Topology-Preserving Scalar Field Optimization for Boundary-Conforming Spiral Toolpaths on Multiply Connected Freeform Surfaces

1. Problem Statement

The paper addresses the challenge of generating high-quality toolpaths for multiply connected freeform surfaces (surfaces with holes or protrusions), which are common in automotive and aerospace components (e.g., headlamp molds).

• Current Limitations: Existing methods (decomposition, boundary offsetting, or standard scalar field optimization) often fail on complex geometries, resulting in:
  • Discontinuities: Toolpaths that break or require frequent retraction.
  • Sharp Turns: Sudden directional changes that cause actuator deceleration and vibrations.
  • Non-Uniform Scallop Heights: Inconsistent surface finish quality.
  • Incomplete Boundary Coverage: Residual material left at edges.
  • Self-Intersections: Paths that cross over themselves, distorting milling loads.
• Core Challenge: Achieving a global optimization that simultaneously ensures path continuity, strict boundary conformity, non-self-intersection, uniform scallop height, and smoothness is mathematically difficult due to conflicting objectives and the topological complexity of the surface.

2. Methodology

The authors propose a Topology-Preserving Scalar Field Variational Optimization framework. The method transforms the toolpath planning problem into a mesh deformation problem governed by specific constraints.

A. Evaluation Criteria

The optimization minimizes a comprehensive energy function $E$ composed of two terms:

1. Scallop Height Uniformity ($E_w$): Based on a symmetrized Itakura–Saito divergence, ensuring the distance between adjacent toolpaths (scallop height) remains constant across the surface.
2. Toolpath Smoothness ($E_k$): Quantified by the integral of the squared geodesic curvature of the iso-level curves, penalizing sharp turns.

• A weighting parameter $\alpha$ balances these two objectives.

B. Initialization via Conformal Slit Mapping

To avoid local minima and ensure a feasible starting point:

• Conformal Slit Mapping: The complex multiply connected surface is mapped to a simply connected domain (a unit disk or annulus) with circular-arc slits representing the internal holes.
• Initial Scalar Field: A scalar field is constructed on this mapped domain where iso-level sets are concentric circles. This guarantees an initial field with non-zero gradients (preventing singularities) and boundary conformity.
• Optimization of Initialization: The mapping parameters (center point and function shape) are optimized to minimize the initial energy, providing a "near-optimal" starting state.

C. Variational Optimization via Mesh Deformation

Instead of updating scalar values directly, the method reformulates the optimization as a mesh geometry deformation process on the mapped domain ($S_H$):

• Topological Equivalence: The scalar field is represented by the radial distance of mesh nodes from the center.
• Constraint Enforcement:
  • Constraint 1 (Non-Singularity): Mesh topology must remain homeomorphic (no triangle inversions). This ensures $\|\nabla T\| > 0$, preventing toolpath branching or singularities. This is enforced by restricting vertex movement within a convex hull defined by angle bisectors of neighboring triangles.
  • Constraint 2 (Boundary Conformity): Vertices on boundary slits must move synchronously in the radial direction to maintain circular boundaries.
• Iterative Process: The algorithm iteratively updates vertex positions along the radial direction based on the gradient of the energy function, strictly adhering to the feasible intervals defined by the convex hulls. This ensures the final toolpaths are continuous, boundary-conforming, and non-self-intersecting.

3. Key Contributions

1. Topology-Preserving Optimization Framework: A novel approach that converts scalar field optimization into a constrained mesh deformation problem. This rigorously enforces non-self-intersection and boundary conformity throughout the iterative process, solving a major limitation of previous scalar field methods on porous surfaces.
2. Robust Initialization Strategy: The use of conformal slit mapping to generate a singularity-free, boundary-conforming initial field. This significantly reduces computational cost by avoiding invalid iterations and converging faster than random or uninitialized starts.
3. Unified Multi-Objective Control: The introduction of a weighting parameter ($\alpha$) allows users to trade off between path smoothness and scallop height uniformity, providing flexibility for different machining requirements.

4. Experimental Results

The method was validated through numerical simulations and physical milling experiments on a robotic system (IRB6600) machining an ABS workpiece.

• Comparison: The proposed method (Path 1) was compared against a state-of-the-art conformal slit mapping-based method (Path 2).
• Performance Metrics:
  • Machining Efficiency: Improved by 14.24% (reduced total time from 543.75s to 466.33s) due to a 20.92% reduction in redundant path length.
  • Scallop Height Uniformity: Improved by 5.70% (lower variance in surface finish).
  • Dynamic Stability: Reduced milling impact-induced vibrations by over 10% (specifically, variance and cumulative impact indicators dropped significantly), leading to smoother actuator operation.
  • Path Quality: Generated continuous, boundary-conforming, and self-intersection-free spiral paths even on surfaces with multiple protrusions, whereas the baseline method exhibited discontinuities and sharp turns.

5. Significance

This research provides a robust solution for high-performance machining of complex components. By bridging the gap between theoretical scalar field optimization and practical manufacturing constraints, the method:

• Eliminates the need for manual decomposition of complex surfaces.
• Ensures high surface quality (uniform scallop height) and process stability (reduced vibration).
• Offers a computationally efficient framework that is applicable not only to milling but potentially to other coverage path planning problems like additive manufacturing and UAV area coverage.

The study demonstrates that preserving topological constraints during scalar field optimization is critical for generating viable, high-quality toolpaths on multiply connected freeform surfaces.