Support Diminution Design for Layered Manufacturing of Manifold Surface Based on Variable Orientation Tracking

Abstract

This article proposes a support diminution design method for layered manufacturing of manifold surface based on variable orientation tracking (VOT). We aim at reducing the external support or upholders to a minimum with maximum possibility theoretically to save material and diminish material stripping effect (MSE), thereby improving the bilateral surface precision either exterior or interior. The cosmic gravity effect criterion is first used to extract surface need support from manifold surface with various materials by considering the balance force involving material characteristics and inclination angle. In the light of this criterion theory, varying the substrate normal orientation (SNO), namely workbench, for each layer in printing coordinate system, may break the balance between gravity and its equilibrium force. Therefore, the optimal SNO can be rigorously calculated using mathematical harmonic analysis among the continuous domain. To serve for the multidegree of freedom (DOF) on account of SNO, a reconfigurable VOT robot with six-axis DOF is developed for 3D printing (3DP). The matched servo controller is successfully implemented to accurate tracking of both orientation and Cartesian coordinates, using forward kinematic chains as well as reverse kinematic tracking. What is more, the end-effector (extruder) is holding perpendicular to the substrate workbench. The physical experiment that takes human external ear auricle, for example, using a layer-based process is implemented via VOT. The MSE due to supporting material can be clearly observed and diminished using an optical microscope. The stripped material from external support via diminution design can be evaluated quantitatively by electronic weighting balance. All of which indicate the findings that external support in 3DP can be virtually reckoned and diminished using VOT rather than the so-called build orientation traversal method. The VOT method upon which we touched can be widely applied to various layered manufacturing of accurate structure, for instance, cantilever, sandwich, and scaffolds in the occasion needing precise curtailment of outer support multimaterial.

Introduction

The 3D printing (3DP) or additive manufacturing (AM) fundamentally uses a material additive process through which products are built on a mostly layer-by-layer basis, via a series of cross-sectional slices, using potential external support structures in general cases, origin from the whatever geometry manifold model. 3DP uses discrete materials such as liquid, powder, granule, and filament to accumulate layers based on the 3D CAD metadata of the object, and relies on the bonding force between the materials to finally produce the diversified object, using various techniques that include fused deposition modeling (FDM), stereolithography appearance, selective laser sintering, selective laser melting, laminated object manufacturing, digital light processing, continuous liquid interface production, and so on. All the manufacturing process is generally carried out on the Earth whereon the cosmic gravity effect (CGE) affects everything.

The last centuries have unconsciously seen a steady increase in average human life expectancy all around the world, particularly due to the advances across antibiotics, vaccines, availability of better health care, and improved hygiene. There exist a huge market and demand in transplantation and reconstruction of organs.^1–3 And the typical way to do that is possibly bio-3DP or bio-AM^4–6 for tissue engineering (TE), such as scaffolds, stents, artificial skin, and prostheses. The increasing requirements on fabrication of issues and even intravital organs worldwide are putting forward a higher request to the original design.

Minimizing external support is an important issue in 3DP to save materials and improve processing quality. Chowdhury et al. identified support facets with simple angle criterion.⁷ Ezair et al. found that the volume of the support is a continuous but nonsmooth function, with respect to the orientation angles.⁸ Extrusion-based bioprinting is a rapidly growing technology that has made substantial progress during the last decade. It has great versatility in printing various biologics, including cells, tissues, tissue constructs, organ modules, and microfluidic devices, in applications from basic research and pharmaceutics to clinics.⁹ Stuart et al. defined and refined the concept of “printability” and reviewed seminal and contemporary studies to highlight the current “state of play” in the field with a focus on bioink composition and concentration, manipulation of nozzle parameters, and rheological properties.¹⁰ 3D-printed scaffolds and 3D bioprinting technique have the potential to develop a fully functional heart construct that can integrate with native tissues rapidly. Wassel et al. fabricated scratch-resistant nonfouling surfaces via grafting nonfouling polymers on the pore walls of supported porous oxide structures where the organic/inorganic nanocomposite films provide scratch-resistant antifouling surfaces.¹¹

Additional support structures are often needed, which leads to material, time, and energy waste. The external support might induce excessive material and energy consumption, meanwhile, reducing the surface accuracy, whether exterior or interior, and even upper surface and lower surface. Therefore, reducing the external support is apparently somewhat of a bottleneck and of great significance to improve the accuracy. Seepersad reviewed the advantages and developments of 3DP.¹² Habib presented an investigation on designing and fabricating possible scaffolds of a different internal structure with desired porosity and surface-area-to-volume ratio for TE by using FDM.¹³ Mirdamadi presented a modified method of embedded bioprinting, which allows maintaining freestanding 3D-printed structures in cell culture conditions for extended periods of time.¹⁴ Walker presented a framework for using isothermal curing kinetics and transient rheological data to 3D print a curing thermoset silicone without support.¹⁵ Hildreth presented that overhanging surfaces often require support structures to be fabricated and minimize thermally induced distortion. Unlike polymer AM processes, soluble sacrificial support materials have not been identified and characterized for metallic materials, as a result, support structures in 3D-printed metals must be removed using additional machining operations.¹⁶ Lefky demonstrated two approaches to dissolve supports. The approaches replace slow machining operations with fast electrochemical bath.¹⁷ Ameen presented a segmentation strategy for the support. The strategy can reduce the support without reducing the accuracy of overhang.¹⁸ Jiang et al. proposed a four-step strategy of multipart production for reducing support consumption. When printing a group of parts in the same build vat or chamber, the strategy optimizes the print orientation for each part, combines every two parts based on the geometries, proposes several possible multipart combinations, and then selects the optimal part positions for fabrication.^19,20

The potential degree of freedom (DOF) of design offered by AM is, however, often limited when printing complex geometries due to an inability to support the stresses inherent within the manufacturing process. Many scholars have studied it from the perspective of geometric calculation theory. Di Angelo presented the objective functions of build direction and calculated the best direction for quality and support with the objective functions.²¹ In addition, Llewellyn-Jones et al. proved the ability to automatically generate curved layer G-code tool paths for arbitrary shapes from a typical CAD model file, which are then printed with a Delta 3D printer. Experimental results showed the efficacy of using curved layers to improve the surface finish of a printed component.²² Xu et al. presented a curved layer-based process planning algorithm for multiaxis printing of an arbitrary freeform solid part.²³ Chen et al. studied so-called curved layer FDM by allowing the sliced layers to have variable thicknesses and adjusting the build direction adaptively concerning the surface normal to print thin-shell models.²⁴ Most articles depend upon solely build orientation to improve accuracy, which is prone to impracticality, whereas the build orientation method seems ineffectual, in many cases particularly for holosymmetric body.

The traditional 3DP devices, such as the most basic XYZ printer and Delta printer, have only three DOFs. The printing nozzle is always perpendicular to the carrier platform, which makes it impossible to print when the printing platform is inclined at a certain angle. It is difficult for the 3-DOF printer to print complex models, especially in mechanical and medical fields. Inspired by computer numerical control and robotics, multi-DOF 3DP devices ensure stiffness and repeatability, and give the possibility of new path generation; this can handle the typical problem of 3DP such as staircase effect (SE), angle deposition, and minimize or avoid the use of support material. For typical printers with gantry orthogonal three DOFs, the feasible way to tackle these problems would be to reduce the thickness of each layer, but this would likewise bring unacceptable production time and support material waste. Morocho and colleagues designed a 3D printer prototype based on a parallel robot, which decreases printing time without losing quality in the final product.²⁵ Dai tried to fabricate 3D models on a robotic printing system equipped with multiaxis motion,²⁶ nevertheless, the theoretical model of mechanism is insufficient. Fiore et al. designed a five-DOF 3D printer based on a five-DOF parallel kinematic manipulator to cope with SE and angle deposition.²⁷ Wu et al. designed a five-DOF wireframe printer with two extra rotation DOFs for the carrier platform and presented a method to print arbitrary meshes.²⁸ However, the more redundant motion axis may lead to rotation tool center point (RTCP) effects and rotation part center point effects, which are absent in the description.

These publications have more or less witnessed the development of 3DP for high-performance requirements. Nevertheless, there still lacks a theoretical method to diminish external support for 3DP of complex manifold surface. Therefore, based on our previous work,^29–31 a support diminution design method for layered manufacturing of manifold surface based on variable orientation tracking (VOT) is herein proposed to reduce the external support or upholders to a minimum with maximum possibility theoretically. The VOT actually means adaptively varying workbench (carrier) normal for each layer to minimize external support till support-free 3DP considering interpolation smoothness using reverse kinematics via more DOFs. The VOT method is carried out on the 3DP system and verified by a physical experiment concerning manifold surface precision measurement and electronic weighting balancing experiment.

Printing Coordinate System of the Manifold Surface Model

The mechanical equipment for AM of manifold surface in the orthogonal printing coordinate system (PCS) is schematically shown in Figure 1. The equipment mainly includes the following: power system, electrical system, mechanical feeding system, carrier workbench, material feeding system, end-effector (EE) (nozzle), and so on. The maximum print strokes of a printer along $x,y,z$ direction are denoted as $x_p,y_p,z_p$, respectively. It's important to note that the layer thickness interval $\left[d_{min},d_{max}\right]$ of the mechanical equipment affects the accuracy of the equipment, to a large extent. For example, the d can be set about 0.02–0.5 mm in FDM.

FIG. 1.

Mechanical equipment for AM of manifold surface in orthogonal PCS. AM, additive manufacturing; PCS, printing coordinate system.

The build orientation has an effect on precision, external support, and printing stroke. Hence, the factors that are affected should be considered in advance. For a nominal geometry, namely polyhedron manifold model M in R³ to be fabricated, let A_i, $i=0,\cdots ,N-1$ be the N facets, with vertices ($a_i,b_i,c_i,$), which are assumed to be ordered counterclockwise on A_i. The axis aligned bounding box (AABB) of M itself is generated to define the scale of the model in a model coordinate system (MCS). The length of the AABB along the $x,y,z$ direction can be represented by $x_b,y_b,z_b$, which should be within the scope of $x_p,y_p,z_p$.

In PCS, abiding by predefined uniform build orientation, to slice the manifold model into planar layers, the normalized height h_n of i-th layer can be calculated hereby to represent the normalized height of the layer:

$$h_n=\frac{z_i}{z_b}{h}_n\in \left(0,1\right].$$ (1)

The total area of surface $S_{object}$ of the object M to be printed equals to the following:

$$S_{object}={\sum }_0^{N-1}{A}_i={\sum }_0^{N-1}\frac{\left|a_i{b}_i\times a_i{c}_i\right|}{2}$$ (2)

The enclosed volume $V_{object}$ of the object M to be printed is equal to the sum of the tetrahedron volume between each patch and the origin, which can be computed via the divergence theorem:

$$V_{object}={\sum }_0^{N-1}{V}_i={\sum }_0^{N-1}\frac{1}{3}{a}_i\cdot n_i={\sum }_0^{N-1}\frac{1}{6}{a}_i\cdot {\widehat{n}}_i$$ (3)

where ${\widehat{n}}_i$ is the outer normal of the triangle, ${\widehat{n}}_i=a_i{b}_i\times a_i{c}_i$, n_i is the outer unit normal, $n_i=\widehat{n_i}∕\left|{\widehat{n}}_i\right|$.

For extrusion-based or layer-based 3DP, there exists the inherent SE. The SE can be manifested by cusp height $\delta$ and volume error V_e. The cusp height can be defined as follows:

$$\delta =d\times \left|cos\alpha \right|$$ (4)

$$\alpha =co{s}^{-1}\frac{n_z\times n_f}{n_z\cdot n_f}\alpha \in \left[0,\pi \right]n_z=\left(SN{O}_x,SN{O}_y,SN{O}_z\right)$$ (5)

where d is layer thickness, α is the included angle between the z normal vector n_z and facet normal vector n_f.

The layered volume V_e can represent the error in cubic space.

$$V_e\left(n\right)={\sum }_{i=1}^{card\left(d\right)}{V_e}_i$$ (6)

where $card\left(d\right)$ is the cardinality of the layer thicknesses set $d$. ${V_e}_i$ is the volume error of i-th layer.

The layer thickness sequence d with all positive real number elements can be denoted by:

$$d=diff\left(Z\right)=\left\{d_1,d_2,d_i,\cdots ,d_{n-1}\right\}{d}_i\in R^{+}\&i\in \left[1,n-1\right]$$ (7)

where diff means first-order forward difference, arbitrary $\forall d_i=z_{i+1}-z_i$, and $\forall d_i\in \left[d_{min},d_{max}\right]$ for $i\in \left[1,n-1\right]$.

Obviously, for uniform slicing, layer thickness is equal to each other, $\forall d_i=0$. For adaptive slicing, layer thickness varies, $\exists d_i\ne 0$.

The adaptive slicing is generally realized via variable thickness using multiple criteria such as cusp height $\delta$ and volume error V_e. Therefore, the adaptive slicing architecture can be expressed in the below mathematical model.

$$\left\{\begin{array}{c}findd\\ mincard\left(d\right)\\ Subjectto:f\left(d\right)\le {\delta }_{max}\\ d\in \left[d_{min},d_{max}\right]\\ \end{array}\right.$$ (8)

where $card\left(d\right)$ is the cardinality of the d,$d\in \left[d_{min},d_{max}\right].$ $f\left(d\right)$ is the constraint function that presents the accuracy requirement, and ${\delta }_{max}$ can be the maximum cusp height of each layer.

External Support Structure of Overhang Material Due to CGE

CGE criterion to extract surface need support

In 3DP, materials can be accumulated without extra supporting structures (i.e., the model can support itself without causing deformation) whenever the material is enough to balance CGE. We hereby proposed CGE criterion to extract surface need support, from a more essential perspective, which satisfies the following:

$$G>max\left(F_b\right)G=mg$$ (9)

where G is the material gravity (N), F_b is the balance force generated from material affected by many factors such as material characteristics and inclination angle. m is mass (kg), g is acceleration due to gravity (N/kg).

The geometric included angle between the model surface and its printing orientation is defined as inclination angle $\alpha$. Most publications use a threshold angle ${\alpha }_{overhang}$, also called the maximal self-support angle to extract overhang region.^7,8,32 However, ${\alpha }_{overhang}$ is usually difficult to be set as a precise arithmetic number owing to the material variation.

$$\alpha =\theta \left(n_f,n_z\right)\alpha \in \left[0,\pi \right]$$ (10)

As we all know, different materials have different properties of solidification strength and van der Waals' (VDW) force at even mesoscopic scale. The imbalance of VDW forces may result in surface tension. The surface tension of nonmetals is mostly smaller than that of metals. On the basis of volume of the fluid method in computational fluid dynamics, the material equations can be derived.³³

$$F_s=\sigma \kappa n_f\delta \left(x\right)$$ (11)

where scalar F_s is the surface tension force (N · m⁻²). $\sigma$ is the coefficient of fluid surface tension (N · m⁻¹),$\sigma \in R^{+}$. $\kappa$ is the curvature of the surface (m⁻¹). n_f is the unit normal of the surface. $\delta \left(x\right)$ is a Delta function concentrated at the material interface.

$$\sigma \propto \mu ∕T$$ (12)

where $\mu$ is dynamic viscosity (N·s·m⁻²), T is temperature (K), and $\propto$ is directly proportional symbol.

The surface tension F_s contributes most of the F_b in some common cases for most materials used in the 3DP, which include the following: acrylonitrile butadiene styrene, polylactide, polyvinyl alcohol, thermoplastic polyurethanes (TPU), polycarbonate, polyamide, acrylonitrile styrene acrylate copolymer, high-impact polystyrene, thermoplastic polymers, polylactic-co-glycolic acid (PLGA), and so on.

Herein taking commonly used PLGA and TPU for comparative instance, some of the physical characteristics of the two are similar, whereas others are quite different. For PLGA, the intrinsic viscosity is about 0.2–2.5 dL/g via an Ubbelohde viscosity meter, the viscosity-average molecular weight of PLGA and the number average molecular weight attain 1–40 w, and the dynamic viscosity $\mu$ is about 30 N·s·m⁻². Whilst for TPU, the tensile strength of TPU is about 35 N/mm², the elongation at break can reach 800% or thereabouts, and the dynamic viscosity $\mu$ is about 20 N·s·m⁻². Therefore, the F_s is different even for the same inclination angle $\alpha$ of the manifold surface.

Accordingly, it is practicable to consider the differences in the properties of materials when designing printing parameters for AM using Eqs. (9–12). Therefore, compared with the inclination angle method, the proposed CGE criterion is more essential and more suitable for various materials (Fig. 2).

FIG. 2.

Balance force F_b is influenced by material property with distinctive molecular structure bonding diagram of PLGA (a) and TPU (b). PLGA, polylactic-co-glycolic acid; TPU, thermoplastic polyurethanes.

Determine the build orientation of each layer via principal component analysis

In the light of this criterion theory, varying the substrate normal orientation (SNO), namely workbench, for each layer in PCS may break the balance between gravity and its balance force. The optimization model can be built for each layer on the basis of Eqs. (9–12)

$$\left\{\begin{array}{c}find{F}_b\\ minf\left(F_b\right)\\ Subjectto:g\left(F_b\right)\le {\delta }_{max}\\ \end{array}\right.$$ (13)

where $F_b$ is balance force (N), $f\left(F_b\right)$ is the objective functions, regarding surface area need support, total support length, and support material volume. $g\left(F_b\right)$ is equivalent transformational constraint function set under threshold ${\delta }_{max}$, which can be print stroke, surface accuracy requirement, and so on.

Academically speaking, the F_b for each layer is an infinite domain. It involves analytic mathematical programming methods, covering harmonic analysis, linear and nonlinear programming and multiobjective programming. And it is obviously difficult to find the optimal solution set precisely, just depends on the discrete method. Therefore, we propose and implement a method for calculating optimal solution in manifold surface by projecting high-dimension feature to low-dimension latitude feature of best fit using principal component analysis (PCA).

Let X_i be vector of i-th feature of the problem to be solved using PCA, the data standardization means subtracting the average $\overline{X_i}$ from the original data X_i. The covariance matrix can be constructed as an n order square symmetric matrix, which satisfies the following:

$$\left\{\begin{array}{c}C={\left(C_{i,j}\right)}_{n\times n}i,j\in \left[1,n\right]\\ \begin{array}{c}\forall C_{i,j}=C_{j,i}\\ \begin{array}{c}{C}_{i,j}=cov\left(X_i,X_j\right)\\ X_i=X_i-\overline{X_i}\\ \end{array}\\ \end{array}\\ \end{array}\right.$$ (14)

where cov is covariance, card (X_i) = m, $m\in Z^{+}$.

The eigenvalue decomposition algorithm is used to obtain eigenvalue and eigenvector of covariance square matrix C. The calculated eigenvalues are ranked and the s largest eigenvalues are picked. The eigenvectors corresponding to the eigenvalues are also selected and combined as U.

$$Y_{m\times s}=X_{m\times n}\times U_{n\times s}s\in \left[1,n\right)$$ (15)

where Y contains the s dimension data of the original vector. On the basis of Y and F_s, the appropriate vectors are calculated. Thus, normal orientation n_z of the fitted feature can be calculated with less extra effort.

Feasible solution ratio of each layer

For each layer in 3DP, after PCA manifold learning using Eqs. (14 and 15), herein we propose the feasible solution ratio ${\xi }_{blue}$, which equals to feasible solution amount divided by overall amount of domain of definition

$${\xi }_{blue}=\frac{card\left({\tilde{F}}_b\right)}{card\left(F_b\right)}\times 100\%{\xi }_{blue}\in \left[0,1\right]$$ (16)

where ${\tilde{F}}_b$ means feasible solution of Eq. (13), and F_b means original domain of definition.

Innovatively, the ${\xi }_{blue}$ provides the theoretical principle to quantize the external support structure material for arbitrary given manifold model. The feasible solution ratio ${\xi }_{blue}$ can indirectly reflect the AM complexity, either geometric shape or material property. The lower the value ${\xi }_{blue}$, the more difficult AM is. Once ${\xi }_{blue}$ is small enough, the so-called support-free would not yet be guaranteed. Varying SNO might cause melt sagging. Therefore, the minimal variation between the successive layers considering interpolation smoothness helps reducing the AM defects.

Variable Orientation of Each Layer to Minimize External Support

Overall conceptual design to realize variable orientation and thickness

It is common knowledge that the simplest and typical 3DP requires only three moving axes, namely DOF. To extend the DOF, the corresponding mathematical model is established and deduced. The coordinate system (CS) of a modular reconfigurable robot is shown in Figure 3.

FIG. 3.

Coordinate systems of VOT robot where $OXYZ$ is the base CS; $O_p{X}_p{Y}_p{Z}_p$ is the PCS; $O_n{X}_n{Y}_n{Z}_n$ is the EE CS;$O_{m1}{X}_{m1}{Y}_{m1}{Z}_{m1}$ is the carrier platform CS;$O_{m2}{X}_{m2}{Y}_{m2}{Z}_{m2}$ is the EE support CS. CS, coordinate system; EE, end-effector; VOT, variable orientation tracking.

Note that $O_n{X}_n{Y}_n{Z}_n$ is at the EE, whose origin is located at the center $O_n$ of the EE, and its initial coordinate axis direction is consistent with the $O_{m2}{X}_{m2}{Y}_{m2}{Z}_{m2}$; $O_{m1}{X}_{m1}{Y}_{m1}{Z}_{m1}$is at two revolving axes of the carrier platform, and its origin is the intersection of two revolving axes;$O_{m2}{X}_{m2}{Y}_{m2}{Z}_{m2}$ is at the EE support, and its origin is the position of the revolving center of the EE on the EE support. The coordinate axis direction of $O_{m2}{X}_{m2}{Y}_{m2}{Z}_{m2}$ is always consistent with $OXYZ$.

It is assumed that the initial state of the printing robot is that the EE support CS is at the origin of $XYZ$. The axis direction of the EE swing axis is the same as that of the Z-axis of the EE support CS. The carrier platform is in the initial horizontal position, and the angles of rotary axes $A\prime$, $A$, $C\prime$ relative to the initial state are all $0^{\circ }$.

Let EE length $L=O_{m2}{O}_n$, and therefore, position vector of the origin $O_{m2}$ in $O_n{X}_n{Y}_n{Z}_n$ is $\overrightarrow{P_{M1}}\left(0,0,-L\right)$, and the position vector of the origin $O_{m1}$ in $OXYZ$ is $\overrightarrow{P_{M0}}\left(P_{MX0},P_{MY0},P_{MZ0}\right)$.

In $O_n{X}_n{Y}_n{Z}_n$, the EE position point coordinate is ${\left[\begin{array}{ccc}0& 0& 0\\ \end{array}\right]}^T$, the EE direction vector is ${\left[\begin{array}{ccc}0& 0& 1\\ \end{array}\right]}^T$, and the direction of $O_p{X}_p{Y}_p{Z}_p$ is in accordance with $O_{m1}{X}_{m1}{Y}_{m1}{Z}_{m1}$. The position vector of $O_p$ in $O_{m1}{X}_{m1}{Y}_{m1}{Z}_{m1}$ is $\overrightarrow{P_{M2}}\left(0,0,H\right)$, where $H$ = $O_{m1}{O}_p$.

Establishment of kinematic chains of VOT robot with six DOFs

Let $\overrightarrow{P_M}\left(P_{MX},P_{MY},P_{MZ}\right)$ be the position vector of $O_{m2}{X}_{m2}{Y}_{m2}{Z}_{m2}$ relative to its initial state. The ${\theta }_{A\prime }$, ${\theta }_A$, and ${\theta }_{C\prime }$ refer to the angles of rotation axis $A\prime$, $A$, $C\prime$ relative to the initial state, respectively (determining the positive direction according to the right-hand helical CS).

The expressions of the direction vector of the EE and the position coordinates of the EE in $O_p{X}_p{Y}_p{Z}_p$ are $\overrightarrow{u}\left(u_x,u_y,u_z\right)$ and $P\left(P_X,P_Y,P_Z\right)$, which can be obtained by one rotation of $O_n{X}_n{Y}_n{Z}_n$ relative to $O_{m2}{X}_{m2}{Y}_{m2}{Z}_{m2}$, one translation of $O_{m2}{X}_{m2}{Y}_{m2}{Z}_{m2}$ relative to $OXYZ$, and two rotational coordinate transformations of $O_{m1}{X}_{m1}{Y}_{m1}{Z}_{m1}$ relative to itself. As shown in Figure 4, according to the PCS, including the position vectors of each CS in its adjacent CS and the motion vectors generated whenever the printing robot moves, two kinematic chains describing the position $P\left(P_X,P_Y,P_Z\right)$ of an EE point and the corresponding direction vector $\overrightarrow{u}\left(u_x,u_y,u_z\right)$ of the EE in the base CS can be obtained.

FIG. 4.

The kinematic chains corresponding to the position $P\left(P_X,P_Y,P_Z\right)$ of the EE point and EE direction vector $\overrightarrow{u}\left(u_x,u_y,u_z\right)$.

Recursive homogeneous coordinate transformation of VOT robot

The transformation formulas corresponding to the coordinates P of the position points of the EE and the direction vector $\overrightarrow{u}$ of the EE both satisfy the following:

where ${}_{O_{m2}}^{O}T=T\left(P_M\right)$ means that the transformation from $O_{m2}$ coordinate-system to $O$ CS requires a translation transformation based on vector $\overrightarrow{P_M}$.

Where ${}_{O_n}^{O_{m2}}T=R\left({\theta }_A\right)T\left(P_{M1}\right)$ means that the transformation from $O_n$ coordinate-system to $O_{m2}$ coordinate-system must first undergo a translation transformation based on vector $\overrightarrow{P_{M1}}$ and then a rotation transformation based on angle ${\theta }_A$.

Where the homogeneous coordinate matrix of ${}^{O_n}{O}_n$ is expressed as ${\left[\begin{array}{cccc}0& 0& 0& 1\\ \end{array}\right]}^T$.

where the homogeneous coordinate matrix of ${}^{O_p}P$ is expressed as ${\left[\begin{array}{cccc}{P}_X& P_Y& P_Z& 1\\ \end{array}\right]}^T$.

Where the homogeneous coordinate matrix of${}^{O_n}\overrightarrow{P_{M1}}$ is expressed as ${\left[\begin{array}{cccc}0& 0& 1& 0\\ \end{array}\right]}^T$.

The homogeneous coordinate matrix of${}^{O_p}\overrightarrow{u}$ is expressed as${\left[\begin{array}{cccc}{u}_X& u_Y& u_Z& 0\\ \end{array}\right]}^T$.

Through matrix operation, the expressions of direction vector $\overrightarrow{u}\left(u_x,u_y,u_z\right)$ of EE and coordinate $P\left(P_X,P_Y,P_Z\right)$ of EE position point can be obtained:

$$\left\{\begin{array}{c}{\left[\begin{array}{cc}{u}_X& u_Y\\ \end{array}\begin{array}{cc}{u}_Z& 0\\ \end{array}\right]}^T=R\left(-{\theta }_{C\prime }\right)\cdot T\left(-P_{M2}\right)\cdot R\left(-{\theta }_{A\prime }\right)\cdot T\left(-P_{M0}\right)\cdot T\left(P_M\right)\cdot R\left({\theta }_A\right)\cdot T\left(P_{M1}\right)\cdot {\left[\begin{array}{cc}0& 0\\ \end{array}\begin{array}{cc}1& 0\\ \end{array}\right]}^T\\ {\left[\begin{array}{cc}{P}_X& P_Y\\ \end{array}\begin{array}{cc}{P}_Z& 1\\ \end{array}\right]}^T=R\left(-{\theta }_{C\prime }\right)\cdot T\left(-P_{M2}\right)\cdot R\left(-{\theta }_{A\prime }\right)\cdot T\left(-P_{M0}\right)\cdot T\left(P_M\right)\cdot R\left({\theta }_A\right)\cdot T\left(P_{M1}\right)\cdot {\left[\begin{array}{cc}0& 0\\ \end{array}\begin{array}{cc}0& 1\\ \end{array}\right]}^T\\ \end{array}\right.$$ (19)

The homogeneous coordinate transformation equations of translation and rotation Eqs. (18 and 19) are all depicted by nonsingular matrix:

$$T\left(P_M\right)=\left[\begin{array}{cccc}1& 0& 0& P_{MX}\\ 0& 1& 0& P_{MY}\\ 0& 0& 1& P_{MZ}\\ 0& 0& 0& 1\\ \end{array}\right]$$ (20)

$$R\left({\theta }_A\right)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& cos{\theta }_A& -sin{\theta }_A& 0\\ 0& sin{\theta }_A& cos{\theta }_A& 0\\ 0& 0& 0& 1\\ \end{array}\right]$$ (21)

$$T\left(P_{M1}\right)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& -L\\ 0& 0& 0& 1\\ \end{array}\right]$$ (22)

$$R\left(-{\theta }_{A\prime }\right)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& cos{\theta }_{A\prime }& sin{\theta }_{A\prime }& 0\\ 0& -sin{\theta }_{A\prime }& cos{\theta }_{A\prime }& 0\\ 0& 0& 0& 1\\ \end{array}\right]$$ (23)

$$T\left(-P_{M2}\right)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& -H\\ 0& 0& 0& 1\\ \end{array}\right]$$ (24)

$$R\left(-{\theta }_{C\prime }\right)=\left[\begin{array}{cccc}cos{\theta }_{C\prime }& sin{\theta }_{C\prime }& 0& 0\\ -sin{\theta }_{C\prime }& cos{\theta }_{C\prime }& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ \end{array}\right]$$ (25)

$$T\left(-P_{M0}\right)=\left[\begin{array}{cccc}1& 0& 0& -P_{MX0}\\ 0& 1& 0& -P_{MY0}\\ 0& 0& 1& -P_{MZ0}\\ 0& 0& 0& 1\\ \end{array}\right]$$ (26)

Kinematic equations of VOT robot

Substituting the homogeneous transformation matrices into Eq. (19), the coordinate transformation equations of the VOT robot can be obtained as follows:

$$\left\{\begin{array}{c}{u}_X=sin{\theta }_{C\prime }\left(-sin{\theta }_{A}cos{\theta }_{A\prime }+sin{\theta }_{A\prime }cos{\theta }_A\right)\\ u_Y=cos{\theta }_{C\prime }\left(-sin{\theta }_{A}cos{\theta }_{A\prime }+sin{\theta }_{A\prime }cos{\theta }_A\right)\\ u_Z=sin{\theta }_{A\prime }sin{\theta }_A+cos{\theta }_{A\prime }cos{\theta }_A\\ P_X=cos{\theta }_{C\prime }\left(P_{MX}-P_{MX0}\right)+sin{\theta }_{C\prime }\left[\begin{array}{c}\left(Lsin{\theta }_A+P_{MY}-P_{MY0}\right)cos{\theta }_{A\prime }+\\ \left(-Lcos{\theta }_A+P_{MZ}-P_{MZ0}\right)sin{\theta }_{A\prime }\\ \end{array}\right]\\ P_Y=-sin{\theta }_{C\prime }\left(P_{MX}-P_{MX0}\right)+cos{\theta }_{C\prime }\left[\begin{array}{c}\left(Lsin{\theta }_A+P_{MY}-P_{MY0}\right)cos{\theta }_{A\prime }+\\ \left(-Lcos{\theta }_A+P_{MZ}-P_{MZ0}\right)sin{\theta }_{A\prime }\\ \end{array}\right]\\ P_Z=-sin{\theta }_{A\prime }\left(Lsin{\theta }_A+P_{MY}-P_{MY0}\right)+\left(-Lcos{\theta }_A+P_{MZ}-P_{MZ0}\right)cos{\theta }_{A\prime }-H\\ \end{array}\right.$$ (27)

Based on the mechanical structure of the VOT robot, the corresponding mathematical model is established and expressed in the form of a CS and kinematic chain group. The coordinate transformation equations of position and pose $P\left(P_X,P_Y,P_Z\right)$and $\vec{u}\left(u_x,u_y,u_z\right)$ to six tracking control parameters ${\left[\begin{array}{cccccc}{\theta }_{A\prime }& {\theta }_A& {\theta }_{C\prime }& P_{MX}& P_{MY}& P_{MZ}\\ \end{array}\right]}^T$ are deduced, and the kinematic model of the VOT robot is then established.

VOT via Reverse Kinematics of Six-DOF Robot

Prioritized quadrant projection for unique reverse kinematics

To determine the unique exact solution of the VOT robot, the prioritized quadrant projection (PQP) judgment criteria are proposed.

If the obstacle avoidance requirement of the printer relative to the movement of the EE is not yet considered, the unique orientation solution can be obtained by the three steps.

First step, with the highest priority, according to the $u_x,$ and $u_y$ components of the direction vector $\overrightarrow{u}\left(u_x,u_y,u_z\right)$ of the EE, the quadrant position of $\overrightarrow{u}$ in the Z-direction overhead view of the PCS is determined, and thus, the unique optimal solution ${\theta }_{C\prime }$ of the carrier platform is determined.

Second step, the angle $\alpha$ between $\overrightarrow{u}\left(u_x,u_y,u_z\right)$ and Z-axis of the PCS is determined according to the components of $u_x,u_{y}and{u}_z$ of the direction vector $\overrightarrow{u}$ of the printer, and the EE swing control parameter ${\theta }_A$ and hotbed workbench swing control parameter ${\theta }_{A\prime }$ are uniquely determined according to the solving principle of swing the printer first and then swing the hotbed workbench.

Third step, according to the method of solving nonhomogeneous linear equations, the unique orientation solution of the 3D space motion control parameter $\overrightarrow{P_M}\left(P_{MX},P_{MY},P_{MZ}\right)$ of the EE support is obtained. Thus, a set of unique orientation solutions ${\left[\begin{array}{cccccc}{\theta }_{A\prime }& {\theta }_A& {\theta }_{C\prime }& P_{MX}& P_{MY}& P_{MZ}\\ \end{array}\right]}^T$ can be obtained sequentially ${\theta }_{C\prime }\to {\theta }_A\to {\theta }_{A\prime }\to P_{MX},P_{MY},P_{MZ}$.

Unique optimal solution of rotation control parameter ${\theta }_{C\prime }$ of carrier platform

The quadrant position of $\overrightarrow{u}$ in the Z-direction overhead view of the printer CS is determined. The quadrant of $\overrightarrow{u}$ is determined by the positive and negative of $u_x,u_y$. The PQP is listed in Table 1.

Table 1.

Prioritized Quadrant Projection Judgment of $\overrightarrow{u}$ and Control Parameter Value

Location depicted in Figure 5	Judgment basis	Value of parameter ${\theta }_{C\prime }$
First quadrant I	$u_X>0,u_Y>0$	$+\beta$
Second quadrant II	$u_X<0,u_{Y}0$	$-\beta$
Third quadrant III	$u_X<0,u_Y<0$	$+\beta$
Fourth quadrant IV	$u_X>0,u_Y<0$	$-\beta$
Original center $O_p$	$u_X=0,u_Y=0$	0
$X+$	$u_X>0,u_Y=0$	$90^{\circ }$
$Y+$	$u_X=0,u_Y>0$	0
$X-$	$u_X<0,u_Y=0$	$90^{\circ }$
$Y-$	$u_X=0,u_Y<0$	0

As shown in Figure 5a, $\overrightarrow{u_{XOY}}$ is the projection vector of $\overrightarrow{u}$ in the top view of $O_p{X}_p{Y}_p{Z}_p;$the $\beta$ refers to the acute angle between projection vector $\overrightarrow{u_{XOY}}$ and Y axis in the top view of $O_p{X}_p{Y}_p{Z}_p.$

FIG. 5.

Prioritized quadrant projection judgment criteria are proposed to obtain a unique orientation solution of reverse kinematics of VOT robot. (a) Projection vector position of $\overrightarrow{u}$in Z-direction overhead view of PCS and (b) location of $\overrightarrow{u}$ in PCS.

Then:

$$\beta =arctan\frac{\left|u_X\right|}{\left|u_Y\right|}\beta \in \left(0,90^{\circ }\right)$$ (28)

So far, the unique optimal solution of the rotational control parameter theta${\theta }_{C\prime }$ of the carrier platform can be determined.

Swing angle ${\theta }_A$ of EE and the swing angle ${\theta }_{A\prime }$ of supporting platform

The angle $\alpha$ between $\overrightarrow{u}$ and the Z-axis of the PCS can be determined. As shown in Figure 5b, $\alpha$ is the positive angle between $\overrightarrow{u}$ and coordinate axis $Z_p$. The equation of calculating $\alpha$ from the relation of trigonometric functions satisfies the piecewise function below:

$$\left\{\begin{array}{cc}\alpha =arctan\frac{\sqrt{{\left|u_X\right|}^2+{\left|u_Y\right|}^2}}{\left|u_Z\right|}& \left(\left|u_Z\right|\ne 0\right)\\ \alpha =90^{\circ }& \left(\left|u_Z\right|=0\right)\\ \end{array}\right.$$ (29)

According to the solution principle of swinging EE before swinging hotbed workbench, the parameters are obtained as listed in Table 2.

Table 2.

Prioritized Quadrant Projection of Tracking Control Parameters ${\theta }_A$ and ${\theta }_{A‘}$

Judgment basis	Quadrant of $\overrightarrow{u}$	Value of parameter ${\theta }_A$	Value of parameter ${\theta }_{A\prime }$
$0^{\circ }\le \alpha \le 45^{\circ }$	Z, I, II, X+, Y+	${\theta }_A=-\alpha$	${\theta }_{A\prime }=0^{\circ }$
	III, IV, X−, Y−	${\theta }_A=+\alpha$	${\theta }_{A\prime }=0^{\circ }$
$45^{\circ }<\alpha <90^{\circ }$	I, II, X+, Y+	${\theta }_A=-45^{\circ }$	${\theta }_{A\prime }=+\left(\alpha -45^{\circ }\right)$
	III, IV, X−, Y−	${\theta }_A=+45^{\circ }$	${\theta }_{A\prime }=-\left(\alpha -45^{\circ }\right)$
$\alpha =90^{\circ }$	I, II, X+, Y+	${\theta }_A=-45^{\circ }$	${\theta }_{A\prime }=+45^{\circ }$
	III, IV, X−, Y−	${\theta }_A=+45^{\circ }$	${\theta }_{A\prime }=-45^{\circ }$

Note: According to the requirement of the posture of the EE, $\left|{\theta }_A\right|+\left|{\theta }_{A\prime }\right|=\alpha$, so ${\theta }_{A\prime }$ can be obtained after determining ${\theta }_A$.

So far, the unique orientation solution of the swing control parameter ${\theta }_A$ of EE and swing control parameter ${\theta }_{A\prime }$ of supporting platform can be determined.

Tracking spatial location of EE support

After obtaining the unique orientation solutions of ${\theta }_{A\prime }$, ${\theta }_A$, ${\theta }_{C\prime }$, the trigonometric function $sin{\theta }_{A\prime }$, cos${\theta }_{A\prime }$, $sin{\theta }_A$, cos${\theta }_A$, $sin{\theta }_{C\prime }$, cos${\theta }_{C\prime }$ can be obtained. By substituting the values into Eq. (27), the three variables of ${\theta }_{A\prime }$, ${\theta }_A$, and ${\theta }_{C\prime }$ can be eliminated, so that the equation can be transformed into a system of ternary inhomogeneous linear equations, and the corresponding solutions of the tracking control parameters $P_{MX}$, $P_{MY}$, and $P_{MZ}$can be further obtained.

The final solution of ${\theta }_{A\prime }$, ${\theta }_A$, ${\theta }_{C\prime }$, $P_{MX}$, $P_{MY}$, and $P_{MZ}$ constitute a set of parameters to control the motion of a VOT robot. Using ${}_{O_{m2}}^{O_n}T$, the unique orientation solution of EE can be further obtained.

By orderly assembling the tracking control parameters of EE among all layers of a 3D-printed object, each DOF can be obtained to drive the machine. Combining with the specific motor drive control mode, the parameter set of motor motion control can be obtained with different accuracy.

After reverse tracking, the Cartesian coordinate trajectory P in PCS (shown in Fig. 4) of each layer can be obtained precisely for 3DP. For arbitrary point set $P\left(x,y,z\right)$, the deviation $D_{VOT}$ can be calculated by two-norm of the difference between the two trajectories to evaluated the tracking precision

$$D_{VOT}=P\left(x,y,z\right)-P_{track}\left(x,y,z\right)i\in \left[1,card\left(P\right)\right]$$ (30)

To further improve the surface quality of the printed model, nonuniform rational B-spline interpolation can be used between adjacent layers to improve manufacturing efficiency.

Numerical Example of 3D Manifold Surface

Variable orientation of manifold surface model

Cochlear implant is recognized as an effective bioelectronic medical equipment to provide a sense of sound to totally severe deafness for restoring or reconstructing deaf people's hearing. The auricle is a highly directional sensitive reflector, and therefore, the surface roughness has an important influence on the perception and location of sound source for a human being. For sound waves with a frequency of about (2, 3 kHz), the wavelength may be close to the auricle size, only then does the auricle begin to collect sound waves. For the sound waves with a higher frequency of about (5, 6 kHz), the auricle's spatial positioning function appears to be obvious. Figure 6 indicates medical magnetic resonance imaging and a regenerated human head with an external ear auricle structure.

FIG. 6.

Medical magnetic resonance imaging (a) and regenerated human head with external ear auricle structure (b).

The to be printed model is human external ear (right ear in Fig. 6b) with a complex surface, which is concerning either the exterior or interior surface. It has no geometric symmetry by itself. The size in Y direction, namely auricle size is 59.14 mm (mm, hereinafter inclusive). The stroke ratio in x,y,z direction of AABB is 2.1005:3.1159:1. The total surface area $S_{object}$ = 5360.3767 mm², total volume of enclosed manifold $V_{object}$ = 13016.4936 mm³, and specific surface area is 0.4118(mm⁻¹). The Euler characteristic of Euclidean manifold is 2, and genus is 0. As a necessary step for AM, the slicing is compared by results in Figure 7.

FIG. 7.

Comparison of uniform and variable geometric slicing of human external ear using same total layers where (a) is about uniform layer thickness and (b) is about variable layer thickness.

Figure 8 shows comparisons of cross-sectional area and volume of each layer of external ear (shown in Fig. 7) under the constraint of same layer amount. About the cross-sectional area in Fig. 8(a), the maximum is optimized from 9.423057e+02 at 31.0345% to 9.422570e+02 at 31.0345%, the minimum is optimized from 1.901384e+01 at 100% to 2.044865e-02 at 100%, and the mean value is optimized from 681.6599 to 674.4360. About volume shown in (b), the maximum is optimized from 6.726971e+02 at 27.5862% to 6.386996e+02 at 31.0345%, the minimum is unchanged, and the mean value is changed from 456.9531 to 457.1288. The trend of the cross-sectional area and volume is similar but not the same, which proves that the layer thickness varies.

FIG. 8.

Comparison of cusp height of each layer for external ear under same layer amount $card\left(d\right)$ (shown in Fig. 7).

Figure 9 demonstrates feasible solution ratio ${\xi }_{blue}$ comparison of each layer for external ear (exhibited in Fig. 7). In Figure 9a, as the dot dash line for material with relative smaller surface tension force F_s and F_b (in Fig. 2b), whereof ${\xi }_{blue}$, the maximum is 2.59% at $h_n=3.13\%$, the minimum is zero at $6.25\%$, mean is 0.30%, standard deviation is 0.55%, and variance is 0. By contrast, as solid line for material with a relative larger surface tension force F_s and F_b (in Fig. 2a), the maximum is 20.41% at 30.3030%, the minimum is 0.84% at12.1212%, mean value is 7.47%, standard deviation is 5.62%, and variance is 0.31%.

FIG. 9.

Variable SNO of each layer for external ear where (a) is about feasible solution ratio ${\xi }_{blue}$ using miscellaneous materials and (b) is about optimal unique SNO for further orientation tracking. SNO, substrate normal orientation.

Meanwhile, in Figure 9b, the blue curve indicated about $SN{O}_x$, maximum is 5.000000e-02 at 14.7059%, $min$ is −6.038462e-01 at 91.1765%, mean is −0.3118, standard deviation is 0.2097, and variance is 0.0440. While the red curve indicated in Figure 9b about $SN{O}_y$, maximum is 4.000000e-01 at 17.6471%, minimum is −8.249827e-01 at 76.4706%, mean is −0.2068, standard deviation is 0.3123, and variance is 0.0975.

The full continuous domain of the SNO can be figured out by using polar coordinates instead of Cartesian coordinates. Figure 10 shows nondiscrete continuous SNOs of schematic four layers in a normalized circular workbench. Note that the solved blue or red regions are concordantly continuous, which proved a continuous analytical calculation without discretization. About (a), ${\xi }_{blue}$ is 7.86%, about (b), ${\xi }_{blue}$ is 36.29%; about (c), ${\xi }_{blue}$ is 49.98%; and about (d), ${\xi }_{blue}$ is 46.91%.

FIG. 10.

Nondiscrete continuous SNOs of four layers in normalized circular workbench where blue region means without external support, whereas red region means indispensable external support. (a) $h_n=2$0%; (b) $h_n=$30%; (c) $h_n=4$0%; and (d) $h_n=$50%.

Kinematic parameters of six-DOF 3DP system

Herein we completed the prototype design of six-DOF 3DP system conceptually, shown in Figures 3–4. By adding extra three DOFs on the traditional three-DOF printer, the workbench orientation can be adjusted to minimize upholders for each layer. Prior to any reverse tracking computation, the mechanical parameters should be predefined to clarify the range of tracking control parameters listed in Table 3. (The rotation direction is determined according to the right-hand helix rule.)

Table 3.

Range of Parameter Variation of Variable Orientation Tracking Robot

Tracking control parameters	Variation range of parameters
Swing angle of hotbed workbench ${\theta }_{A\prime }$	$\left[-45^{\circ },+45^{\circ }\right]$
Swing angle of EE ${\theta }_A$	$\left[-45^{\circ },+45^{\circ }\right]$
Rotation angle of the carrier platform ${\theta }_{C\prime }$	$\left.\left(-180^{\circ },+180^{\circ }\right.\right]$
X-direction displacement of EE support $P_{MX}$ (xp)	$\left[0,400mm\right]$
Y-direction displacement of EE support $P_{MY}$(yp)	$\left[0,300mm\right]$
Z-direction displacement of EE support $P_{MZ}$(zp)	$\left[0,400mm\right]$

EE, end-effector.

By iterative convex optimization, all layer tracked orientation angle of six-DOF robot is shown in Figure 11. About rotational parameters with $h_n=20\%$, the ${\theta }_{A\prime },{\theta }_A,{\theta }_{C\prime }$ are, respectively, in degree (hereinafter inclusive) (31.5501°, −31.5501°, 73.9264°). About layer with $h_n=30\%$, the ${\theta }_{A\prime },{\theta }_A,{\theta }_{C\prime }$ are (−28.9217°, 28.9217°, −50.8207°). About trajectory with $h_n=40\%$, the ${\theta }_{A\prime },{\theta }_A,{\theta }_{C\prime }$ are (10.2194°, −10.2194°, 56.3099°). About trajectory with $h_n=50\%$, the ${\theta }_{A\prime },{\theta }_A,{\theta }_{C\prime }$ are (5.5706°, −5.5706°, 88.5312°).

FIG. 11.

All layered normalized SNO and its tracked orientation angle of six-DOF robot. DOF, degree of freedom.

Figure 12 exhibits the layered trajectory to be tracked of different layers in MCS. About (a), the loop amount is 141, the total trajectory length is 1847.3317 with point amount $card\left(P\right)=2079$, and the amount of inflection point is 154; about (b), the loop amount is 98, the total trajectory length is 1871.5387 with $card\left(P\right)=2$108, and the amount of inflection point is 100; about (c), the loop amount is 95, the total trajectory length is 1956.96 with $card\left(P\right)=2$134, and the amount of inflection point is 109; about (d), the loop amount is 158, the total trajectory length is 1773.9663 with $card\left(P\right)=$1976, and the amount of inflection point is 185.

FIG. 12.

Layered trajectory to be tracked of different layers in MCS with (a) $h_n=2$0%; (b) $h_n=$30%; (c) $h_n=4$0%; and (d) $h_n=$50%. MCS, model coordinate system.

VOT of the manifold model via six-DOF 3DP printer

Based on the VOT method, the reverse tracked contour trajectory in PCS of different layers is shown in Figure 13. The tracked results are all with Cartesian coordinates instead of rough shape, which manifest the systematic and rigorous superiority of the proposed VOT. In arbitrary given layer, the Z coordinate varies instead of being a constant, which proves that the build orientation is variable layer by layer. The translational parameters $P_{MX},P_{MY},P_{MZ}$ of each layer are consistent with those in PCS, which illustrate correctness of the result evidently. Regarding tracking deviation$D_{VOT}$ in (a), the maximum is 0.1214 mm at (176.2518, 193.3152, 25.2085) and the mean is 0.0703 mm. In (b), the maximum is 0.1950 mm at (214.9948, 232.3589, 22.8487) and the mean is 0.1195 mm. In (c), the maximum is 0.1200 mm at (202.5838, 178.5030, 40.7932) and the mean is 0.1145 mm. About (d), the maximum is 0.1298 mm at (214.2566, 175.9546, 49.5009) and the mean is 0.1070 mm. It should be further emphasized that the accuracy between the original theoretical trajectory and the reversely tracked trajectory points can be distinctly improved by using smaller electrical servo drive step size. The metadata is generously shared within the Supplementary Table S1.

FIG. 13.

Reverse tracked contour trajectory of different layers in PCS where (a) $h_n=2$0%; (b) $h_n=$30%; (c) $h_n=4$0%; and (d) $h_n=$50%.

The kinematic simulation of the proposed VOT robot can be further verified using multibody dynamics software ADAMS™. The simulation time can be 60 s and the simulation step size can be set as 0.1 s. The amount of simulation points should be in accordance with $card\left(P\right)$. The driver function uses a combination of STEP of functions. It can simulate the movement of displacement via step and velocity of an s-shaped curve. The trajectory of simulation is consistent with that of theoretical calculation with an error of <1e-6.

The process of fabricating manifold surface using six-DOF robot via VOT is shown in Figure 14. The tracked orientations can be validated by Figure 11. It is important to note that the EE (extruder) is holding perpendicular to the substrate workbench, which reflects the accuracy of inverse kinematic calculation.

FIG. 14.

Fabricating manifold surface using six-DOF robot via VOT where (a–d) are successively about 20%, 30%, 40%, and 50%.

VOT result comparison

Beware that the external support structure varies multifariously, for instance, cylinder type, tree type, fractal tree type, and bionic structure. Herein we use the typical cylinder type to evaluate the surface need support, which is akin to feasible solution ratio ${\xi }_{blue}$. Figure 15 shows comparisons of external support material consumption before and after using VOT. The covering surface area, which needs the external support can be reduced from 1252.7480 to 843.6293 mm² with a ratio of 32.66%, and the mandatory consumed material of the external support can be reduced from 33180.0334 to 10078.6089 mm with a ratio of 69.62%. The proposed VOT method can be reversely deployed to evaluate the build orientation in terms of external support material metering.

FIG. 15.

Comparisons of external support material consumption where (a, b) are before and after using support diminution design, respectively.

Physical Experiment of VOT Printing

Physical fabricating via extrusion-based AM

The stereolithography apparatus is extrusion-based manufacturing. The desktop apparatus size is relatively larger than the maximum space. The layer thickness can be as small as 0.02, 0.05, 0.1 mm. The resolution precision size in X/Y direction can be 0.1 mm. The power supply is alternating current (AC) 220 V. The ambient temperature is 25°C with 55% relative humidity. The nominal voltage electrical motor for rotating DOF is direct current (DC) 24 V. The printing process using extrusion-based 3DP is shown in Figure 16a and b. The stripped material from external support via diminution design can be evaluated quantitatively. The material weight (mass) has been measured by an electronic weighting balance using high-precision strain-type weighing sensor with antielectromagnetic interference, shown in Figure 16c. As is known metrology to all, the capacity range and resolution are two contradictory and mutually restrictive indexes. Herein the capacity is 1 kg, meanwhile the resolution ratio is 0.1 g (1‱), maximum permissible error 0.2 g, whose power supply is nominal AC 230 V ± 10% 50 ± 1 Hz 4 W or DC 6 V/4 Ah/20 HR rechargeable battery. Regarding the communication functions, it can connect with computer to collect and calculate data or support remote control to generate ASCII text file, using standard RS 232C serial bidirectional data communication interface (16 pins), with Baud rate 600-9600. The photograph (Fig. 16d) in macromode qualitatively revealed the effect of material stripping on the fabricated surface.

FIG. 16.

Fabricating process for high surface accuracy where (a, b) are about fabricating machine; (c) is electronic weighting balance, and (d) is a fabricated artificial auricle.

Manifold surface precision measurement

The surface roughness of the 3DP fabricated specimen can be measured by surface profilometer, either an optical microscope (OM) or EM. The wavelength of visible light ranges about 380–780 nm. Optical methods of measuring surface roughness have many advantages such as imaging stability, rapid response, noncontact, nondestructive, and area-averaging. Hence, we use the OM (Label.1) to observe and evaluate material stripping effect (MSE) with distinct sampling regions. The relevant physical experiment of material surface topography is shown in Figure 17. The specimen (Label.4) is insulating with big enough electrical conductivity.

FIG. 17.

Physical experiment of material surface topography using OM. OM, optical microscope.

During measurement via using OM, a little bit different from the scanning electron microscope, the region should be approximately 2D planar, Otherwise, the focus area will be reduced, dues to incomplete spotlight. The measured surface topography comparison of the fabricated specimen using OM is shown in Figure 18. The MSE due to supporting material can be clearly observed. The slight defocus of the image nicely proves that the region being photographed is rather a curved surface than a plane. By external support diminution design, the vast majority of surface can be optimized (decreased) from at least about Ra 12.469 μm to no more than Ra 7.681 μm.

FIG. 18.

Surface topography comparison of the fabricated specimen using OM where the magnification is 100 × times for either before (a) and after (b) using support diminution design.

Innovation comparison

The innovation comparison between the others' published articles and the proposed VOT method is conspicuously listed in Table 4.

Table 4.

Innovation Comparison Between the Published Articles and the Proposed Variable Orientation Tracking Method

Innovation concerns	The other published articles	The proposed VOT method
Orientation	“Build orientation” traversal may contribute little to minimization of overall external support and is prone to impracticality (simple angle criterion in Fig. 7 of Chowdhury et al.7 and other Ezair et al.8). Zero-support theory is absent in the section “Materials and Methods” (Walker et al.15).	Proposed CGE criterion (Eq. [9]) instead of solely inclination angle method. Variable SNO for each layer using PCA via Eqs. (13–15). Layered minimization instead of overall minimization.
External support	“Dissolvable support” has strict restrictions (Hildreth et al.16 and Lefky et al.17). “Support-free” may fail because the EE (extruder) cannot hold perpendicular to the substrate workbench (Figs. 1 and 9 in Dai et al.26).	Build theoretical optimization model Eq. (13). Quantitative calculation instead of so-called support-free via feasible solution ratio ${\xi }_{blue}$ using Eq. (16). The proposed method can be expansively applied to different AM processes using various materials, including SLS and SLA.
Multi-DOF mechanical equipment	Results are sketchy for dimensional synthesis of five DOFs (Fig. 3 in Fiore et al.27). Only five-DOF and the kinematic model is not provided (Fig. 2 in Wu et al.28).	Six DOFs shown in Figures 3 and 4 reversed from $\left(SN{O}_x,SN{O}_y,SN{O}_z\right)$ and EE position $\left(P_X,P_Y,P_Z\right)$ in the Tracking Spatial Location of EE Support section. Precise trajectory point tracking using multiaxis kinematic parameter set $(P_{MX}$,$P_{MY}$,$P_{MZ},{\theta }_A,{\theta }_{A\prime }{\theta }_{C\prime })$ shown in Figure 13 (data sharing) and Figure 14.
Physical experiment verification	Further experiments on fabricated curved surface are not given (Fig. 11. in Chen et al.24). Product quality is not measured (Figs. 25 and 26 in Xu et al.23).	MSE is observed and diminished shown in Figures 14 and 15. Weight comparison of stripped material from external support shown in Figure 16c. VOT contributes greatly to support diminution.

AM, additive manufacturing; CGE, cosmic gravity effect; DOF, degree of freedom; MSE, material stripping effect; PCA, principal component analysis; SLA, stereo lithography appearance; SLS, selective laser sintering; SNO, substrate normal orientation; VOT, variable orientation tracking.

Conclusions

Support diminution design method for layered manufacturing of manifold surface based on VOT is proposed

The CGE criterion is first used to extract surface need support from manifold surface with various materials by considering the balance force such as material characteristics and inclination angle. The CGE criterion lays a theoretical foundation for the VOT method. This CGE criterion may inspire more following research because of its profound commonality and adaptability. The important scientific discovery is that varying the SNO, namely workbench, for each layer in PCS may break the balance between gravity and its equilibrium force. The optimal SNO for each layer is calculated using continuous mathematical harmonic analysis on surface. The external support in 3DP then can be exactly calculated and reduced based on rigorous mathematical theory to reduce external support or upholders and further to protect the manifold bilateral surface. The proposed method can be reversely used to evaluate the build orientation in terms of external support material metering. This article facilitates fabricating high-precision multimaterial 3DP and serves to even enhance ecological 3DP.

A reconfigurable VOT robot with six-axis DOF and the matched servo controller are developed simultaneously for 3DP

A reconfigurable VOT robot with six-axis DOF is developed for 3DP. The forward kinematic model of the six-DOF 3DP robot is built via recursive homogeneous coordinate transformation considering the RTCP effects. The PQP judgment criteria are presented to obtain a unique orientation of reverse kinematics of VOT robot. What is more, the EE (extruder) is holding perpendicular to the substrate workbench. The accuracy between the original theoretical trajectory and the reversely tracked trajectory points can be distinctly adjusted by using various electrical servo drive step sizes. The VOT results are given with coordinate values in PCS, which enhances and highlights the rigor of scientific computation process.

The physical experiment of manifold surface 3DP using layer-based process is implemented using VOT

The physical experiment, which takes human external ear auricle, for example, using the layer-based process, is implemented via VOT. Moreover, the physical experiment involves material property experiment, material roughness experiment, mechanical servo drive test, position and orientation measurement, and so on. The MSE due to supporting material can be clearly observed and diminished using OM. The stripped material from external support via diminution design can be evaluated quantitatively by the electronic weighting balance. All of which indicate the findings that external support in 3DP can be putatively reckoned and diminished using rather VOT than the so-called build orientation traversal method. The covering surface area of the external support is reduced with a ratio of up to 32.66%, and the mandatory consumed material of the external support is reduced with a ratio of 69.62%. The proposed method provides a relatively scientific quantification for minimizing external support material rather than the so-called support-free concept. It can be expectantly applied to various AM of more accurate 3D manifold structures such as cantilever issues and even intravital organs.

In future, the VOT method and its machine prototype robot will be developed toward high performance with intelligent adaptation, together with printing more complex manifolds, mechanical parts, and surface curved thin end-use objects. We expect this VOT approach will promote and arouse more ongoing research, which spans intelligently adaptive slicing, multimaterial external-supporting diminution, geometry trajectory generation, infill structure topology optimization, and even electromechanical control.

Nomenclature

$n_p$

the unit normal vector of the substrate carrier platform

$OXYZ$

the base coordinate system

$O_p{X}_p{Y}_p{Z}_p$

the printing coordinate system (PCS)

$O_n{X}_n{Y}_n{Z}_n$

the end-effector (EE) coordinate system

$O_{m1}{X}_{m1}{Y}_{m1}{Z}_{m1}$

the carrier platform coordinate system

$O_{m2}{X}_{m2}{Y}_{m2}{Z}_{m2}$

the EE support coordinate system

$L$

the distance between $O_{m2}$ and $O_n$

$\overrightarrow{P_{M0}}$

the position vector of the origin $O_{m1}$ in $OXYZ$

$\overrightarrow{P_{M1}}$

the position vector of the origin $O_{m2}$ in $O_n{X}_n{Y}_n{Z}_n$

$\overrightarrow{P_{M2}}$

the position vector of $O_p$ in $O_{m1}{X}_{m1}{Y}_{m1}{Z}_{m1}$

$H$

the Euclidean distance from $O_p$ to $O_{m1}$

$\overrightarrow{P_M}$

the position vector of $O_{m2}{X}_{m2}{Y}_{m2}{Z}_{m2}$ relative to its initial state

${\theta }_{A\prime }$

the angle of rotation axis $A\prime$relative to its initial state

${\theta }_A$

the angle of rotation axis$A$ relative to its initial state

${\theta }_{C\prime }$

the angle of rotation axis $C\prime$ relative to its initial state

$\overrightarrow{u}\left(u_x,u_y,u_z\right)$

the direction vector of the end-effector (EE) in $O_p{X}_p{Y}_p{Z}_p$

$P\left(P_X,P_Y,P_Z\right)$

the position coordinates of the end-effector (EE) in $O_p{X}_p{Y}_p{Z}_p$

$\overrightarrow{u_{XOY}}$

the projection vector of $\overrightarrow{u}$ in the top view of $O_p{X}_p{Y}_p{Z}_p$

$\alpha$

the positive acute angle between $\overrightarrow{u}$ and coordinate axis $Z_p$

$\beta$

the acute angle between projection vector $\overrightarrow{u_{XOY}}$ and axis $Y_p$ in the top view of $O_p{X}_p{Y}_p{Z}_p$

Authors' Contributions

J.X. initiated the essential innovations and finalized the article. M.G. carried out the analytic mathematical programming and theoretical computation among the continuous domains. X.F. participated in the mechanical precision experiment. Z.S. developed the six-DOF robot and relevant matched servo controller. K.W. verified the feasibility and practical superiority of diminishing material. S.Z. guided the team research as PI (Principle Investigator). J.T. made constructive suggestions to the work as Chief Scientist. The all metadata of Figure 13 are shared for the readers for verification or calculation.

Author Disclosure Statement

No competing financial interests exist.

Funding Information

The work presented in this article is funded by the National Natural Science Foundation of China (nos. 51935009; 51775494; 51821093), the Zhejiang Provincial Research and Development Project of China (no. LGG21E050020), and the National Key Research and Development Project of China (no. 2018YFB1700701).

Supplementary Material

Supplementary Table S1