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

TL;DR

This work introduces a universal differentiable framework for multi-axis process planning by representing both deposition layers and toolpaths as implicit neural fields $f_l$ and $f_p$, modeled with sinusoidally activated networks (SIRENs). By embedding collision avoidance directly into the field-generation stage and jointly optimizing layer geometry and toolpath topology, the approach enables explicit control over toolpath curvature, spacing, and directionality across additive and subtractive processes, including milling. The authors analyze how SIREN frequency and loss design influence singularities and topology, proposing masking strategies and non-normalized direction losses to enable topological transitions while maintaining manufacturability. The method is validated through numerical case studies and physical fabrication on a 6-DoF robotic system, showing improved collision-free, self-supporting prints and continuous, direction-aligned fiber paths with reduced toolpath jumps. Overall, the framework unifies layer and toolpath optimization in a differentiable pipeline, offering explicit collision handling and broad applicability to both 3D printing and rough milling, with potential for integration into broader differentiable design workflows.

Abstract

Existing curved-layer-based process planning methods for multi-axis manufacturing address collisions only indirectly and generate toolpaths in a post-processing step, leaving toolpath geometry uncontrolled during optimization. We present an implicit neural field-based framework for multi-axis process planning that overcomes these limitations by embedding both layer generation and toolpath design within a single differentiable pipeline. Using sinusoidally activated neural networks to represent layers and toolpaths as implicit fields, our method enables direct evaluation of field values and derivatives at any spatial point, thereby allowing explicit collision avoidance and joint optimization of manufacturing layers and toolpaths. We further investigate how network hyperparameters and objective definitions influence singularity behavior and topology transitions, offering built-in mechanisms for regularization and stability control. The proposed approach is demonstrated on examples in both additive and subtractive manufacturing, validating its generality and effectiveness.

Figures (33)

• Figure 1: This figure summarizes our work. In the computational domain defined by an input model (a), we employ sinusoidally-activated neural networks (b) to represent implicit scalar-field functions (c), where the level sets of these scalar fields form the basis for generating layers and toolpaths. In our approach, collision-avoidance is directly enforced during layer generation to ensure manufacturability in multi-axis systems. Similarly, toolpath geometry can be simultaneously controlled during layer generation to accommodate applications like continuous carbon-fiber deposition. In (d), we demonstrate support-free fabrication of the Fertility model with integrated collision avoidance. The same framework can be generalized to support 5-axis rough milling -- e.g., the Cup model as shown in (e). Finally, we illustrate the effectiveness of our method in mechanical reinforcement by spatially deposited continuous carbon fiber in (f), enabled by toolpath-level geometry control.
• Figure 2: The figure illustrates our unified representation of both layers and toolpaths in multi-axis manufacturing for a part (a). We define (b) a layer scalar field $f_l$ and (c) a path scalar field $f_p$ over the part geometry. The corresponding level sets of these fields are denoted by $l_l$ and $l_p$, respectively (d, e). Layers are represented by the level set $l_l$, while toolpaths are obtained as contours formed by the intersection of $l_l$ and $l_p$ located on $l_l$ as shown in (f).
• Figure 3: Pipeline of the algorithm for neural field-based process planning: (a) illustrates the structure of the neural network used for all field representations in our work. Each unit $f$ is a multilayer perceptron (MLP) with sinusoidal activation, commonly referred to as a SIREN sitzmann_implicit_2020. The input to the network is a 3-dimensional vector representing a coordinate in $\Re^3$ , and the output is a scalar value corresponding to the field at that location. The spatial derivatives such as the gradient ($\nabla f$) and the Hessian ($\mathbf{H}_f$) can be computed directly at any point. These quantities form the standard three-part output of our network: [$f, \nabla f, \mathbf{H}_f$]. $\Theta$ denotes the network parameters that are optimized during the process. We also hyper-parameterize the network depth, the frequency-scaling inside the activation sine functions, which are chosen differently for the layer, toolpath, and model networks. (b) illustrates the overall pipeline of our method. For a given part ($\Omega$), which itself can be represented as a signed distance field (${sdf}_{model}$), we employ two field networks corresponding to layers and toolpaths. From these networks, we obtain the field values and their derivatives, which are then used to define a set of requirements expressed as loss functions. The loss functions are categorized into regularization losses (e.g., curvature), functional losses (e.g., directional alignment), and collision losses. The combined sum of these individual losses constitutes the total loss, which is minimized to determine the network parameters representing the layers and toolpaths.
• Figure 4: Illustration of the proposed collision detection scheme. The shaded gray region represents the printed part, while the coloured curves (blue, green, yellow) denote successive layers. The tool (pink) is positioned at a point $\mathbf{x}$ on the current layer ($f_l(\mathbf{x}) = c$) and oriented along the layer normal $\mathbf{n}_{f_l}$. Sampling points $\mathbf{y}_1$ and $\mathbf{y}_2$ on the tool correspond to the cases of collision ($f_l(\mathbf{y}_1) < c$) and non-collision ($f_l(\mathbf{y}_2) > c$), respectively. We denote the tool configuration at $\mathbf{x}$ as $\mathcal{T}_{\mathbf{x}}$, with $\mathbf{y}_1, \mathbf{y}_2 \in \{\mathcal{T}_{\mathbf{x}}\}$.
• Figure 5: An illustration of our inter-layer distance control strategy. The blue and green lines represent two iso-surfaces with scalar values $c$ and $c+\delta c$ respectively. The magnitude of the gradient is the rate of growth of the field and can be used to estimate inter-layer spacing. Minimizing the differences in gradient magnitude encourages uniform layer spacing. This illustration is intended for conceptual clarity and is not mathematically exact. The values are used inside the square brackets (e.g. $[\|\nabla f_l\|]$) to indicate that the expressions inside the brackets can be used as an measure of those distances and does not mean the exact value.