Implicit Neural Field-Based Process Planning for Multi-Axis Manufacturing: Direct Control over Collision Avoidance and Toolpath Geometry

1. Problem Statement

Multi-axis manufacturing (both additive and subtractive) offers significant advantages over traditional planar methods, such as support-free fabrication, improved surface finish, and the ability to align material deposition with principal stress directions (e.g., for continuous carbon fiber reinforcement). However, current process planning methods face three critical limitations:

• Indirect Collision Avoidance: Existing methods typically handle collisions indirectly by limiting local surface curvature or through post-processing steps. These approaches often fail to detect global collisions (where the tool intersects the part far from the deposition point) and cannot guarantee a globally valid, collision-free configuration.
• Decoupled Toolpath Optimization: Toolpath geometry (spacing, turning angles, continuity) is usually generated in a separate, discrete step after layer generation. This decoupling prevents the co-optimization of layers and toolpaths, leading to suboptimal mechanical performance and manufacturability issues (e.g., sharp turns in fiber paths).
• Discrete Representations: Most methods rely on discrete volumetric structures (voxels or tetrahedral meshes), which require interpolation for derivative estimation and lack the differentiability needed for gradient-based joint optimization.

2. Methodology

The authors propose a unified, differentiable framework based on Implicit Neural Fields to solve these problems.

Core Representation

• Implicit Scalar Fields: Instead of discrete meshes, the method represents both manufacturing layers ($f_l$) and toolpaths ($f_p$) as continuous scalar fields defined over the 3D domain.
  • Layers: Defined as the level sets (iso-surfaces) of the layer field $f_l(\mathbf{x})$.
  • Toolpaths: Defined as the iso-contours formed by the intersection of the layer field $f_l$ and the path field $f_p$ on the layer surface.
• Neural Architecture: The fields are represented by SIREN (Sinusoidal Representation Networks), which use sinusoidal activation functions. This architecture is chosen for its ability to represent high-frequency signals and, crucially, their higher-order derivatives (gradients and Hessians) via automatic differentiation.

The Optimization Pipeline

The process is formulated as a multi-objective optimization problem where network parameters are updated to minimize a total loss function ($L_{total}$). The pipeline involves:

1. Field Evaluation: Sampling points in the domain to evaluate $f_l$, $f_p$, and their spatial derivatives ($\nabla f$, $H f$).
2. Loss Computation: Calculating specific loss terms based on manufacturing constraints.
3. Gradient Descent: Updating network weights to minimize the total loss, simultaneously refining layers and toolpaths.

Key Loss Functions

The framework integrates various constraints directly into the optimization:

• Collision Avoidance ($L_{cl}$): A novel, differentiable formulation. Instead of checking discrete points, the method samples points on the tool volume and ensures they lie in regions where the layer field value is strictly greater than the current layer value (for AM) or outside the target part (for milling). This allows the optimizer to "push" layers away from collisions directly.
• Directional Alignment ($L_{df}$): Enforces toolpaths to align with a prescribed vector field (e.g., principal stress directions) by minimizing the cross-product between the toolpath tangent and the target direction.
• Geometric Regularization:
  • Curvature ($L_{cr}$): Penalizes high mean and Gaussian curvatures of layers and geodesic curvature of toolpaths to prevent kinematic limits and printing defects.
  • Spacing ($L_{ds}$): Controls the gradient norm of the fields to ensure uniform layer thickness and toolpath spacing.
• Support-Free & Setup ($L_{sf}, L_{setup}$): Enforces overhang constraints and optimizes the part's orientation relative to the build platform.

3. Key Contributions

1. Direct Collision Avoidance: The first method to embed global collision avoidance directly into the implicit field generation stage via a differentiable loss, eliminating the need for heuristic post-processing.
2. Joint Layer-Toolpath Optimization: A unified formulation that co-optimizes layer geometry and toolpath topology simultaneously, allowing for explicit control over toolpath spacing, turning angles, and continuity.
3. Implicit Neural Field Framework: Demonstrates the use of SIRENs for multi-axis process planning, enabling the direct computation of complex geometric quantities (normals, curvatures, geodesic curvature) required for manufacturing constraints.
4. Singularity and Topology Analysis: Provides an analytical study on how network hyperparameters (frequency scaling) and loss definitions influence singularities and topology transitions, offering mechanisms to control field smoothness and stability.
5. Generality: The framework is validated on both Additive Manufacturing (multi-axis 3D printing with continuous carbon fiber) and Subtractive Manufacturing (5-axis rough milling).

4. Results and Experiments

The authors validated the approach through numerical simulations and physical fabrication on a 6-DoF robotic arm and a 5-axis CNC machine.

• Support-Free & Collision-Free Printing:
  • On complex models (e.g., "Fertility," "4C"), the method successfully generated collision-free, support-free layers.
  • Comparison: Compared against "Neural Slicer" and "S3-Slicer," the proposed method avoided severe tool-part collisions that other methods missed, which could not be resolved via post-processing.
• Continuous Carbon Fiber (CCF) Reinforcement:
  • Applied to a "T-Bracket" and "S-Curve" model.
  • Performance: The method generated continuous fiber paths with fewer jumps and turns compared to high-density toolpath methods.
  • Mechanical Results: Parts printed with the optimized paths showed 33.9% higher stiffness and comparable breaking force while using 18.5% less carbon fiber than baseline methods.
• Multi-Axis Milling:
  • Demonstrated on a "Cup" model, the method automatically generated rough-milling layers that avoided tool collisions with the part geometry, outperforming anchor-based strategies that require manual intervention.
• Ablation Studies:
  • Showed that derivative-based distance control (minimizing the spatial derivative of the gradient norm) yields more uniform layer spacing than fixing the gradient norm to unity.
  • Demonstrated that non-normalized loss formulations are essential for handling singularities and achieving stable convergence in directional alignment tasks.

5. Significance

This work represents a paradigm shift in multi-axis manufacturing process planning. By moving from discrete, post-processed workflows to a continuous, differentiable, and implicit neural field framework, the authors achieve:

• True Co-Optimization: The ability to balance conflicting objectives (e.g., strength alignment vs. collision avoidance vs. toolpath smoothness) in a single optimization loop.
• Robustness: The elimination of "collision-prone" local minima that plague traditional methods, ensuring manufacturability by design rather than by correction.
• Unified Approach: A single mathematical formulation applicable to both material addition (printing) and material removal (milling), paving the way for integrated hybrid manufacturing workflows.
• Scalability: The use of neural networks allows the representation of complex geometries without the memory overhead of high-resolution meshes, while the differentiable nature enables integration with broader design optimization frameworks.

In summary, this paper provides a foundational step toward "end-to-end" differentiable manufacturing, where the physical constraints of the machine and the material are mathematically embedded into the generation of the toolpath itself.