In current AM implementations the design choice of a 2.5-axis machine configuration is built into the process specification. Since the printing head (or scanning mirror) can only scan planar layers, the nominal geometry must be approximated (i.e. sliced) with 2D contours, equally spaced along a predetermined build direction. For circular aperture extruders, the slicing orientation ultimately determines the overall build accuracy of the 3D printed part, because it directly influences both the dimensional accuracy of the finishes part along the slicing direction [30], and the extruder’s reachability in each slice (see Examples 1.2.2 and 1.2.4).
Although the dimensional accuracy along the slicing direction cannot be improved substantially in a classic layer-by-layer deposition (because of the fixed layer height), the layer’s contour accuracy can be significantly improved by using non-circular extruders executing general planar motions (i.e. 3-DOF, two translations and one rotation). This opens two completely new and unexplored research topics, namely:
• What is the theoretical geometry of a part printed with an arbitrary shaped extruder, given the desired 3D model and some arbitrarily chosen machine configuration?
• How can we compute suitable motions of the extruder (possibly with a non- circular aperture) to match the nominal geometry?
To answer the question of the maximum build volume, I propose analyzing the motion’s parametric space (see Section 2.2) to identify the particular machine configurations in which the extruder aperture is permanently contained in the interior
of the desired 2D contour. Section 2.3 shows that this approach not only works for virtually any geometric representation but also applies to fully 3D geometry and not only planar layers. In other words, using the framework presented in this thesis, we can calculate the maximum theoretical volume of a 3D part regardless of the particular kinematic configuration of the AM machine. This way we can decouple the machine architecture of an FDM printer from the fusing process, allowing us to fit extruders to complex kinematic chains such as 5-axis machining centers, industrial robotic arms or even over-actuated linkages (such as snake robots). This in return opens exciting new manufacturing techniques including non-planar layers or possibly no layers at all in the traditional sense.
Build time is also captured in this framework. The extruders proposed for robotic 3D printing include a non-circular nozzle with high aspect ratio, such as an ellipse or a thick slit. By using printing heads with rotational degrees of freedom, the asymmetric aperture can be conveniently oriented during the extrusion process not only to resolve small features of the target geometry but also to cover large areas of the layer’s interior in as few motions interpolations as possible, and thus maintaining or even reducing the overall build time.
Motion planning with fully 3D path generation in a static domain is a well- known research topic, with many practical applications including robot path planning, autonomous driving, cartography, etc. In Additive Manufacturing, motion planning has been drastically simplified by leveraging the axial symmetry of the extruder nozzle, or laser beam respectively, and the discretization of the target geometry using planar slices. Under these assumptions, coverage path planning [37][65], which is the natural choice for this application, is solved by simply calculating planar curve
offsets of the layers’ boundary. The introduction of non-circular extruders will render all current motion planning algorithms ineffective because geometrically asymmetric apertures are not rotation invariant. Therefore, any motion planners relying on classic morphological operators such as the Minkowski sum [152] must be replaced with more general operators (for example the configuration product [130]).
This thesis presents in Chapter 3 a novel method of computing the path of circular extruders in non-planar layers, and argue that the same path planning technique generalizes to arbitrary shaped extruders. In 2D, the trajectory generation relies on computing the medial axis of a planar arbitrary shape (i.e. the boundary of a sample layer). By a careful manipulation of the radial function, we can prescribe a twist angle (or heading) of the extruder, which will maximize the surface coverage and at the same time, resolving small features such as edges and corners. This motion planning policy corresponds to computing a dense infill, similar in appearance to curve offsets but mathematically different, known in practical applications as concentric infill.
Motion planning with fully three dimensional tool paths is analyzed in this thesis assuming that no infill will be deposited (i.e. the so-called vase build). This extra level of difficulty ensures that the method described here generalizes to arbitrary complexity kinematics. For this problem I first show an analysis of coverage planning assuming a circular aperture extruder. This will be applied initially to traditional FDM architectures with a 2.5-axis machine configuration. Here we are only interested in generating the model’s boundary and infill. Computing the infill in a 6DOF printing configuration is generally not required unless specific requirements are imposed to increase the mechanical rigidity of the printed model. I will then show a path planning algorithm posed as an optimization problem where the objective function models a
convolution which implicitly measures the overlap volume between subsequent traces of the extruder.
I propose computing the 6D motion in a canonical 2D space, where we perform a surface parameterization [35][70][98] of the input 3D geometry, producing a planar and non-self-intersecting mesh. The choice of conformal mapping over authalic or isometric is due the low distortion of local geometry (theoretically non-existent). I then compute the entire 2D mesh deformation as a scalar field and use it to correct the position and geometry of the extruder aperture in this canonical space. We then obtain the point trajectories of the extruder so that its aperture moves on 3D geodesic offset curves, relative to a seeding curve (usually the surface boundary), or an isolated seeding point. Finally, the extruder’s orientation is computed point-wise using the local normal information of the original input model and by maintaining a controlled air gap between neighboring deposition traces.