Spatial Distance Constraint

• Spatial Distance Constraint is a formal condition that enforces required separations between geometric objects via Minkowski operations.
• It models minimum and maximum distance requirements in configuration space to determine feasible regions for object placement or motion.
• Applications in robotics, CAD, packaging, and manufacturing utilize these constraints to ensure collision-free and optimized spatial arrangements.

A spatial distance constraint is a formal condition placed on the relative positions of geometric objects, ensuring specified separations or non-overlapping between them within a space. Such constraints are foundational in spatial planning tasks across robotics, CAD, manufacturing, and packaging, as they mathematically encode requirements like minimum or maximum allowable distances, directional separations, or configuration restrictions for geometric objects. Spatial distance constraints are systematically modeled and computed using Minkowski operations on the configuration space, enabling the derivation of feasible regions for spatial arrangement and motion.

1. Mathematical Foundations: Configuration Space and Translational Distance Constraints

Spatial planning addresses the problem of determining feasible placements or motions of a movable object $B$ relative to a fixed object or obstacles $A$, formalized within a configuration space (C-space). In this context, each possible configuration (e.g., translation, rotation) of $B$ corresponds to a unique point in a typically Euclidean space $\mathbb{R}^n$, where $n$ is 2 or 3 for common applications.

The essential translational distance constraint, commonly the non-overlap or separation condition, is formulated as: $$d_1(B + p, A) = \inf_{a \in A,\, b \in B} \| (b + p) - a \| > 0,$$ where $p$ is a translation vector. Enforcing non-overlap translates to the requirement that

$$(B + p) \cap A = \emptyset \quad \Leftrightarrow \quad p \notin A \oplus (-B),$$

with $-B$ denoting the reflection of $B$.

General spatial distance constraints may impose bounds—not only on the minimum but also on the maximum separation, along specific directions, or with respect to rotational or dynamic spacing (see Section 5).

2. Minkowski Operations and Constraint Encoding

Minkowski operations provide the primary analytic tool to map geometric constraints into C-space regions:

• Minkowski Sum: $A \oplus B = \{a + b \mid a \in A,\, b \in B\}$
• Minkowski Difference: $A \ominus B = \{x \mid x + B \subseteq A\}$

For a minimum distance constraint $\delta > 0$, the forbidden configuration region becomes: $p \in A \oplus (-B) \oplus K_\delta,$ where $K_\delta$ is a ball of radius $\delta$. The feasible region is its complement.

Arbitrary composite constraints are formed by Boolean combinations of distance conditions: $$\nu_I(B+p, A) = ( [d_1(B+p, A) \leq \delta_1] \odot_1 [d_1(B+p, A) \geq \delta_2] ) \odot_2 \cdots,$$ with $\odot_i$ as logical connectives (AND/OR).

Parametric families—such as parallel pencils $(_1(\epsilon, K, A) = A \oplus K$ for offset, or $A \ominus K$ for erosion)—flexibly represent requirements like containment, minimum/maximum separation, and more.

3. Construction of Feasible Regions

Given spatial constraint $\nu$, the set of admissible translations is: $$N(B, A) = \{ p \in \mathbb{R}^n \mid \nu(B + p, A) \text{ holds} \}$$ with each basic or composite constraint mapped to a specific C-space region through corresponding Minkowski operations.

Examples:

• Minimum separation $\delta$:

$N(B, A) = [A \oplus (-B) \oplus K_\delta]^c$

• Annular feasible regions (minimum $\delta_1$, maximum $\delta_2$):

$$N(B, A) = [A \oplus (-B) \oplus K_{\delta_2}] \setminus [A \oplus (-B) \oplus K_{\delta_1}]$$

When multiple obstacles $\mathcal{A} = \{A_1, \dotsc, A_n\}$ and a region $R$ are present, the feasible region is computed as: $$FP_B(\mathcal{A}, R) = (R \ominus B) \setminus \bigcup_{i=1}^{n} (A_i \oplus (-B))$$

4. Applications in Robotics, CAD, Packaging, and Manufacturing

Spatial distance constraints are prevalent in:

• Robotics: Defining and navigating collision-free spaces, enforcing clearance, and enabling robust manipulability via C-space obstacle mapping.
• CAD/Nesting: Arranging parts within geometric tolerances, ensuring required separations, and optimizing layouts for manufacturability.
• Packaging/Packing Optimization: Efficient space utilization by characterizing allowed object placements through complex Minkowski operations, accommodating buffer zones and custom limits.

Complex real-life constraints—such as those involving multiple minimum and maximum separations, direction-dependent gaps, and containment—are systematically represented and solved through the combinatorial Minkowski framework.

5. Extensions: Directional, Rotational, and Dynamic Constraints

The spatial distance constraint formalism is extensible:

• Directional Constraints: Enforcing separation only in specified directions by restricting Minkowski computations to vector intervals (see Section 9.3).
• Rotational Constraints: Mapping rotations to translations in polar coordinates; Minkowski sums/differences apply after transforming $A$ via $(r, \theta+\phi)$.
• Dynamic/Moving Objects: Representing object configurations in four-dimensional space-time (e.g., union of time-parameterized Minkowski sums), enabling planning in dynamic settings.

6. Computational and Algorithmic Considerations

To implement spatial distance constraints in practice, the general procedure is:

1. Express constraints as Minkowski-set Boolean operations,
2. Compute C-space regions using BRep, CSG, or LRRep representations and robust geometric algorithms,
3. Combine regions per logical structure to generate the final feasible set,
4. Apply outcome to optimization, placement, or motion planning as appropriate to the context.

Efficient computation hinges on high-performance algorithms for Minkowski sums/differences and their Boolean combinations, particularly when handling nontrivial geometries or higher dimensions.

7. Summary Table: Common Constraints and C-Space Mapping

Constraint Type	C-Space Representation
Non-overlap ($d_1(B, A) > 0$)	$p \notin A \oplus (-B)$
Min distance ($d_1(B, A) \geq \delta$)	$p \notin A \oplus (-B) \oplus K_\delta$
Directional ($d_u(B, A) > \delta$)	Partial Minkowski sum along direction $u$
Rotational	Translation in polar, then Minkowski sum/difference
Complex (e.g., min/max, containment)	Correct Boolean combination of Minkowski sums/diffs
Dynamic situations	4D Minkowski sum over object trajectory

8. Illustrative Examples and Figures

Typical feasible regions produced under spatial distance constraints are depicted in the referenced figures of the original work—e.g., non-simply-connected C-space regions resulting from complex (combined min/max) separation requirements in layouts involving multiple obstacles.

Figures 16a, 14dd, and 14ee (see Sections 8 and 9) display these mappings in contexts such as robotic workspace clearance and piece nesting in package design.

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Spatial distance constraints, when encoded via Minkowski operations in configuration space, provide a rigorous and unified methodology for expressing, computing, and applying geometric requirements on the relative position of objects. This machinery underpins not only basic non-overlap but also sophisticated spatial relations such as allowable proximities, directions, rotations, and dynamic trajectory-based separations, with direct, practical relevance to real-world design, manufacturing, and motion planning challenges. Robust and computable algorithms for these constraint mappings are central tools in enabling efficient, accurate, and flexible spatial reasoning across engineering domains.