Vibration Compensation of Delta 3D Printer with Position-varying Dynamics using Filtered B-Splines

1. Introduction

Delta robots are increasingly favored in Fused Filament Fabrication (FFF) 3D printing due to their superior speed capabilities compared to traditional serial-axis designs. However, this speed advantage is often undermined by undesirable vibrations that degrade part quality, a problem exacerbated by the robot's coupled, position-dependent (nonlinear) dynamics. While feedforward control techniques like Filtered B-Splines (FBS) have successfully suppressed vibration in serial printers, their direct application to delta printers is computationally prohibitive. This paper addresses this bottleneck by proposing an efficient methodology to implement FBS-based vibration compensation on delta 3D printers.

2. Methodology

The proposed approach tackles computational challenges through a three-pronged strategy designed to make real-time, model-based feedforward control feasible on resource-constrained printer controllers.

2.1 Offline Parameterization of Position-Dependent Dynamics

The position-varying elements of the delta robot's dynamic model are pre-computed and parameterized offline. This involves creating a compact representation (e.g., using polynomial or spline fits) of how inertial and Coriolis/centrifugal terms change across the workspace. During online operation, the full dynamic model at any point can be reconstructed efficiently by evaluating these pre-defined parameterized functions, rather than computing complex kinematics and dynamics from scratch.

2.2 Real-Time Model Computation at Sampled Points

Instead of generating a new dynamic model for every setpoint along a toolpath—a process that would be too slow—the controller computes models only at strategically sampled points along the trajectory. The control input between these sampled points is then generated using interpolation techniques. This significantly reduces the frequency of the most computationally intensive operations.

2.3 QR Factorization for Computational Efficiency

The core of the FBS method involves solving a linear system of equations to compute the pre-filtered reference trajectory. This requires a matrix inversion, which is computationally heavy. The paper proposes using QR factorization to solve the system more efficiently. QR decomposition ($\mathbf{A} = \mathbf{Q}\mathbf{R}$) transforms the problem into solving $\mathbf{Rx} = \mathbf{Q}^T\mathbf{b}$, which is computationally cheaper and more numerically stable than direct inversion, especially for the structured matrices common in this application.

3. Technical Details & Mathematical Formulation

The dynamics of a delta robot can be represented as a Linear Parameter-Varying (LPV) system due to its position-dependent inertia and coupling. The standard FBS approach inverts a dynamic model to pre-shape the reference command. For a discrete-time system, the output $y[k]$ is related to the input $u[k]$ through a transfer function. The FBS method designs a filter $F(z)$ such that when applied to a B-spline defined reference $r[k]$, the actual output tracks the desired trajectory $y_d[k]$ closely: $y[k] \approx G(z)F(z)r[k] = y_d[k]$. This requires solving for the filter coefficients, which involves inverting a matrix derived from the system's Markov parameters.

The computational challenge arises because for a delta robot, the plant model $G(z, \theta)$ varies with the position $\theta$. The matrix to be inverted, $\mathbf{H}(\theta)$, becomes position-dependent: $\mathbf{H}(\theta)\mathbf{f} = \mathbf{y}_d$. The proposed method approximates this as $\mathbf{H}(\theta_i)\mathbf{f} \approx \mathbf{y}_d$ at sampled positions $\theta_i$, and uses QR factorization ($\mathbf{H}(\theta_i) = \mathbf{Q}_i\mathbf{R}_i$) to solve for $\mathbf{f}_i$ efficiently at each sample. The filter for intermediate points is interpolated from these sampled solutions.

4. Experimental Results & Performance

4.1 Simulation Results: Computational Speedup

Simulations compared the proposed method against a controller using the exact, continuously updated LPV model. The proposed method—combining offline parameterization, model sampling, and QR factorization—achieved a computation time reduction of up to 23 times, while maintaining tracking accuracy within 5% of the exact method. This demonstrates the method's effectiveness in overcoming the primary computational bottleneck.

4.2 Experimental Validation: Print Quality & Vibration Reduction

Experiments were conducted on a delta 3D printer. The proposed controller was compared against a baseline controller using a single Linear Time-Invariant (LTI) model identified at one position in the workspace.

• Print Quality: Parts printed at various locations on the build plate showed significant quality improvements with the proposed controller. Features were sharper, with reduced ringing and ghosting artifacts common in high-speed delta printing.
• Vibration Measurement: Accelerometer data recorded during printing confirmed the source of quality improvement. The proposed controller reduced vibration amplitudes by more than 20% across the workspace compared to the baseline LTI controller.

Chart Description (Implied): A bar chart would likely show vibration amplitude (in g's) on the Y-axis for different print positions (X-axis), with two bars per position: one for the Baseline LTI controller (higher) and one for the Proposed FBS controller (significantly lower). A line graph overlay could depict the computation time per trajectory segment, showing a flat, low line for the proposed method versus a high, variable line for the exact LPV method.

5. Analysis Framework & Case Example

Framework for Evaluating Real-Time Control Feasibility:
When adapting a computationally intensive algorithm (like full LPV FBS) for a resource-constrained platform (like a 3D printer's ARM-based microcontroller), a systematic analysis is required:

1. Bottleneck Identification: Profile the algorithm to find the most time-consuming operations (e.g., matrix inversion, full dynamic model computation).
2. Approximation Strategy: Determine which computations can be approximated (e.g., model sampling vs. continuous update) or pre-computed (offline parameterization) with minimal performance loss.
3. Numerical Optimization: Replace generic routines with optimized ones for the specific problem structure (e.g., QR factorization for structured matrices).
4. Validation: Test the simplified algorithm against the original in simulation for fidelity, then on hardware for real-time performance and practical efficacy.

Case Example - Applying the Framework:
For this delta printer project: The bottleneck was the online inversion of a position-dependent matrix. The approximation strategy was to compute models only at sampled trajectory points. The numerical optimization was employing QR factorization. Validation showed a 23x speedup with maintained accuracy, proving feasibility.

6. Future Applications & Research Directions

• Broader Robotic Applications: This methodology is directly applicable to other parallel robots (e.g., Stewart platforms, SCARA-like systems) and serial robots with significant configuration-dependent flexibility, where real-time model-based control is challenging.
• Integration with Learning-Based Methods: The offline parameterized model could be enhanced or adapted online using Gaussian Process regression or Neural Networks to account for unmodeled dynamics or wear, as seen in advanced adaptive control research from institutions like MIT's CSAIL.
• Cloud-Edge Co-Processing: The most computationally heavy offline parameterization and trajectory pre-planning could be offloaded to a cloud service, with the lightweight sampled-model and QR solver running on the printer's edge device.
• Standardization in Firmware: The principles could be integrated into open-source 3D printer firmware (e.g., Klipper, Marlin) as a premium feature for high-speed delta and CoreXY printers, democratizing access to advanced vibration compensation.

7. References

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4. Edoimioya, N., & Okwudire, C. (2021). Filtered B-Splines for Vibration Compensation on Serial 3D Printers: A Review and Implementation Guide. Mechatronics.
5. Codourey, A. (1998). Dynamic modeling of parallel robots for computed-torque control implementation. The International Journal of Robotics Research.
6. Angel, L., & Viola, J. (2018). Fractional order PID for torque control in delta robots. Journal of Control Engineering and Applied Informatics.
7. MIT Computer Science & Artificial Intelligence Laboratory (CSAIL). (2023). Adaptive and Learning-Based Control Systems. [Online]. Available: https://www.csail.mit.edu

8. Original Analysis & Expert Commentary