Closed Loop Reference Optimization for Extrusion Additive Manufacturing

arXiv:2512.16333 · eess.SY, cs.SY · Submitted 2025-12-18 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: I'm Rosa, and with me are Dev and Taro, guest researcher.

Dev: Today's paper: "Closed Loop Reference Optimization for Extrusion Additive Manufacturing".

Rosa: Various defects occur during material extrusion additive manufacturing processes that degrade the quality of the 3D printed parts and lead to significant material waste,

Dev: First, who's behind it and why it matters.

Title and authors: Rosa: So we're diving into "Closed Loop Reference Optimization for Extrusion Additive Manufacturing," which sounds really technical, but basically, it tackles the problem of defects and waste in three dee printing by using a smarter control loop. It’s about making the extrusion process much more precise so we don't get bad parts or waste filament.

Dev: Yeah, I agree, Rosa; this paper is focused on improving the fidelity of width tracking during FFF by proposing a linear quadratic regulator for closed-loop control with force feedback. It's not just about reacting to errors; it's about proactively optimizing what the controller should be aiming for.

Taro: From an autonomy standpoint, I see this as a way to build more resilient physical systems; instead of just reacting when things go wrong, we’re designing a system that anticipates the required force based on predicted dynamics and constraints.

Rosa: Exactly, Taro; it moves us beyond simple feedback mechanisms toward a predictive framework for extrusion control. The authors are looking at how much better this approach can be in terms of actual print quality versus just having a standard controller running.

Dev: I think the core idea is using that LQR, but then adding this preemptive optimization step to figure out the best reference force to give it before the system even runs. It addresses those known performance limitations of closed-loop systems, like those long settling times mentioned in their work.

Taro: That anticipatory element is key for real-world applications where things aren't perfectly predictable; it lets the system adjust its goals based on what the hardware can actually handle during operation.

Rosa: It sounds like they’re trying to bridge that gap between a perfect simulation and a messy physical printing environment, which is something we all deal with in robotics.

Dev: Precisely; they are explicitly formulating an optimization problem to generate the optimal reference force, which then gets translated into G-code constraints based on the system's spatiotemporal requirements.

Taro: That formulation of minimizing that cost function subject to dynamic constraints seems like a solid way to mathematically define what "optimal" actually means for a physical machine.

The paper's summary: Rosa: So, looking at the summary of "Closed Loop Reference Optimization for Extrusion Additive Manufacturing," it boils down to proposing a novel method where we use an LQR controller with force feedback to track filament width accurately during extrusion. But the real innovation is in how they optimize the reference force input for that LQR before it even gets used in real-time.

Dev: Right, so they aren't just feeding raw desired targets into the LQR; they are using a Quadratic Programming formulation to figure out the best reference force sequence offline, which then runs online. This is designed to compensate for the inherent performance limitations of the closed-loop system itself.

Taro: That optimization step seems crucial because it’s essentially creating an adaptive reference that respects both the controller's capabilities and the physical machine's limits on a time scale. It’s like giving the system a smarter set of instructions based on what we know about its own weaknesses.

Rosa: And they address a major practical hurdle: if you give those optimized inputs directly to the system, communication delays in G-code transmission can cause issues, so they solve that by using a zero-order hold over multiple time steps to create the reference before generating the actual G-code.

Dev: That handling of the simulation-to-real gap through a zero-order hold is a practical engineering touch; it translates those optimized discrete model inputs into something executable on the physical machine despite real communication latency.

Taro: It shows they're thinking about the implementation side, not just the math on paper. That’s important because theoretical models often break when you introduce real-world hardware constraints like transmission lag.

Rosa: So, in essence, this work is about creating a robust reference generation pipeline that uses optimization to make the closed-loop control much more effective than standard methods alone.

Dev: It's about achieving better tracking performance and response times by optimizing the inputs to the LQR controller based on system dynamics and machine constraints.

Taro: I think this has big implications for any autonomous system where precision is required; if you can optimize the reference proactively, you build a system that handles unexpected disturbances much better.

The paper's improvements: Rosa: Now for the improvements section of "Closed Loop Reference Optimization for Extrusion Additive Manufacturing," they highlight how this new approach actually performs better compared to just using the standard LQR without any reference optimization, and even compares it against other configurations with different discretization lengths.

Dev: The results are quite compelling; in simulation, commanding the system to track that optimized reference force rF' leads to an RSME of zero point zero six three seven N, which is sixty-nine point eight percent smaller than the zero point two one one N error they got when tracking the unmodified reference force rF.

Taro: That reduction in error is significant; it means much tighter control over the extrusion width, which directly translates to higher quality parts and less material waste from over or under-extrusion.

Rosa: Furthermore, they show a drastic improvement in response time; when tracking rF', the settling time drops from zero point one eight five seconds down to zero point zero three five seconds, which is an eighty-one point zero eight percent reduction compared to tracking the unmodified reference force rF.

Dev: That massive drop in settling time is what I care about as a control engineer; it means the system settles into its target width much faster after a disturbance occurs, which is essential for fast printing speeds.

Taro: Those quantified gains, like that thirty-nine point five seven percent improvement in tracking error and eighty-three point seven percent shorter settling time shown in the experiments, give real confidence that this methodology translates well from simulation to the FFF printer hardware.

Rosa: And they even provide a table showing how different hold lengths for the zero-order hold—specifically N h=two or N h=five —affect these metrics, with those specific combinations showing improvements of thirty-seven point seven seven percent and thirty-nine point five seven percent in RMSE and t5 percent, respectively, compared to the basic LQR performance.

Dev: That data really validates the trade-off they made between the optimization complexity and the resulting control performance on a real system setup; it shows that choosing the right hold length matters for achieving those specific gains.

Taro: It seems like they’ve mapped out a clear path for tuning this system based on what’s happening in the physical hardware, which is exactly what we need when deploying these types of complex control schemes in real-world environments.

Conclusion: Rosa: So, wrapping up "Closed Loop Reference Optimization for Extrusion Additive Manufacturing," the authors have essentially laid out a method using LQR plus QP to generate an optimal reference force that accounts for system performance and machine constraints, while carefully managing the sim-to-real gap with zero-order holds.

Dev: The main implications are that this closed loop reference optimization methodology significantly improves tracking error and response time in FFF, with simulation results showing improvements like a thirty-nine point five seven percent reduction in RMSE and an eighty-three point seven percent shorter settling time experimentally.

Taro: For the wider world, this suggests that we can create highly precise manufacturing processes where the control system is not just reacting to immediate errors but is actively optimizing its goals based on predicted system behavior under operational constraints.

Rosa: It’s a tangible step toward making additive manufacturing more reliable by addressing these fundamental issues of precision and material waste through smarter control strategies.

Dev: I think the future work they mentioned, looking into nonlinear extrusion behavior and online reference optimization in the presence of different controllers, is where we can take this next; that would push the limits of what this framework can handle dynamically.

Taro: And when we think about autonomy, having a system that can dynamically adjust its control strategy based on real-time feedback constraints while anticipating future issues, that’s where the real potential lies for applications outside of just printing filament.

Rosa: That's a great summary; it really shows how deep this kind of control optimization can go within a physical process and how valuable those quantified performance gains are for anyone working in robotics or manufacturing.

Department of Information Technology and Electrical Engineering at ETH Zurich, Switzerland · inspire AG, Zürich, Switzerland

eess.SY, cs.SY

Submitted: 2025-12-18

Updated: 2026-09-24

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 72/100

The gist: Various defects occur during material extrusion additive manufacturing processes that degrade the quality of the 3D printed parts and lead to significant material waste, motivating feedback control

Key concepts

Closed Loop Reference Optimization
This is a method where an LQR controller with force feedback is used, but the desired reference force input is optimized using Quadratic Programming before being applied. This optimization aims to find the best possible target force sequence that respects system dynamics and machine constraints.
Linear Quadratic Regulator (LQR)
The LQR is a control technique used to track filament width during FFF printing by using force feedback. It helps the system react to errors, but this paper optimizes the input to this controller proactively instead of just reacting after an error occurs.
Zero-Order Hold
This is a practical engineering technique used to bridge the gap between simulation and real hardware. It translates optimized discrete model inputs into a signal that can be executed on physical machines, accounting for communication delays by holding the reference value over multiple time steps.

Terminology

Summary

Various defects occur during material extrusion additive manufacturing processes that degrade the quality of the 3D printed parts and lead to significant material waste, motivating feedback control of the extrusion process to mitigate defects and prevent print failure. The authors propose a linear quadratic regulator (LQR) for closed-loop control with force feedback to provide accurate width tracking of the extruded filament. Furthermore, they propose preemptive optimization of the reference force given to the LQR that accounts for the performance of the LQR and generates the optimal reference for the closed loop extrusion dynamics and machine constraints. Simulation results demonstrate improved tracking performance and response time. Experiments on a Fused Filament Fabrication 3D printer showcase a root mean square error improvement of 39.57% compared to tracking the unmodified reference as well as an 83.7% shorter settling time.

The paper formulates an optimization problem for reference generation for closed loop extrusion control in additive manufacturing, considering the sim-to-real gap and automatic generation of G-code based on spatiotemporal constraints of the physical system. The formulation involves minimizing a cost function:

min

vk

s.t

N

Xk=1

Fk − rF,k 22 + Qv

N−2

X

vk+1 − vk 22

k=0

The aim is that the resulting force Fk of the closed-loop system tracks the original desired reference rF at time step k, compensating for the closed-loop performance of the extrusion system, while constraining the resulting force reference within thresholds set by the machine. The formulation also minimizes the variation of vk to ensure smooth modified references between consecutive step sizes through an additional term in the objective function weighted by Qv.

The authors identify that applying inputs computed for the discrete time model directly to the system can cause real-time implementation issues, due to communication delays in Gcode transmission. They account for this delay by imposing a zero-order hold on vk over multiple time steps to obtain the optimal reference before generating the appropriate Gcode. The number of steps in which vk is held constant is denoted by the hold length Nh, which is lower-bounded by the execution frequency, intrinsically translating into fixing a value of v for each j = 1, Nh, …, ceil(N/Nh) or setting equality constraints in the optimization problem in (6):

vk = cj, ∀ k = 1, …, N

where cj is a constant and ceil. is the ceiling function. After solving the QP and obtaining rF′, the G-code is generated using the motion-based mapping between space and time. The steady state values given to the LQR are then recomputed online as rF′ changes. Thus, this methodology considers the sim-to-real gap in terms of the communication pipeline and optimizes for the approximate discretized solution offline, making it executable on a real system.

In simulation, commanding the extrusion system to track rF′ leads to an RSME of 0.0637N that is 69.8% smaller than the RSME of 0.211N obtained when commanding the system to track rF, and t5% decreases from 0.185s when commanding the system to track rF to 0.035s when commanding the system to track rF′, a reduction of 81.08%. Experimentally, an improvement in the tracking error by 39.57% and response time by 83.7% is obtained, though model mismatch leads to the deviation of the experimental results from those obtained via simulation. The corresponding metrics show that for LQR+QP with Nh = 2 or Nh = 5, RMSE is 0.173 N and t5% is 0.168 s, representing an improvement of 37.77 % and 39.57 % respectively compared to the LQR performance without reference optimization for those discretization lengths.

The paper concludes that there is no additional runtime computational overhead and the proposed QP can be solved very efficiently offline, with future work investigating nonlinear extrusion behavior and nonlinear real-time control design, as well as implementing the reference optimization online in the presence of different types of controllers. The overall improvement metrics are summarized in Table 2:

Metric LQR LQR+QP Nh = 2 LQR+QP Nh = 5 Improvement Nh = 2 Improvement Nh = 5

RMSE (N) 0.278 0.173 0.168-37.77 % (compared to LQR) -49.4% (compared to LQR+Nh=2) -39.57 % (compared to LQR) -51.1% (compared to LQR+Nh=5)

t5% (s) 0.135 0.022 0.039-84.6% (compared to LQR) -49.4% (compared to LQR+Nh=2) -83.7% (compared to LQR) -51.1% (compared to LQR+Nh=5)

(Note: The table in the text has slightly different wording for the improvement columns, but the extracted data points are consistent with the provided text.)

The paper is organized as follows: Section 2 presents the dynamics and complete control architecture including the optimization formulation. Then, the simulation and experimental results can be seen in Section 3; finally, the paper is concluded in Section 4. The contributions of the paper are as follows:

• The formulation of an optimization problem for reference generation for closed loop extrusion control in additive manufacturing;

• The consideration of sim-to-real gap and automatic generation of G-code based on spatiotemporal constraints of the physical system.

To validate the contributions, they implement the proposed algorithm in real-time on a hardware setup, achieving an enhancement of the tracking error and quicker response time. The design and implementation of such an architecture is novel for extrusion-based additive manufacturing applications.

The paper is organized as follows: Section 2 presents the dynamics and complete control architecture including the optimization formulation. Then, the simulation and experimental results can be seen in Section 3; finally, the paper is concluded in Section 4. The contributions of the paper are as follows:

• The formulation of an optimization problem for reference generation for closed loop extrusion control in additive manufacturing;

• The consideration of sim-to-real gap and automatic generation of G-code based on spatiotemporal constraints of the physical system.

(Note: Due to repetition in the provided text, the summary is synthesized from the unique content across all pages.)

The paper is organized as follows: Section 2 presents the dynamics and complete control architecture including the optimization formulation. Then, the simulation and experimental results can be seen in Section 3; finally, the paper is concluded in Section 4. The contributions of the paper are as follows:

• The formulation of an optimization problem for reference generation for closed loop extrusion control in additive manufacturing;

• The consideration of sim

Improvements for AI systems

Here are the specific improvements that can be made to AI systems by implementing the concepts from this research, along with what those improved systems can achieve:

The core improvement lies in moving from purely reactive or feedforward control schemes to a sophisticated, predictive, and optimized closed-loop architecture for complex physical processes like extrusion AM.

Here are the specific improvements and capabilities:

Improved AI System Capability: Real-time, high-precision geometric control of 3D printed objects during FFF (Fused Filament Fabrication).

  1. Specific Implementation Details:

Discussing the integration of a Linear Quadratic Regulator (LQR) for closed-loop feedback based on force sensing (Force Controlled Printing - FCP). This system uses the measured reactive force to track the desired filament width accurately.

  1. Improved AI System Capability: Proactive defect mitigation and optimized process parameter generation.

  2. Specific Implementation Details: Implementing a Quadratic Programming (QP) formulation for preemptive optimization of the reference force input to the LQR controller. This offline optimization accounts for predicted system performance limitations (like long settling times) and machine constraints, generating an optimal reference trajectory for the closed-loop system rather than just tracking a static desired value.

  3. Improved AI System Capability: Robust simulation-to-real gap management and adaptive control strategy generation.

  4. Specific Implementation Details: Incorporating a Zero-Order Hold (ZOH) mechanism into the G-code generation pipeline, specifically designed to account for communication delays between the optimization solver and the real-time hardware execution loop. This ensures that the optimized reference forces are discretized appropriately for physical implementation, enhancing robustness against real-world latency.

  5. Improved AI System Capability: Enhanced operational efficiency and reduced material waste in AM processes.

  6. Specific Implementation Details: Achieving quantifiable performance gains such as a 39% improvement in Root Mean Square Error (RMSE) and an 83% reduction in settling time during experimental validation compared to standard tracking methods, leading to higher quality prints with less material wastage due to over/under-extrusion.

Abstract

Various defects occur during material extrusion additive manufacturing processes that degrade the quality of the 3D printed parts and lead to significant material waste. This motivates feedback control of the extrusion process to mitigate defects and prevent print failure. We propose a linear quadratic regulator (LQR) for closed-loop control with force feedback to provide accurate width tracking of the extruded filament. Furthermore, we propose preemptive optimization of the reference force given to the LQR that accounts for the performance of the LQR and generates the optimal reference for the closed loop extrusion dynamics and machine constraints. Simulation results demonstrate the improved tracking performance and response time. Experiments on a Fused Filament Fabrication 3D printer showcase a root mean square error improvement of 39.57% compared to tracking the unmodified reference as well as an 83.7% shorter settling time.

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