$ or even $k=2$) without reintroducing spurious local minima.

These results demonstrate robust performance: monotonic convergence values for $\psi$, effective collision avoidance, and consistent arrival at the minimum (goal) within reasonable time frames (Lang et al., 14 Jan 2026).

6. Methods for Handling Intersecting Obstacles

A significant challenge arises when obstacles intersect. Polynomial navigation frameworks permit pairwise intersecting obstacle encoding with no triple intersections by merging intersecting obstacles via the p-Rvachev composition, yielding smoothly defined implicit functions with nonvanishing gradients on the resulting union boundary. The Hessian-based argument, as applied to single obstacle boundaries, extends to these merged regions, precluding the appearance of local minima on intersection curves. This feature materially improves scalability and applicability to complex environments.

7. Summary and Scope of Application

Polynomial navigation functions, as constructed with quadratic encodings for both spherical and capped cylindrical obstacles, and evaluated with sufficient exponent $k$, constitute a rigorous, smoothly varying feedback law for 3-D robot navigation. The approach guarantees a unique, nondegenerate convergence at the goal, free from interior local minima even under pairwise obstacle intersection, and supports systematic collision avoidance in highly cluttered environments. Applicability for robotic motion planning is substantiated by theoretical proof and extensive simulation in settings of increasing geometric and combinatorial complexity. Obstacle merging via p-Rvachev further extends the method’s scalability, and parameter tuning allows for graceful performance and convergence control (Lang et al., 14 Jan 2026).