Singularity-Aware Motion Planning for Multi-Axis Additive Manufacturing

1. Introduction

Multi-axis additive manufacturing (MAAM) represents a significant evolution beyond conventional planar layer-based 3D printing. By enabling material deposition along dynamically varied directions (e.g., along surface normals), MAAM systems offer solutions to long-standing issues such as the need for support structures, weak inter-layer strength, and staircase artifacts on curved surfaces. However, this increased geometric freedom introduces complex motion planning challenges, particularly when realizing designed toolpaths on hardware platforms that typically combine three translational axes with two rotational axes.

1.1 Problem of Motion Planning in MAAM

The core challenge lies in the nonlinear mapping between the workpiece coordinate system (WCS), where the toolpath is designed, and the machine coordinate system (MCS), which controls the physical actuators. A smooth, uniformly sampled toolpath in WCS can be mapped to a highly discontinuous motion in MCS when the tool orientation approaches vertical—a region known as kinematic singularity. In filament-based AM, this discontinuity disrupts the stable extrusion flow, leading to over-extrusion or under-extrusion, which manifests as surface artifacts and compromises mechanical integrity. Unlike in CNC milling where motion can be paused, AM requires continuous motion and must adhere to strict speed constraints ($f_{min} \leq v_{tip} \leq f_{max}$) dictated by the extruder's physical limits. Furthermore, collision avoidance must be integrated into the planning process.

2. Background and Related Work

2.1 Multi-Axis Additive Manufacturing Systems

Various hardware configurations exist, including systems with a tilting-rotating worktable (e.g., 3+2 axis) or a robotic arm (6-DOF). These systems enable support-free printing of overhangs by aligning the deposition direction with the surface normal.

2.2 Toolpath Generation for Curved Layers

Research has focused on generating non-planar, curved-layer toolpaths to optimize strength and surface finish. However, the physical realization of these complex paths is often overlooked.

2.3 Singularity in Multi-Axis CNC Machining

Singularity is a well-known issue in 5-axis CNC machining, where the tool axis aligns with a rotary axis, causing a mathematical discontinuity in the inverse kinematics solution. Traditional CNC solutions often involve toolpath modification or reparameterization, but they cannot be directly applied to AM due to the need for continuous extrusion and bounded speed.

3. Proposed Methodology

3.1 Problem Formulation

The input is a toolpath defined as a sequence of waypoints $\mathbf{W}_i = (\mathbf{p}_i, \mathbf{n}_i)$ in WCS, where $\mathbf{p}_i$ is position and $\mathbf{n}_i$ is the nozzle orientation (typically the surface normal). The goal is to find a corresponding motion sequence in MCS, $\mathbf{M}_j = (x_j, y_j, z_j, A_j, C_j)$ for a typical 5-axis machine (XYZAC), that:

1. Avoids kinematic singularities or manages their effects.
2. Maintains continuity to ensure unbroken extrusion.
3. Keeps the nozzle tip speed within $[v_{min}, v_{max}]$.
4. Avoids collisions between the print head and the part.

3.2 Singularity-Aware Motion Planning Algorithm

The paper proposes an algorithm that identifies singular regions in the toolpath (e.g., where the normal vector's vertical component is near 1). Instead of naively sampling waypoints uniformly in WCS, it performs adaptive sampling and local toolpath optimization in these regions. This might involve slight deviations in orientation or a re-timing of the motion to smooth out the discontinuous jumps in the rotary axes ($A$, $C$), thereby preventing sudden changes in the nozzle tip velocity.

3.3 Integrated Collision Avoidance

The motion planner integrates a sampling-based collision checker. When a potential collision is detected during the planning of a singularity-avoiding motion, the algorithm iteratively adjusts the toolpath or the machine posture until a collision-free and singularity-managed solution is found.

4. Technical Details and Mathematical Formulation

The inverse kinematics for a typical 5-axis machine with a tilting-rotating table (AC axes on the table) can be expressed. The tool orientation vector $\mathbf{n} = (n_x, n_y, n_z)$ in WCS is mapped to rotary angles $A$ (tilt) and $C$ (rotation). A common formulation is:

$A = \arccos(n_z)$

$C = \operatorname{atan2}(n_y, n_x)$

The singularity occurs when $n_z \approx \pm 1$ (i.e., $A \approx 0^\circ$ or $180^\circ$), where $C$ becomes undefined—a gimbal lock situation. The Jacobian matrix relating joint velocities to tool tip velocity becomes ill-conditioned here. The paper's algorithm likely monitors the condition number of this Jacobian or the value of $n_z$ to detect singular regions. The core of the planning involves solving an optimization problem that minimizes a cost function $J$:

$J = \alpha J_{continuity} + \beta J_{speed} + \gamma J_{singularity} + \delta J_{collision}$

where $J_{continuity}$ penalizes discontinuities in MCS motion, $J_{speed}$ ensures tip speed bounds, $J_{singularity}$ penalizes proximity to singular configurations, and $J_{collision}$ is a collision penalty. Weights $\alpha, \beta, \gamma, \delta$ balance these objectives.

5. Experimental Results and Analysis

5.1 Experimental Setup

The method was validated on a custom 5-axis 3D printer (XYZ translation, AC rotary table) fabricating models like the Stanford Bunny with curved layers.

5.2 Fabrication Quality Comparison

Figure 1 (Referenced from PDF): Shows a clear visual comparison. The bunny printed with conventional planning (Fig. 1a) exhibits severe surface artifacts (over-/under-extrusion) in regions circled, corresponding to areas where the surface normal is near vertical (singular region). The bunny printed with the proposed singularity-aware planning (Fig. 1c) shows significantly smoother surfaces in those same regions. Fig. 1b visually highlights the waypoints located in the singular region in yellow, demonstrating the algorithm's detection capability.

5.3 Motion Continuity and Speed Analysis

Plots of rotary axis angles ($A$, $C$) and calculated nozzle tip speed over time would show that the proposed method smooths the near-discontinuous jumps in the rotary angles observed in the conventional method. Consequently, the nozzle tip speed remains within the stable extrusion window $[v_{min}, v_{max}]$, whereas the conventional method causes speed spikes or drops to near-zero, directly explaining the extrusion defects.

6. Analysis Framework: A Non-Code Case Study

Scenario: Printing a dome-shaped object with a vertical axis of symmetry.
Challenge: The apex of the dome has a vertical normal ($n_z=1$), placing it directly in a singular configuration. A spiral toolpath from the base to the apex would naively cause the C-axis to spin uncontrollably as it approaches the top.
Proposed Method Application:

1. Detection: The algorithm identifies waypoints within a threshold (e.g., $n_z > 0.98$) as the singular region.
2. Planning: Instead of forcing the tool to point exactly vertically at the apex, the planner may introduce a slight, controlled tilt (e.g., $A=5^\circ$) for a few layers around the apex. This keeps the C-axis well-defined.
3. Optimization: The toolpath in this region is re-timed to ensure the nozzle moves at a constant, optimal speed, and the slight geometric deviation is compensated for in the adjacent non-singular path to maintain overall shape fidelity.
4. Outcome: A smooth, continuous motion is achieved, resulting in a dome with a consistent surface finish at the apex, free of blobs or gaps.

7. Application Outlook and Future Directions

• Advanced Materials & Processes: This planning is critical for printing with continuous fiber composites or concrete, where flow control is even more sensitive to motion discontinuities.
• Integration with Generative Design: Future CAD/CAE software could incorporate "manufacturability constraints" based on this singularity model during the generative design phase, avoiding designs that are inherently difficult to print smoothly on multi-axis systems.
• Machine Learning for Path Planning: Reinforcement learning agents could be trained to navigate the complex trade-off space between singularity avoidance, speed maintenance, and collision avoidance more efficiently than traditional optimization.
• Standardization & Cloud Slicing: As multi-axis printing becomes more accessible, cloud-based slicing services could offer singularity-optimized toolpath planning as a premium feature, similar to how supports are optimized today.

8. References

1. Ding, D., et al. (2015). A review on 5-axis CNC machining. International Journal of Machine Tools and Manufacture.
2. Chen, X., et al. (2021). Support-Free 3D Printing via Multi-Axis Motion. ACM Transactions on Graphics.
3. ISO/ASTM 52900:2021. Additive manufacturing — General principles — Terminology.
4. Müller, M., et al. (2022). Real-time trajectory planning for robotic additive manufacturing. Robotics and Computer-Integrated Manufacturing.
5. The MathWorks, Inc. (2023). Robotics System Toolbox: Inverse Kinematics. [Online] Available: https://www.mathworks.com/help/robotics/ug/inverse-kinematics.html

9. Original Analysis & Expert Commentary