## Abstract

This paper presents a novel two degrees-of-freedom planar parallel manipulator (PPM) designed for infinite-axis 3D printing, alongside tools for facilitating future design iterations. Unlike traditional gantry-supported designs used in infinite-axis 3D printing, which impose significant mass movement requirements, the examined new design prioritizes reducing overall weight to enhance speed potential at the cost of a reduced work area. In this innovative approach, the PPM effectively reduces weight by decoupling the motion of the hot end from that of the motor. Motors are attached to the frame, controlling a system of pulleys, and connecting arms to drive the hot-end's motion. Due to the length of the arms, the hot end will be unable to fully explore the entire printing plane. Verification of the angled PPM for 3D printing involved developing kinematic and dynamic equations, conducting finite element analysis on critical components, and testing a completed prototype. A metaheuristic optimization method was employed to derive optimal design parameters, focusing on optimizing the arm length of the connectors while maximizing dynamic performance. Considerations included the usable workspace and the angle between the connecting arm and end-effector. The final prototype validated the stability and rigidity of the PPM during movement, indicating its viability for 3D printing. The results presented in this paper demonstrate the capabilities of using an angled PPM in infinite 3D printing, providing fundamental knowledge crucial for future designs involving this innovative mechanism.

## 1 Introduction

The development of new 3D printing technologies has been rapidly growing in the past few decades, largely due to the expiration of the patents surrounding the technology. Currently, ASTM recognizes seven distinct categories of 3D printing technologies that include binder jetting, directed energy deposition, material extrusion, material jetting, powered bed fusion, sheet lamination, and vat photopolymerization [1]. These categories were created to differentiate 3D printers based on the processes they use to create a 3D object. Each process uses different techniques that allow for the utilization of a wide range of materials and precision. The aerospace, automotive, food, healthcare, construction, textile/fashion, and electronic industries have all been disrupted by the advent of 3D printing [2]. Some marvels of 3D printing include custom medical implants [3], fabrication of soft robotics [4], automotive parts [5], 3D printed houses [6], 3D printed jewelry [7], and even 3D printed rockets [8].

In 2017, the company BlackBelt released the first commercially available 3D printer featuring an angled gantry design mounted on a rotating bed, as depicted in Figs. 1 and 2 [9]. This new design, referred to as a conveyor belt 3D printer, introduced the ability to continuously print objects of infinite length. The process of printing at an angle is functionally identical to traditional extrusion 3D printers. Instead of building layers up vertically, layers are stacked on top of each other at an angle as depicted in Fig. 2. Fast forward to 2021, and there are only two commercially available conveyor belt 3D printers—the BlackBelt and the Creality CR-30 [10]. Both printers rely on the same hot end mounted gantry design. The hot end being mounted to a gantry necessitates moving a significant mass during printing, posing limitations on printing speed. The lack of diversity with the design of commercially available belt 3D printers, points to a substantial opportunity for innovation.

Motivated by the aim to create a faster alternative, this paper proposes a new implementation of a planar parallel manipulator (PPM) as a weight reduction design to replace the conventional gantry design. This innovative approach aims to decrease the moving weight by eliminating the need for a cross beam and fixing the axis motors stationary to the frame. A system of pulleys and arms will facilitate the movement of the hot end. The primary objective of this paper is to validate the feasibility of utilizing an angled PPM and introduce a methodology for generating an optimized design. However, comparisons between types of conveyor belt 3D printers will not be investigated, as this is beyond the scope of the current research.

### 1.1 Literature Review.

Given that belt 3D printing, also known as infinite-axis 3D printing, is still in its infancy with very few commercial offerings, there are no peer-reviewed articles covering the design and abilities of these printers. Most of the development of these printers has been achieved by 3D printing enthusiasts who publish their final designs in an open-source format [11–13]. The most relevant academic research available on this topic only details the capabilities of traditional selective compliance assembly robot arm (SCARA) and delta-style 3D printers, as those types of printers use parallel manipulators as a means of achieving motion. These articles provide insight into the governing principles of parallel manipulators used for 3D printing. Unfortunately, research surrounding the use of parallel manipulators for 3D printing focuses on SCARA/delta printers. There is no examination of a hybrid approach where bed movement is used [14–18].

However, there are a series of papers published that detail the use of a PPM for the translational motion of an industrial-sized milling machine [19–22]. The PPM detailed in these papers consists of a symmetric design whereby the end-effector is attached to two arms, which are attached to a carriage as shown in Fig. 3. The arms are attached using revolute joints and the carriage is attached to the frame via a prismatic joint. In the papers, this design was mounted in an upright orientation. This design can be angled and fitted with a hot end, enabling its adaptation for creating an infinite 3D printer.

## 2 Design

To create a functioning infinite 3D printer, the PPM shown in Fig. 3 will need to be attached to the frame of the printer at an angle. In this design, the PPM will be attached at 45 deg with respect to the build plane. Figure 4 shows a computer-aided design (CAD) representation of the final design. The kinematic and dynamic derivations of this PPM are detailed in the following sections.

### 2.1 Kinematics.

*C*, can be written as

*B*(

_{i}*i =*1, 2) can be written as

*P*(

_{i}*i =*1, 2) can be written as

*r*represents the length of the end-effector and

*2R*is the width of the frame.

### 2.2 Inverse Dynamics.

Dynamic equations for an upright PPM have been previously established [19,20]. In these works, the authors independently derive these equations using two distinct methods—the Lagrangian and the virtual work principle. However, these derivations cannot be directly applied when the PPM is at an angle. The problem is fundamentally different now that gravity is influencing the mechanism outside the plane of motion.

To address this angled scenario, Kane's method will be applied to create the equations of motion. Kane's method offers the advantage of systematically analyzing a system without introducing excessive complexity, while also allowing for modifications even after the derivation is complete. This unique approach enables the PPM to be initially analyzed as an unconstrained system. Subsequently, additional constraints can be applied post-derivation to formulate the entire scenario, resulting in the finalized dynamic equations. For this analysis, the simplified unconstrained PPM in Fig. 5 will be used as a reference.

#### 2.2.1 Partial Velocities.

*P*that allows the arms to rotate about the $n^3$-axis and

*P*

_{1}is free to move along the $n^2$-axis. The generalized speeds can be written as

The transformation matrix between the *N* and *A* frame of reference is given in Table 1, while the transform between *N* and *B* is given in Table 2. It is assumed that the points on the system are massless and the bodies, *A* and *B*, have the same mass (*m*) and length (*L*). The center of mass for body *A* is located at *L*_{cm}. The center of mass for body *B* is located at *L* − *L*_{cm}. The partial velocities ** V_{r}** and the partial angular velocities

*ω**for the PPM are recorded in Table 3.*

_{r}$n^1$ | $n^2$ | $n^3$ | |
---|---|---|---|

$a^1$ | $cos(q2)$ | $\u2212sin(q2)$ | $0$ |

$a^2$ | $sin(q2)$ | $cos(q2)$ | 0 |

$a^3$ | 0 | 0 | 1 |

$n^1$ | $n^2$ | $n^3$ | |
---|---|---|---|

$a^1$ | $cos(q2)$ | $\u2212sin(q2)$ | $0$ |

$a^2$ | $sin(q2)$ | $cos(q2)$ | 0 |

$a^3$ | 0 | 0 | 1 |

$n^1$ | $n^2$ | $n^3$ | |
---|---|---|---|

$b^1$ | $cos(q3)$ | $sin(q3)$ | 0 |

$b^2$ | $\u2212sin(q3)$ | $cos(q3)$ | 0 |

$b^3$ | 0 | 0 | 1 |

$n^1$ | $n^2$ | $n^3$ | |
---|---|---|---|

$b^1$ | $cos(q3)$ | $sin(q3)$ | 0 |

$b^2$ | $\u2212sin(q3)$ | $cos(q3)$ | 0 |

$b^3$ | 0 | 0 | 1 |

r = 1 | r = 2 | r = 3 | |
---|---|---|---|

$\omega rA$ | 0 | $n^3$ | 0 |

$\omega rB$ | 0 | 0 | $b^3$ |

$VrP1$ | $n^2$ | 0 | 0 |

$VrA\xaf$ | $n^2$ | $Lcma^2$ | 0 |

$VrP2$ | $n^2$ | $La^2$ | 0 |

$VrB\xaf$ | $n^2$ | $La^2$ | $(L\u2212Lcm)b^2$ |

$VrP3$ | $n^2$ | $La^2$ | $Lb^2$ |

r = 1 | r = 2 | r = 3 | |
---|---|---|---|

$\omega rA$ | 0 | $n^3$ | 0 |

$\omega rB$ | 0 | 0 | $b^3$ |

$VrP1$ | $n^2$ | 0 | 0 |

$VrA\xaf$ | $n^2$ | $Lcma^2$ | 0 |

$VrP2$ | $n^2$ | $La^2$ | 0 |

$VrB\xaf$ | $n^2$ | $La^2$ | $(L\u2212Lcm)b^2$ |

$VrP3$ | $n^2$ | $La^2$ | $Lb^2$ |

#### 2.2.2 Translational and Angular Accelerations.

**and angular acceleration**

*a***for the elements on the PPM can be constructed by relating the accelerations of two points fixed on a rigid body. The accelerations for the PPM are given as follows:**

*α*#### 2.2.3 Generalized Inertial Forces.

*r =*1, 2, 3) for body

*β*in the system is

*N*for a rigid body

*β*, $a\u2217$ is the acceleration of the center of mass of

*β*in

*N, M*is the total mass of

*β*,

**is the central dyadic of**

*I**β*,

**is the inertial force for**

*R***β*in

*N*, and

**is the inertia torque for**

*T***β*in

*N*. The generalized inertial force for the system is

*N*is the total number of bodies in the system.

#### 2.2.4 Generalized Active Forces.

*P*

_{1}. The equation for the generalized active forces is

where *k* is the total number of active forces acting on the system.

#### 2.2.5 Equation of Motion for Unconstrained Planar Parallel Manipulator.

#### 2.2.6 Equation of Motion for Constrained Planar Parallel Manipulator.

*P*

_{3}. This constrains the system so that

*P*

_{3}can only move in the

*n*

_{2}-axis like

*P*

_{1}. To derive the new equations of motion, a new generalized velocity

*u*

_{4}is selected to represent the velocity of

*q*

_{4}.

*m*

_{4}is the mass of

*P*

_{4}. Note that the new constraint has given rise to two dependent generalized speeds. Equations (22) and (23) can be rewritten in terms of the independent generalized speeds. This will give form to

*n =*4 is the number of independent generalized speeds (prior to constraints), and

*m =*2 is the number of independent constraints applied. From there, the new equation of motion for the constrained system can be written as

*r =*1,…,

*n*−

*m*.

### 2.3 Performance Indicators.

The performance of the planar parallel manipulator is assessed using two key indicators in this study: the percentage of not usable vertical workspace (PNUVW) and the global conditioning index (GCI). PNUVW measures the efficiency of the mechanism in utilizing the available space, with a lower value indicating better performance. On the other hand, GCI serves as an indicator of dynamic performance, where a higher value is indicative of superior results.

#### 2.3.1 Percentage of Not Usable Vertical Workspace.

*y*-axis $(yw)$ must be determined with the following equation:

*h*is the total height of the PPM, and

*d*is the distance from the carriage tower to the workspace where the end-effector cannot reach. The value of

*d*can be written as

*x*-direction. The PNUVW can be represented as the following ratio:

#### 2.3.2 Global Conditioning Index.

*W*is the workspace of the end-effector.

## 3 Design Analysis/Optimization

This section will focus on the analysis and optimization of the PPM. For the analysis portion, a finite element method is utilized to understand the deflection and stress that the arms will be experiencing under static loading conditions assuming larger than expected load values to reflect the maximum potential loads during motion. For the optimization portion, a metaheuristic approach will take place to determine a range of desirable design variables that result in optimized performance indicators.

### 3.1 Finite Element Analysis Complexity Reduction.

The PPM arms are attached to the prismatic joints via carriage and revolute joints. When the PPM is at an angle, the revolute joints will experience an axial load, and some amount of deformation is expected to occur in the arms. Finite element analysis (FEA) was performed under static loading conditions of the PPM to determine the bearing loading conditions and the deflection in the arms. The static load values envelop possible max dynamical loading conditions by overestimating the static loading variables.

To perform a quick examination of the mechanism, the system can be reduced to the representative version as shown in Fig. 7. Figure 7 shows that the system can be modeled as a two-element truss problem experiencing an out-of-plane force applied at the node attaching both elements together.

*K*is the stiffness matrix,

*U*is the displacement vector, and

*F*is the force vector.

*K*matrix be a 12 × 12 matrix. The global

*K*matrix is expected to be 18 × 18 matrix because of the arrangement of all the elements. This means that both

*U*and

*F*will be a vector with a size of 18. For this problem, there are two fixed ends where the elements are restricted to experience no deformation at those nodes. Because of these boundary conditions, the global matrix can be reduced to a 6 × 6 matrix, and the force vector can also be reduced to a size of 6. Now the deformation vector can be calculated by the following equation:

### 3.2 Finite Element Analysis Via ansys.

The above method presents a simplified scenario that can be analyzed without any supporting software. However, given the reduction in complexity, the accuracy of the result is also slightly diminished. ansys is used to generate more accurate results because it has the capability to create a mesh of an object and break the object into thousands of different elements/nodes. ansys is then able to numerically solve the system of equations under given boundary conditions. For an accurate representation of the problem, the CAD model of the PPM was imported into ansys and the material properties for the arms and end-effector were specified as carbon fiber and polyethylene terephthalate respectively. All six arms were selected to be a fixed support on the face they would be attached to the carriage. Finally, a gravitational force was implemented at a 45-deg rotation, and a point mass representing that hot end was defined. The point load was estimated at double the actual mass to account for any additional loads caused by the acceleration of the machine. Figures 8 and 9 show the results.

From the simulation, it can be shown that the reaction at the fixed support is about 4 N from Fig. 8, and the maximum deflection is about 0.6 mm from Fig. 9. The 0.6 mm deflection can be considered negligible considering the change in length would result in a 0.2% difference. This indicates that the stiffness of the mechanism would impart a negligible amount of slack to the system resulting in no noticeable impact on 3D printing accuracy.

From the simulation, it shows that the bearings are expected to experience about 4 N of combined loading. The concern is that most of the loading experienced is axial loading and not normal loading. The bearings that are used have an inner diameter of 3 mm, an outer diameter of 10 mm, and a thickness of 4 mm. The bearing's spec sheet specifies a maximum allowable dynamic load of 628 N and a maximum static load of 216 N [27]. Importantly, the maximum dynamic and static load conditions exceed the expected static loading conditions by 157 and 54 times, respectively. Hence, it is reasonable to assume that the selected ball bearing will be suitable for this application, even when subjected to dynamic conditions typical in 3D printing and experiencing combined axial loading.

### 3.3 Metaheuristics Optimization.

*L*) and the width of the frame (

*R*), the multi-objective genetic algorithm is employed and implemented with matlab. For this implementation, the cost functions used were the performance indicators. The bounds for the design variables are

Equation (37) is used to ensure that the length of the arms can span the entire width of the PPM. Equation (38) is used to ensure that the PPM does not approach a singularity configuration when at the edge of the workspace. The specified parameters passed into the algorithm were *r =* 25 mm, *x _{w} =* 220, and

*h =*700.

#### 3.3.1 Optimization Results.

The Pareto front in Fig. 10, and the plot compared $\alpha $ to the PNUVW in Fig. 11 were created using the scenario described above and the multi-objective genetic algorithm from matlab's optimization toolbox.

The Pareto front in Fig. 10 represents the most efficient solutions given the competing optimum objectives. Figure 10 should be used as a reference for determining what parameters to use as there is a tradeoff between performance and spatial efficiency. Figure 11 was included to visualize the optimum results in terms of $\alpha $. This figure is important for understanding how close the PPM is to achieving singularity when at the boundary of the workspace.

With the results of the optimization presented, there is no single solution for what values result in the best possible PPM. Instead, design desires need to be taken into consideration before an optimum design can be selected. For creating an infinite-angled 3D printer, the entire footprint of the machine is an important factor to consider. Machines that are very big are cumbersome and can get in the way. Therefore, a recommendation is made to select a solution that prioritizes the efficient use of space. This would lead to the selection of the first point on the Pareto front. Comparing this selection to Fig. 11, the parameters would result in $\alpha =6.2deg$. This value is comparatively small, so the next point on the Pareto front is recommended. This selection would result in $\alpha =11.54deg$, a more desirable result. The parameters resulting from the optimization would be *L =* 320 mm and *R =* 221.17 mm. This configuration would result in the best possible performance of the machine.

## 4 Prototype and Testing

This section details the construction of the first prototype along with the methodology used during the testing of the machine. The prototype discussed in this section had been built prior to the exploration of optimized parameters. The design variables used for the construction of the prototype can be found in Table 4.

### 4.1 Prototype Construction.

The preliminary design for the angled PPM was modeled using Inventor. The model includes the ability to change the configuration of the manipulator to see how the PPM can move in the given space. Figures 12 and 13 show an example of the CAD model displaying different end-effector configurations. This was helpful in determining component size and potential interference that may occur prior to building the prototype.

The frame was constructed out of 20 × 20 mm aluminum extrusion and was attached together using a combination of 3D printed parts and steel brackets. The frame for the PPM mechanism was attached at a 45 deg angle to the base frame of the 3D printer using an angled bracket and additional side plates to increase frame rigidity.

The PPM consisted of linear rails, bearings, carbon fiber rods, stepper motors, a pulley system, and 3D printed parts. The arms of the mechanism featured carbon fiber rods with a 3D printed attachment where a bearing was press fitted into. Additionally, the prototype featured a tensioning system between two of the arms on either side of the end-effector. This system was included to reduce any sag in the end-effector. The controller for the system was a repurposed Anet A8 control board. The assembled prototype is shown in Fig. 14.

### 4.2 Prototype Firmware/Software.

The firmware used to operate the machine is based on a fork of Marlin 1.1.8, an open-source firmware widely adaptable and utilized by various 3D printers. The firmware employed on the angled PPM printer is a derivative of Stuart's firmware [28], with modifications made to accommodate the angled attachment of the PPM and the extensive capabilities of the rotating bed.

To operate the machine, G-code is necessary to communicate to the firmware and instruct the end-effector's movements. The BlackBelt Cura (BBC) program was used to generate G-code used for the test. BBC is a free program provided by the BlackBelt company that is used to convert 3D models into the code used by angled 3D printers.

### 4.3 Testing Methodology.

With the completion of the prototype, testing was done to examine the dynamic performance of the machine. The following test was conducted to closely examine the acceleration of the carriages and the end-effector as the machine is programmed to run a series of ellipses of decreasing size. With the machine operating at high-speeds, the acceleration values collected will provide insight into the stability and rigidity of the system. If the high-speed testing results in irregular and large peaks in acceleration in unexpected directions, this will indicate poor performance for a 3D printer. Large spikes in acceleration would induce vibrations that could cause visible defects on the printed part. Ideal performance would exhibit acceleration only in the axis of motion, with clearly defined acceleration peaks and little noise. This would represent consistent circular motion where the end-effector is constantly experiencing a change of acceleration.

To test the dynamic performance of the angled PPM, an accelerometer was attached to three different test points: (1) the end-effector, (2) the right carriage, and (3) the left carriage. The accelerometer used was an ADXL335, and the data were collected using an Arduino Uno. According to the datasheet, the sensor is capable of measuring acceleration with a minimum full-scale range of ±3 g [29]. The accelerometer was calibrated using the table the printer was sitting on as the reference point. To calibrate a specific direction, the accelerometer was laid flat on the table, aligning the positive direction of the calibrating axis downwards. The readout of the accelerometer was then tuned to output 1 g. This process was repeated for each axis.

The accelerometer was attached flush to the surface of the test point so that the positive *x*-direction was aligned along the angled plane to the ground, the positive *y*-direction oriented to the right of the frame, and the positive *z*-direction was perpendicular to the angled frame downward. To initiate movement for testing, G-code was generated to move the end-effector in a series of decreasingly sized ellipses. Additionally, the motor controlling the rotating bed was unplugged to eliminate the possibility of induced vibrations from an external source. Due to resource limitations, acceleration values could only be collected from one test point at a time. A total of 15 tests were conducted, with each test point undergoing five tests. The results of the testing were then averaged for each position, and the final plot of the accelerations can be found in Sec. 5.

## 5 Results and Discussion

Prototype testing results are shown in Figs. 15–17. These findings align with the expected outcomes. Figure 15 illustrates consistent levels of acceleration for the *z*-axis throughout the testing duration, due to its orthogonality to the plane of motion. Moreover, the findings depicted in Fig. 15 indicate that the end-effector exhibits stiffness, displaying minimal oscillation during rapid movements. This characteristic is highly desirable for 3D printing processes, as it contributes to enhanced precision and accuracy in the final printed results.

The outcomes depicted in Figs. 16 and 17 align with expectations but unveil an intriguing phenomenon. In both the left and right carriages, the acceleration in the *Y* and *Z* directions should ideally manifest as constant flat lines, given that the carriage can only travel along the frame. However, Fig. 16 reveals occasional peaks in *Y*-axis accelerations, introducing irregularities in comparison to the smoother profile observed in Fig. 17. This suggests that the carriage deviates from its linear path along the frame toward the outside or center of the frame.

This deviation is likely attributed to the linear rails incorporating roller bearings. As the end-effector moves laterally and undergoes directional changes, inertia exerts a force on the PPM, compelling it to move away from the anchor point. The minimal play present between the bearings allows the carriage to experience momentary changes in accelerations. To mitigate this phenomenon, an alternative attachment for the prismatic joint is proposed. Utilizing V-slot roller wheels could offer a more effective solution by functioning as grippers, enhancing adherence to the frame, and minimizing undesired lateral movements.

Throughout the testing phase, the stepper motors emitted a discernible whine, accompanied by an audible “stepping” sound that persisted the entire time. The absence of an optimized firmware package to efficiently drive the machine might have influenced the performance of the PPM. The evident lack of smooth movement is present at all testing points, manifested as noisy and jagged accelerations. This erratic movement implies a stop-and-go pattern as the printer traveled. Addressing these issues could be achieved through the implementation of updated software.

## 6 Concluding Remarks and Future Work

This paper introduces a novel belt 3D printer design featuring an angled PPM instead of the traditional gantry design. The new design significantly reduces the moving weight, reducing inertia, and facilitating higher accelerations and dynamic performance. However, it is essential to acknowledge that this design alteration introduces additional unused space, resulting in a larger footprint of the 3D printer. To address the tradeoff between spatial efficiency and dynamic performance, an optimization using the multi-objective genetic algorithm was employed. The outcomes of this optimization effort yielded a Pareto front, delineating optimal design parameters based on key performance indicators—specifically, the GCI for dynamic performance and the PNUVW for spatial utilization. A static FEA evaluation conducted under conservative loading assumptions reveals the impact of the design on bearing load and arm deflection within the PPM. Importantly, findings suggest that these factors are unlikely to compromise the performance of the angled PPM. Finally, prototype testing substantiates the stability and rigidity of the angled PPM during motion, providing empirical support for the proposition that this innovative approach holds promise for advancing belt 3D printing technology. This comprehensive investigation highlights the potential viability of the angled PPM as a progressive and suitable solution in the realm of belt 3D printing.

The work done in this paper highlights the novelty in kinematic and dynamic formulation and optimization of planar parallel manipulator needed for the design of a 3D printer. While these findings mark a significant contribution, the journey toward a competitive alternative to the prevalent Cartesian gantry-designed infinite 3D printers remain ongoing. Future endeavors will focus on fine-tuning the control system, enhancing overall construction, and implementing optimal design parameters. Through the refinement of the machine, direct comparisons with various belt printers can be conducted, allowing for a comprehensive evaluation of the effectiveness of the angled PPM design. In essence, this work lays the groundwork for continued exploration into advanced 3D printing technologies.

## Acknowledgment

Saint Martin's University and the Hal and Inge Marcus School of Engineering provided access to the Robotics and Manufacturing labs. This access was instrumental in enabling the fabrication and testing of a live prototype.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.