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Results

.

We demonstrate the approach using an illustrative 3D example. A near-net shape along with a cutting tool to remove supports at dislocation features is shown in Fig. 1.

We assume that the part is fixtured properly (not shown here, similarly to Fig. [fig_metal]) such that the removal sequence of the support components does not impact its stability.

At every round, Algorithm [alg_recursive] identifies the outer layer of supports that can be peeled off, using FFT-based convolution (Sections 4.1 and 4.2) as illustrated in Fig. 4. For each layer, solving the TSP on the graph of fibers (Section 5) gives a sequence of configurations to visit, while asserting all of them will contact a distinct dislocation feature without collision.

The OMPL motion planner is invoked to find a collision-free tool-path to fracture the dislocation features one after another (Section 5.2), as illustrated in Fig. 5.

Figures 2 and 3 show running times for computing $`{\mathsf{C}}-`$space obstacles and solving TSP, respectively.

Identifying Removable Supports

Removing supports by cutting off their contact regions with the part is an efficient alternative to machining the entire support material using a traditional milling process. For the latter, in the same way that one would remove material from a full raw stock , one can clean out the support material by sweeping the tool within the support material over the continuum of accessible configurations. In our approach, we need only finite contact configurations to peel off the support components. This assumption is critical in our ability to efficiently compute the free space for an evolving near-net shape.

At the beginning, it is likely that only a subset of support components are removable in this manner with a given tool. Intuitively, we may imagine a forest of support components (e.g., columns) in a near-net shape, where the occluded columns are inaccessible until the columns along the periphery are removed.

Our approach uses known properties of the $`{\mathsf{C}}-`$space of relative rigid transformations (detailed in Sections 4.1 and 4.2) to automatically identify the maximal collection of supports that are guaranteed to be removable from a near-net shape in an intermediate state. Briefly, the algorithm proceeds as follows: The $`{\mathsf{C}}-`$space obstacle is computed explicitly and all contacting but non-colliding configurations of the tool are extracted. A support component is deemed removable iff all of its dislocation features are accessible through contact, meaning that there exists at least one configuration in the boundary of the $`{\mathsf{C}}-`$space obstacle that corresponds to each dislocation feature. All removable support components are collected and designated for removal at the current “round” of the post-process plan. After removing this collection from the near-net shape, the $`{\mathsf{C}}-`$obstacle reduces in size, more dislocation features become accessible, hence more support components become removable. The algorithm recursively identifies a sequence of removable supports that are peeled off by the cutting tool to finally converge to the desired part.

Note that this process is independent of the precise order and tool-path ¹ in which the collected support components of a given round are removed in practice. A manufacturability test is also encoded in the algorithm, which can identify early on if the near-net shape includes support components that are inherently unreachable even after removing the outer layers.

Constructing Configuration Space Obstacles

We formulate the support removal problem for $`\mathrm{d}-`$dimensional shapes (typically, $`\mathrm{d}=2`$ or $`3`$). Throughout the paper, we will use illustrations in 2D for building intuition, and show 3D results in Section 6.

Queries about the interference of a moving body (e.g., the tool assembly) $`T \subseteq {\mathds{R}}^\mathrm{d}`$ against a stationary set of 3D obstacles (e.g., the near-net shape) $`N \subseteq {\mathds{R}}^\mathrm{d}`$ are conceptualized as evaluating membership predicted against the $`{\mathsf{C}}-`$space obstacle $`\mathcal{O}(N,T) \subseteq {\mathsf{C}}`$ which partitions the $`{\mathsf{C}}-`$space of relative rigid transformations $`{\mathsf{C}}:= {\mathrm{SE}(\mathrm{d})}`$ into three disjoint regions representing free, contacting, and interfering configurations respectively. These regions correspond to the exterior, boundary, and interior of the $`{\mathsf{C}}-`$space obstacle, respectively.

Constructing the obstacle $`\mathcal{O}(N,T)`$ is a critical step in our algorithm. Practical approaches use the fact that the particular group structure of $`{\mathrm{SE}(\mathrm{d})} = {\mathrm{SO}(\mathrm{d})} \rtimes {\mathds{R}}^\mathrm{d}`$ (i.e., the Lie group of rigid transformations) allows it to be factored into two simpler and lower-dimensional subgroups; namely, $`{\mathrm{SO}(\mathrm{d})}`$ (for rotations) and $`{\mathds{R}}^\mathrm{d}`$ (for translations). Hence, we can represent each rigid transformation $`\tau \in {\mathrm{SE}(\mathrm{d})}`$ as a tuple $`\tau \equiv (r, {\mathbf{t}})`$ with $`{\mathbf{t}}\in \mathbb{R}^\mathrm{d}`$ and $`r \in {\mathrm{SO}(\mathrm{d})}`$. In 2D, the former is a 2D vector while the latter is represented by a angle $`\theta \in [0, 2\pi)`$. In 3D, the former is a 3D vector while the latter can be represented in a number of different ways (e.g., orthogonal matrices, quaternions, axis-angle, Euler angles, etc.).

Let us denote a rotation of $`T \subseteq {\mathds{R}}^\mathrm{d}`$ by $`r \in {\mathrm{SO}(\mathrm{d})}`$ by $`T_r \subseteq {\mathds{R}}^\mathrm{d}`$. Assuming that we are only dealing with solids (i.e., compact-regular seminalaytic sets in $`{\mathds{R}}^\mathrm{d}`$) , the $`{\mathsf{C}}-`$obstacle $`\mathcal{O}(N,T)`$ may be expressed as follows:

\begin{equation}
    \mathcal{O}(P,T) = \bigcup_{r \in {\mathrm{SO}(\mathrm{d})}} \mathsf{i}(N \oplus (-T_r)),
    \label{eq_obs}
\end{equation}

where $`\mathsf{i}(\cdot)`$ represents the topological interior operator. The pointset $`(N \oplus (-T_r))`$ is the topological closure of the translational $`{\mathsf{C}}-`$space obstacle of $`N`$ with $`T_r`$, i.e., the $`{\mathsf{C}}-`$obstacle for a fixed orientation $`r \in {\mathrm{SO}(\mathrm{d})}`$. It is well-known that this pointset may be calculated in terms of a Minkowski sum. ² The $`{\mathsf{C}}-`$space for all rotations is thus obtained by unifying the different translational $`{\mathsf{C}}-`$spaces, each forming a slice of the $`{\mathsf{C}}-`$obstacle. In practice, $`\mathcal{O}(N,T)`$ may be approximated as a stack of a finite number of $`r-`$slices, indexed by rotations sampled in $`{\mathrm{SO}(\mathrm{d})}`$. This approach is commonly used to calculate $`{\mathsf{C}}-`$space obstacles for planar motions .

Notice that $`(N \oplus (-T_r))`$ represents the closure of the translational $`{\mathsf{C}}-`$space obstacle and its boundary, denoted by $`\partial(N \oplus (-T_r))`$ represents all the translations of the moving body (fir the fixed orientation) that cause surface contact but no volumetric interference with $`P`$. The contact configurations $`\mathcal{C}(N,T) \subseteq {\mathrm{SE}(\mathrm{d})}`$ are therefore obtained as:

\begin{equation}
    \mathcal{C}(P,T) = \bigcup_{r \in {\mathrm{SO}(\mathrm{d})}} \partial(N \oplus (-T_r)).
\end{equation}

The contact space is normally a lower-dimensional submanifold of $`{\mathrm{SE}(\mathrm{d})}`$, which makes its computation challenging. In practice, we approximate $`\mathcal{C}(N,T)`$ with a slightly “thickened” set, i.e., a narrow (but measurable) region surrounding the boundary of the $`{\mathsf{C}}-`$space obstacle.

Querying Contact Configurations

Our approach uses the fact that the Minkowski sum $`(A \oplus B) \subseteq {\mathds{R}}^\mathrm{d}`$ of a pair of arbitrary solids $`A, B \subseteq {\mathds{R}}^\mathrm{d}`$ corresponds to the support (i.e., $`0-`$superlevel set) of the convolution $`({\mathbf{1}}_A \ast {\mathbf{1}}_B)`$ of the indicator functions $`{\mathbf{1}}_A, {\mathbf{1}}_B: {\mathds{R}}^\mathrm{d}\to \{0, 1\}`$ of the two sets. ³ The advantage of this approach is that the convolution may be implemented in terms of Fourier transforms. ⁴ When the sets are sampled uniformly on a regular grid, the fast Fourier transform (FFT) provides an efficient and scalable implementation of each translational $`{\mathsf{C}}-`$space obstacle (i.e, $`r-`$slice of $`\mathcal{O}(P,T)`$):

\begin{equation}
    (r, {\mathbf{t}}) \in \mathcal{O}(N, T) \quad\text{iff}\quad ({\mathbf{1}}_N \ast {\mathbf{1}}_{-T_r})
    ({\mathbf{t}}) > 0, \label{eq_obs_conv}
\end{equation}

A configuration $`\tau \equiv (r, {\mathbf{t}})`$ is inside the obstacle if the convolution does not vanish for at least one orientation $`r \in {\mathrm{SO}(\mathrm{d})}`$. Note that the convolution of nonnegative functions is always nonnegative, hence the free space is defined implicitly by and equality test:

\begin{equation}
    (r, {\mathbf{t}}) \not\in \mathcal{O}(P, T) \quad\text{iff}\quad ({\mathbf{1}}_N \ast {\mathbf{1}}_{-T_r})
    ({\mathbf{t}}) = 0. \label{eq_free_conv}
\end{equation}

The convolution provides a field of overlap measure values between $`P`$ and $`T_r`$ over the translational $`{\mathsf{C}}-`$space. In other words, $`({\mathbf{1}}_N \ast {\mathbf{1}}_{-T_r})({\mathbf{t}}) = \mu^\mathrm{d}[N \cap (T_r+{\mathbf{t}})]`$ measures the volume of intersection (if any) between $`P`$ and the moved body $`T_\tau := (T_r + {\mathbf{t}}) = \{{\mathbf{x}}+ {\mathbf{t}}~|~ {\mathbf{x}}\in T_r\}`$. Here, $`\mu^\mathrm{d}[\cdot]`$ represents the Lebesgue $`\mathrm{d}-`$measure (e.g., area for $`\mathrm{d}= 2`$ and volume for $`\mathrm{d}= 3`$) . The queried configuration is in the $`{\mathsf{C}}-`$space obstacle (resp. free space) iff this measure is greater than (resp. equal to) zero.

The contact space $`\mathcal{C}(P, T)`$ consists of configurations at which the overlap measure is critically zero, meaning that there exists an infinitesimal change to the configuration that can make it nonzero. To approximate $`\mathcal{C}(N, T)`$ in a computable fashion, we implicitly define an ‘$`\epsilon-`$contact’ space using the following predicate:

\begin{equation}
    (r, {\mathbf{t}}) \in \mathcal{C}_\epsilon(N, T) \quad\text{iff}\quad 0 < ({\mathbf{1}}_N \ast
    {\mathbf{1}}_{-T_r}) ({\mathbf{t}}) < \epsilon. \label{eq_ont_conv}
\end{equation}

Using a small the threshold $`\epsilon > 0`$, one may assume that $`\mathcal{C}_\epsilon(N, T)`$ closely approximates $`\mathcal{C}(N, T)`$ from a measure-theoretic standpoint.

To summarize, if the $`r-`$slices are computed and stored beforehand (as a finite number of convolutions), it is possible to query the overlap measure in constant time at any sampled translation and rotation by interpolation against the stack of convolution fields. In particular, $`\mathcal{C}(N,T)`$ is numerically approximated as the set of all transformations $`(r,{\mathbf{t}}) \in {\mathrm{SE}(\mathrm{d})}`$ where $`0 < ({\mathbf{1}}_N \ast {\mathbf{1}}_{-T_r})({\mathbf{t}}) < \epsilon`$ for some tolerable interference volume $`\epsilon

0`$.

Figure 6 illustrates in 2D how the computed stack of convolutions for different $`r-`$slices for sampled rotations is used to classify a given tool configuration as free, contact, or colliding. In 3D, the same exact method applies, except that each convolution is a 3D field and the rotations are samples in 3D using any number of standard uniform or pseudo-uniform/random sampling methods .

Algorithm [alg_contact] describes a simple (and parallel) procedure to compute the $`\epsilon-`$contact space for a finite sample $`\{r_s\}_{1 \leq s \leq n_1} \subset {\mathrm{SO}(\mathrm{d})}`$ of orientations. The convolution is faster to compute on a regular sample $`\{{\mathbf{t}}_s\}_{1 \leq s \leq n_2} \subset {\mathds{R}}^\mathrm{d}`$ of translations using uniform FFTs , which requires a regular axis-aligned sample (i.e., voxelization) of the part and rotated tool. The procedure takes $`O(n_1 n_2 \log n_2)`$ which is very close to linear time in the number of sampled configurations $`n := n_1 n_2`$. Note that this procedure is called only once per round of the recursive outer-loop (Algorithm [alg_removal]). See the results in Fig. 2 of Section 6.

Note that storing the entire 6D set of $`{\mathsf{C}}-`$space obstacle $`\mathcal{O}(N,T)`$ for 3D parts is not practical. However, the contact space $`\mathcal{C}(N,T)`$, approximated by $`\mathcal{C}_\epsilon(N,T)`$, is a sparse subset of the $`{\mathsf{C}}-`$space that can be precomputed and queried later in constant-time.

Identifying Removable Support Components

In this Section, we reason about the tool assembly’s free space to automatically construct a maximal collection of removable support components. This collection is defined as the set of all support components that may be removed by the tool assembly without interfering with the near-net shape, except by the tool tip at the fracture points.

Let $`P, S \subseteq {\mathds{R}}^\mathrm{d}`$ denote the part and support structure, respectively, which only contact over their common boundary denoted by $`F := (P \cap S) = (\partial P \cap \partial S)`$. The initial near-net shape is obtained as $`N := (P \cup S)`$.

The support structure commonly consists of many connected components (called hereafter support components):

\begin{equation}
    S = \bigcup_{1 \leq i \leq n_S} S_i, \quad(S_{i} \cap S_{i'}) = \emptyset
    \quad\text{if}~ i \neq i', \label{eq_components}
\end{equation}

where $`S_i \subseteq {\mathds{R}}^\mathrm{d}`$ stands for the $`i`$$`^\text{th}`$ component ($`1 \leq i \leq n_S`$). We assume that a given support component is removable with a given cutting tool if the tip of the tool can access and fracture all contact regions between the support component and the part boundary in one or more collision-free configurations, i.e., positions and orientations of the tool at which the tool does not interfere with the part or other support components. The contact region $`F = (P \cap S) = (\partial P \cap \partial S)`$ is also decomposed into its connected components (hereafter called dislocation features):

\begin{equation}
    F = \bigcup_{1 \leq j \leq n_F} F_j, \quad(F_{j} \cap F_{j'}) = \emptyset
    \quad\text{if}~ j \neq j'. \label{eq_features}
\end{equation}

where $`F_j \subseteq {\mathds{R}}^\mathrm{d}`$ stands for the $`j`$$`^\text{th}`$ feature ($`1 \leq i \leq n_F`$). Note that the indexing scheme for ([eq_components]) and ([eq_features]) are different because every support component is connected to the part via two or more dislocation features hence $`n_F \geq 2 n_S`$. The nomenclature is illustrated in Fig. 7 for a 2D example.

Support structures are typically designed to have small contact regions with the part’s surface to enable easier fracturing and removal and to minimize surface roughness after removal. Note that sometimes support components are constructed from one location on the part to another, but it is preferred that they are constructed connect the part’s surface to the machine’s base plate. Moreover, one support component can be connected to the part at multiple locations, e.g., when they are designed to branch out from a main trunk. To accommodate the most general condition, we do not make any simplifying assumption on the shape, connectivity, number, or layout of support components and their dislocation features.

Let us position the tool in its original configuration in such a way that the cutter tip (or some point on the cutting surface) coincides with the origin of the coordinate system used as a reference frame to quantify the rigid transformations of the tool assembly $`T \subseteq {\mathds{R}}^\mathrm{d}`$ with respect to the near-net shape $`N \subseteq {\mathds{R}}^\mathrm{d}`$ at a given round. This choice ensures that a transformation $`\tau \equiv (r, {\mathbf{t}}) \in {\mathrm{SE}(\mathrm{d})}`$ brings the tool’s cutter (and not any other point on the tool) to the point $`{\mathbf{t}}\in {\mathds{R}}^3`$ in the Euclidean $`\mathrm{d}-`$space where the near-net shape resides. Although this choice is not necessary from a theoretical standpoint, it simplifies the accessibility analysis by querying the pre-computed $`{\mathsf{C}}-`$space map.

In a $`\mathrm{d}-`$dimensional space, the tool can operate with all $`\mathrm{d}(\mathrm{d}+1)/2`$ degrees of freedom, $`\mathrm{d}(\mathrm{d}-1)/2`$ of which are for rotations and the remaining $`\mathrm{d}`$ are for translations. We can define projections from $`{\mathrm{SE}(\mathrm{d})} = {\mathrm{SO}(\mathrm{d})} \rtimes {\mathds{R}}^\mathrm{d}`$ to $`{\mathrm{SO}(\mathrm{d})}`$ and $`{\mathds{R}}^\mathrm{d}`$, respectively, that map every rigid transformation $`\tau \equiv (r, {\mathbf{t}}) \in {\mathrm{SE}(\mathrm{d})}`$ to $`r \in {\mathrm{SO}(\mathrm{d})}`$ and $`{\mathbf{t}}\in {\mathds{R}}^\mathrm{d}`$:

\begin{equation}
    \begin{tikzcd}[column sep=0.3em]
    & {\mathrm{SE}(\mathrm{d})} \arrow[dl, "\pi_1"'] \arrow[dr, "\pi_2"] & \\
    {\mathrm{SO}(\mathrm{d})} & & {\mathds{R}}^\mathrm{d}
    \end{tikzcd}
    \begin{array}{l}
        \pi_1(\tau) = \pi_1(r, {\mathbf{t}}) := r, \\
        \\
        \pi_2(\tau) = \pi_2(r, {\mathbf{t}}) := {\mathbf{t}}, 
    \end{array} \qquad\qquad\qquad
\end{equation}

Notice that the second projection depends on the earlier choice of origin. Clearly, these maps are not invertible as functions. Nonetheless, we can define liftings that take every $`r \in {\mathrm{SO}(\mathrm{d})}`$ and $`{\mathbf{t}}\in {\mathds{R}}^\mathrm{d}`$ to a subspace of $`{\mathrm{SE}(\mathrm{d})}`$: ⁵

\begin{equation}
    \begin{tikzcd}[column sep=0.1em]
        & \mathcal{P}({\mathrm{SE}(\mathrm{d})}) & \\
        {\mathrm{SO}(\mathrm{d})} \arrow[ur, "\pi_1^{-1}"] & & {\mathds{R}}^\mathrm{d}\arrow[ul, "\pi_2^{-1}"']
        \end{tikzcd}
        \begin{array}{l}
            \pi_1^{-1}(r) = \big\{ (r, {\mathbf{t}}) ~|~ {\mathbf{t}}\in {\mathds{R}}^\mathrm{d}\big\}, \\
            \\
            \pi_2^{-1}({\mathbf{t}}) = \big\{ (r, {\mathbf{t}}) ~|~ r \in {\mathrm{SO}(\mathrm{d})} \big\}, \\
        \end{array}
\end{equation}

By extension, we can define set functions; for example, $`\pi_2^{-1}: \mathcal{P}({\mathds{R}}^\mathrm{d}) \to \mathcal{P}({\mathrm{SE}(\mathrm{d})})`$ such that $`\pi_2^{-1}(F)`$ assigns a copy of the entire $`{\mathrm{SO}(\mathrm{d})}`$ to every point $`{\mathbf{t}}\in F`$:

\begin{equation}
    \pi_2^{-1}(\Omega) := \{\pi_2^{-1}({\mathbf{t}}) ~|~ {\mathbf{t}}\in F\} \cong ({\mathrm{SO}(\mathrm{d})}
    \times F).
\end{equation}

Given a dislocation feature $`F_j \subset P`$, the set of all configurations at which it can be touched by the tool tip can be obtained in terms of the above projection/lifting maps:

\begin{equation}
    \mathpzc{f}_j := \mathpzc{f}(F_j, N, T) := \pi_2^{-1}(F_j) \cap \mathcal{C}(N, T).
\end{equation}

In other words, every dislocation feature is lifted to the $`{\mathsf{C}}-`$space by pairing it with all possible orientations. The pairing results in the set of all possible configurations that bring the tool tip to the dislocation feature (at different orientations). Among them, the ones that are contained within the contact space $`\mathcal{C}(N, T)`$ are collision-free, hence they represent the set of all accessible configurations. Hereafter, we refer to $`\mathpzc{f}_j \subset {\mathrm{SE}(\mathrm{d})}`$ as the contact fiber for the dislocation feature. Once again, for computational purposes, we can approximate the contact fibers via $`\epsilon-`$fibers, using the measurable $`\epsilon-`$contact region:

\begin{equation}
    \mathpzc{f}_j \approx \mathpzc{f}_\epsilon(F_j, N, T) := \pi_2^{-1}(F_j) \cap
    \mathcal{C}_\epsilon(N, T).
\end{equation}

In practice, the dislocation features are small enough to assume that touching any point on their circumference is sufficient to fracture it. As a result, we can simply project the fiber from $`{\mathrm{SE}(\mathrm{d})}`$ to $`{\mathrm{SO}(\mathrm{d})}`$ to collect all accessible orientations at all positions within the feature:

\begin{equation}
    \mathpzc{f}_j^\ast := \mathpzc{f}^\ast(F_j, N, T) = \pi_1 \big( \pi_2^{-1}(F_j) \cap
    \mathcal{C}(N, T) \big).
\end{equation}

whose $`\epsilon-`$fiber approximations are given as expected by:

\begin{equation}
    \mathpzc{f}_j^\ast \approx \mathpzc{f}^\ast_\epsilon(F_j, N, T) = \pi_1 \big(
    \pi_2^{-1}(F_j) \cap \mathcal{C}_\epsilon(N, T) \big).
\end{equation}

Figure 8 illustrates the fibers for a 2D example in which the dislocation features are contracted to points. From an algorithmic perspective, each $`\epsilon-`$fiber can be hashed into a list of sampled orientations anchored at a different dislocation features. A dislocation feature is accessible iff its fiber is nonempty. As a result, we observe that a given support component is removable iff every one of its dislocation features has a nonempty fiber.

Algorithm [alg_removal] describes the procedure for finding the contact fibers in a single round of recursive support removal planning. The procedure performs $`O(n_F)`$ queries on every one of the convolution fields precomputed and stored for $`n_1`$ sampled orientations, resulting in a total of $`O(n_F n_1)`$ queries—each convolution-read taking constant time. This analysis assumes that we need $`O(1)`$ queries per each dislocation feature, because each feature is sampled by one or at most a few point(s), which is reasonable. Every orientation that is accessible is pushed into a stack that represents the fiber for the queried feature.

Note that before performing the queries, we can shortlist the $`n_F`$ features rapidly to accessible ones by projecting the $`\epsilon-`$contact space to the Euclidean $`\mathrm{d}-`$space (i.e., losing orientation information) and finding its common points with the dislocation features:

\begin{equation}
    F_j ~\text{is accessible iff}~ \pi_2(\mathcal{C}(N, T)) \cap F_j \neq \emptyset.
\end{equation}

This means that there exists at least one point in the dislocation feature $`F_j`$ that belongs to the projected contact space. The projection ensures that there exists at least one non-colliding orientation at which the said point is accessible. Note that the projected contact space can be obtained by unifying the translational contact spaces $`\partial(N \oplus (-T_r))`$ for different orientations:

\begin{equation}
\pi_2(\mathcal{C}(N, T)) = \bigcup_{r \in {\mathrm{SO}(\mathrm{d})}} \partial (N \oplus (-T_r)),
\end{equation}

Once again, the contact space can be approximated by the $`\epsilon-`$contact criterion. As such, the above early test for accessibility can be rapidly computed by summing up the indicator functions of the translational contact spaces, which, in turn, are obtained as the $`(0, \epsilon)-`$interval level set of the convolution fields for different $`r-`$slices. The resulting pointset is a single field over the $`\mathrm{d}-`$space with a narrow support, containing all positions in the vicinity of the near-net shape’s boundary that can be accessed by at least one orientation. For every point $`{\mathbf{t}}\in {\mathds{R}}^\mathrm{d}`$, we accumulate the convolution value over $`r-`$slices for which the condition $`0 < ({\mathbf{1}}_N \ast {\mathbf{1}}_{-T_r})({\mathbf{t}}) < \epsilon`$ holds. This incurs no additional computation cost because as the convolutions are precomputed, their $`(0, \epsilon)-`$interval level sets can be extracted and accumulated on-the-fly. The result is a field with a narrow-band support over the $`\mathrm{d}-`$space that quantifies non-colliding orientations of a given point. A 2D example of the accumulated and projected field of contact measures is illustrated in Fig. 8 (c) (base field). A dislocation feature is accessible iff it has at least one point at which this function is nonzero. Note that this early test counts the accessible orientations; however, it does not tell us which orientations (i.e., the fiber).

Recursive Support Removal Rounds

It is likely that multiple support components are removable from the near-net shape at a given intermediate state. Intuitively, every round of the recursive algorithm peels off all removable components to expose a new set of components. The process is repeated until either all supports are removed, or the part is deemed non-manufacturable because some supports are inaccessible. See the results in Fig. 4 of Section 6.

At round$`-\mathtt{t}`$ of the support removal algorithm, ⁶ the near-net shape is $`N^{\mathtt{t}} := (P \cup S^{\mathtt{t}})`$ with initial conditions $`N^{\mathtt{0}} = N`$ and $`S^{\mathtt{0}} = S`$. Both $`N^{\mathtt{t}}`$ and $`S^{\mathtt{t}}`$ are monotonically reduced (in terms of set containment) from one round to the next, i.e., $`N^{\mathtt{t+1}} \subseteq N^{\mathtt{t}}`$ and $`S^{\mathtt{t+1}} \subseteq S^{\mathtt{t}}`$.

Let $`\mathbf{I}^\mathtt{t}`$ represent the indices for the maximal collection of removal supports at a given round, i.e., for every $`i \in \mathbf{I}^\mathtt{t}`$, the support component $`S_i \subseteq S^\mathtt{t}`$ is removable at round$`-\mathtt{t}`$. If $`\mathbf{J}(i)`$ represents the indices of the dislocation features for the same support component, then $`i \in \mathbf{I}^\mathtt{t}`$ iff for every $`j \in \mathbf{J}(i)`$, the fiber $`\mathpzc{f}_j`$ is nonempty, i.e.,

\begin{equation}
    \mathbf{I}^\mathtt{t} = \big\{ 1 \leq i \leq n_S ~|~ \forall j \in
    \mathbf{J}(i): \mathpzc{f}_j \neq \emptyset \big\}. \label{eq_indices}
\end{equation}

The remaining support for the next round is computed as: $`S^{\mathtt{t+1}} = S^{\mathtt{t}} - \bigcup_{i \in \mathbf{I}^\mathtt{t}} S_i`$. The algorithm continues until either of two termination criteria can happen:

1. $`N^{\mathtt{t}} = P`$, i.e., $`S^{\mathtt{t}} = \emptyset`$, which implies successful removal of the entire support; or

2. $`N^{\mathtt{t}} = N^{\mathtt{t+1}}`$ and $`S^{\mathtt{t}} = S^{\mathtt{t+1}}`$ which indicates that the remaining support components cannot be reached.

Figure 9 illustrates a few rounds of recursive support removal for a 2D example. The $`{\mathsf{C}}-`$space obstacle and contact space change as more supports are removed.

Necessary and Sufficient Conditions

Algorithm [alg_recursive] illustrates an implementation of the recursive algorithm. All potentially accessible support components are identified by checking if $`\pi_2(\mathcal{C}_\epsilon(N, T)) \cap F_j \neq \emptyset`$ for every $`j \in \mathbf{I}_i`$ for every remaining support component $`S_i \subseteq S^\mathtt{t}`$. If at any given round, no supports are identified for removal, the algorithm returns a failure message indicating that the support structure is inherently not removable. In other words, this test provides a necessary condition for the AM post-process to be feasible. While the test is quite useful in finding the order in which the support components should be fractured, it does not provide a sufficient condition. For example, the test does account for the situations where there is a collision-free final configuration to touch the dislocation feature, but there may not exist a collision-free path in the $`{\mathsf{C}}-`$space that brings the tool from its initial configuration to the final pose. This situation can happen if the free space $`(\mathcal{O}(N,T))^c`$ is not path-connected, and the initial and final configurations are located in different connected components. Figure 10 illustrates an example of a false-positive. Such scenarios are rare if the support structure is designed properly.

Although analyzing path-connectedness is important, $`{\mathrm{SE}(\mathrm{d})}`$ is high-dimensional for $`\mathrm{d}\geq 3`$ and the free space topology can be highly complex. As opposed to reasoning about the topology, we can use a sampling-based motion planner (e.g., based on probabilistic roadmaps ) to find a collision-free path from a reasonable starting configuration in $`(\mathcal{O}(N,T))^c`$ to at least one configuration in the fiber.