Effective Sampling for Robot Motion Planning Through the Lens of Lattices

TL;DR

The paper addresses finite-time guarantees for motion-planning under sampling-based methods by translating lattice geometry into deterministic $(\delta,\varepsilon)$-complete sample sets. It leverages the $A_d^*$ lattice to construct highly efficient coverings, derives explicit sample-complexity and collision-check bounds, and demonstrates substantial practical speedups over both deterministic grids and random sampling in high-dimensional scenarios. The key contributions include a general lattice-to-sample transformation, tight bounds on sample and collision-check complexity, and an implicit-A* planning framework that exploits lattice regularity for batch-style planning. The results indicate that $A_d^*$-based sampling dramatically reduces computation while maintaining comparable solution quality, enhancing the practical applicability of deterministic sampling in complex robotics tasks.

Abstract

Sampling-based methods for motion planning, which capture the structure of the robot's free space via (typically random) sampling, have gained popularity due to their scalability, simplicity, and for offering global guarantees, such as probabilistic completeness and asymptotic optimality. Unfortunately, the practicality of those guarantees remains limited as they do not provide insights into the behavior of motion planners for a finite number of samples (i.e., a finite running time). In this work, we harness lattice theory and the concept of $(δ,ε)$-completeness by Tsao et al. (2020) to construct deterministic sample sets that endow their planners with strong finite-time guarantees while minimizing running time. In particular, we introduce a highly-efficient deterministic sampling approach based on the $A_d^*$ lattice, which is the best-known geometric covering in dimensions $\leq 21$. Using our new sampling approach, we obtain at least an order-of-magnitude speedup over existing deterministic and uniform random sampling methods for complex motion-planning problems. Overall, our work provides deep mathematical insights while advancing the practical applicability of sampling-based motion planning.

Figures (10)

• Figure 1: Sample sets within a fixed disc in $\mathbb{R}^2$, derived from the lattices $\mathbb{Z}^2, D_2^*$ and $A^*_2$, which yield $(\delta,\varepsilon)$-complete guarantees for the same values of $\delta$ and $\varepsilon$. The set $\mathcal{X}_{\mathbb{Z}_2}^{\delta,\varepsilon}$ can be viewed as a tessellation of space using cubes. The set $\mathcal{X}_{D_2^*}^{\delta,\varepsilon}$ is obtained by placing a (rescaled) standard grid, and then placing another point in the middle of each cube. The set $\mathcal{X}_{A_2^*}^{\delta,\varepsilon}$ can be viewed as a rescaled hexagonal grid as each point is surrounded by a hexagon whose vertices are points in the set. Note that the density of $\mathcal{X}_{\mathbb{Z}^2}^{\delta,\varepsilon}$ and $\mathcal{X}_{D^*_2}^{\delta,\varepsilon}$ is the same, and higher than the density of $\mathcal{X}_{A^*_2}^{\delta,\varepsilon}$.
• Figure 2: $(\delta,\varepsilon)$-complete sample sets in $\mathbb{R}^3$ derived from the lattices $\mathbb{Z}^3, D_3^*$ and $A^*_3$. Note the sets $\mathcal{X}_{D_d^*}^{\delta,\varepsilon},\mathcal{X}_{A_d^*}^{\delta,\varepsilon}$ coincide for $d=3$, and diverge for $d\geq 4$. Note that the density of $\mathcal{X}_{D^*_3}^{\delta,\varepsilon}$ and $\mathcal{X}_{A^*_3}^{\delta,\varepsilon}$ (also known as the Body-Centered Cubic structure in crystallography), and is lower than the density of $\mathcal{X}_{\mathbb{Z}^3}^{\delta,\varepsilon}$.
• Figure 3: Visualization of embedding the lattice $A_2^*$ originally defined in $\mathbb{R}^3$ onto $\mathbb{R}^2$ via the mapping $T$. The blue rectangle represents the plane $H$, where the corresponding $A_2^*$ lattice points are drawn in red. The points are generated by taking integer vectors in $\mathbb{R}^d$ and applying the mapping $G^t$. $H$ and $A_2^*$ is reflected onto the plane $H_0=\{x_3=0\}$ using the mapping $PG^t$ (denoted by the green rectangle). The third dimension is removed via the mapping $E$ to yield the embedding of $A_2^*$ in $\mathbb{R}^2$.
• Figure 4: A sample-complexity plot for the sample sets $\mathcal{X}_{\mathbb{Z}^d}^{\delta,\varepsilon},\mathcal{X}_{D_d^*}^{\delta,\varepsilon}$, and $\mathcal{X}_{A_d^*}^{\delta,\varepsilon}$ with $\delta=1$ and $\varepsilon=2$. The dashed line represents the theoretical approximation (\ref{['eq:sample bounds']}), where the asymptotic error term $P_d$ is excluded. The solid line depicts the practical value, i.e., the number of lattice points within the $r^*$-ball in practice. Missing values are due to memory limitations.
• Figure 5: A collision-check complexity plot for the sample sets $\mathcal{X}_{\mathbb{Z}^d}^{\delta,\varepsilon},\mathcal{X}_{D_d^*}^{\delta,\varepsilon}$, and $\mathcal{X}_{A_d^*}^{\delta,\varepsilon}$ with $\delta=1$ and $\varepsilon=2$. The dashed line represents the theoretical approximation (\ref{['eq:CC_thm']}), where the asymptotic error term $P_d$ is excluded.

Theorems & Definitions (15)

• Definition 1: ($\delta,\varepsilon$)-completeness tsao2020sample
• Definition 2
• Lemma 1: Completeness-cover relation tsao2020sample
• Definition 3: Lattice
• Definition 4: Lattice generator
• Definition 5: $\mathbb{Z}^d$ lattice
• Definition 6: $D_d^*$ lattice
• Definition 7: $A^*_d$ lattice
• Definition 8
• Theorem 1