Research on integrated motion and vibration control methods for heated nozzles in space additive manufacturing equipment

2.1. Basic working principle of the thermal nozzle

The working principle of FDM additive manufacturing is based on heating a thermoplastic material to its molten state and then precisely extruding it through a thermal nozzle to construct three-dimensional objects layer by layer. The process begins with the filament being fed into the heated nozzle by a feeding mechanism, where it is melted and subsequently extruded along a predefined path. Each layer of molten material solidifies upon cooling and is deposited sequentially to form the complete object[23].

FDM systems are widely used in prototyping and functional part fabrication due to their simple structure, user-friendly operation, and broad material compatibility. These characteristics make FDM particularly suitable for environments with limited space, such as homes and educational institutions. Owing to their ease of use and low cost, PID control is widely used in FDM devices due to its simplicity and reliability. As illustrated in Fig. 1.

Fig. 1Schematic diagram of the working structure of a FDM additive manufacturing system

2.2. Dynamic modeling of delta-type additive manufacturing equipment

Dynamic analysis is essential for both performance evaluation and motion control of parallel robotic systems, as it characterizes the relationship between joint actuation and system motion states. Forward dynamics analysis is primarily used for system simulation and performance validation, whereas inverse dynamics provides the theoretical basis for controller design. An accurate dynamic model not only enables deeper insight into the robot’s dynamic behavior but also supports the development of high-speed, high-precision trajectory planning and control algorithms. The dynamic configuration of the Delta-type mechanism considered in this study is illustrated in Fig. 2.

Fig. 2Schematic diagram of the delta-type additive manufacturing structure: 1 – drive motor; 2 – linkage; 3 – thermal nozzle actuator; 4 – lead screw; 5 – base

To characterize the high-frequency micro-vibration behavior of the thermal nozzle during space-based FDM printing, the nozzle and its supporting end-effector platform are modeled as an equivalent lumped mass m, as shown in Fig. 3. A local Cartesian coordinate system is established at the center of mass to describe translational motions along the horizontal $x$-axis, $y$-axis and the vertical $z$-axis. For vibration suppression analysis, the system is further simplified as a three-degree-of-freedom lumped-parameter model.

The mechanical compliance of the Delta mechanism, including the flexibility of the three driving limbs and the upper platform, is represented by three independent linear elastic and damping elements. The equivalent stiffness and damping coefficients along each axis are denoted as $k_x$, $k_y$, $k_z$ and $c_x$, $c_y$, $c_z$, respectively. During printing, the nozzle is subject to external excitations – such as structural vibrations, actuator-induced disturbances, and base micro-vibrations – which are lumped as generalized forces $F_x\left(t\right)$, $F_y\left(t\right)$, and $F_z\left(t\right)$.

Under the small-amplitude vibration assumption, the dynamics of the nozzle can be linearized and formulated as a three-degree-of-freedom lumped-parameter model:

These equations can be written in matrix form as:

where $q=[x,y,z{]}^T$ denotes the nozzle displacement vector, $M$, $C$, $K$ are the equivalent mass, damping and stiffness matrices, and $F\left(t\right)=\left[F_x\right(t),F_y(t),F_z(t){]}^T$ represents the generalized disturbance forces.

This 3-DOF dynamic model provides the physical foundation for the subsequent ES-FXLMS vibration suppression algorithm and establishes a clear link between the Delta mechanism and its vibration-response characteristics.

Although the mass $m$ can be obtained numerically, the computation is complex and inefficient. Therefore, this study derives the dynamic model of the Delta mechanism using the principle of virtual work.

The principle of virtual work applied to $n$ rigid bodies can be expressed as follows:

where, $m_i$ and $I_i$ denote the mass and inertia matrix of the $i$ rigid body, respectively; ${\ddot{x}}_i$ represents the linear acceleration of its center of mass; ${\omega }_i$ and ${\dot{\omega }}_i$ are the angular velocity and angular acceleration; $F_i$ and $T_i$ denote the external force and torque applied to the rigid body; $\delta x_i$ and $\delta {\varphi }_i$ are the corresponding virtual displacements.

Fig. 3Simplified three-degree-of-freedom lumped-parameter dynamic model of the thermal nozzle, including equivalent mass, stiffness, damping, and external disturbance forces

For virtual displacements such as $\delta x_i$ and $\delta {\varphi }_i$, they can be transformed into functions of the virtual joint displacements through the Jacobian matrix. Thus, we obtain:

Therefore, since Eq. (4) holds for any virtual displacement $\delta q$, it follows that:

The applied forces include the joint driving torque $\tau$ as well as other external forces $F_{i,ext}$ and external torques $T_{i,ext}$. Thus, we have:

Thus, the dynamic equation can be expressed as follows:

Given that the primary objective of spaceborne FDM equipment is to suppress micro-vibrations induced by the coupling between motion and external disturbances, the fundamental cause of these vibrations lies in the intrinsic dynamic characteristics of the structure, which remain unchanged in a microgravity environment. According to existing studies on the dynamics of space structures [24], the principle of virtual work is a commonly used and effective approach for analyzing micro-vibration problems in large flexible spacecraft and their appendages. Therefore, the dynamic model established in this work provides a reasonable and reliable basis for investigating the motion and vibration control of space-based FDM systems during on-orbit operation.

In this study, the full multibody dynamics of the Delta mechanism were first derived using the principle of virtual work. This complete model reveals the sources of inertia, coupling, and external disturbances.

However, for the purpose of vibration suppression, the nozzle operates in a small-amplitude micro-vibration region. Therefore, the full nonlinear dynamics are linearized around the operating point and reduced to an equivalent 3-DOF translational mass-spring-damper model:

This linearized model is used throughout the controller design and stability analysis. Within the small-displacement range, the matrices $M$, $C$, and $K$remain constant, and the system behaves as a lightly damped second-order system with all poles in the left-half complex plane. The PID controller further shifts the closed-loop poles leftward, ensuring zero steady-state error and improved damping. Although nonlinear effects (Coriolis, centrifugal, geometric nonlinearities) are neglected here, they will be evaluated through future Hardware-in-the-loop (HIL) experiments to confirm the applicability of the linear model.

Although the three actuated limbs of the Delta mechanism are driven independently, the parallel kinematic structure inherently introduces inter-axis coupling. Such coupling may potentially affect the end-effector trajectory accuracy, especially under high-speed and large-amplitude motion. To quantitatively assess the influence of coupling terms within the workspace considered in this study, a sensitivity-based analysis was carried out around a representative operating point near the center of the printing workspace.

Small unit perturbations were applied independently to each joint variable $q_j$, and the resulting changes in the Cartesian nozzle position $x_i$ were recorded. On this basis, normalized sensitivity coefficients were defined as:

where $S_{ii}$ represents the primary sensitivity of axis $i$ to its own joint motion, and $S_{ij}$ ($i\ne j$) reflects the cross-coupling effect from joint $j$ to axis $i$. A value of $S_{ij}$ close to zero indicates weak coupling, whereas a value comparable to $S_{ii}$ would imply strong coupling. The computed normalized sensitivity matrix for the analyzed operating region is summarized in Table 2.

The results indicate that all off-diagonal sensitivity coefficients $S_{ij}$ ($i\ne j$) remain below 0.05, less than 5 % of the corresponding primary sensitivities. This is consistent with the modal sensitivity analysis and dynamic simulations performed, where the off-diagonal elements of both the Jacobian matrix and the inertia matrix were found to be less than 5 % of the corresponding diagonal elements. These findings quantitatively confirm that, within the motion ranges considered in this work, the inter-axis coupling is weak.

Because the coupling terms contribute only a small fraction to the overall trajectory deviation, their effect on tracking accuracy can be regarded as second-order. Under the present operating conditions, this influence is therefore negligible. Therefore, independent PID control is adopted for each axis in this study. This not only simplifies the controller structure and parameter tuning but also improves real-time implementation efficiency without sacrificing trajectory accuracy in the tested workspace.

Nevertheless, it should be emphasized that the proposed framework is not limited to independent control. For future applications involving larger workspaces, higher speeds, or heavier loads, where coupling may become more pronounced – the current scheme can be extended to a multivariable coordinated control structure, for example by incorporating model-based decoupling or adaptive multivariable compensation to further enhance global trajectory accuracy and dynamic consistency.

3.1. Overall architecture design

This paper proposes an integrated control method that combines PID-based motion control with FXLMS-based adaptive vibration suppression, tailored to the motion characteristics and vibration mitigation requirements of the thermal nozzle in Delta-type additive manufacturing systems. The overall system architecture consists of a motion control loop and a vibration suppression loop, which are responsible for precise trajectory tracking and real-time vibration attenuation, respectively. The effectiveness of the proposed control strategy is validated through co-simulation with an Adams model. As illustrated in Fig. 4.

Fig. 4Block diagram of the integrated motion and vibration control system for the thermal nozzle in delta-type additive manufacturing equipment

3.2. PID motion control

To achieve precise trajectory tracking of the thermal nozzle in three-dimensional space for space-based additive manufacturing equipment, this study adopts a closed-loop feedback strategy based on a PID controller in the motion control loop. As shown in Fig. 5, the PID controller offers a simple structure, strong real-time performance, and ease of tuning, making it effective in improving both system response speed and steady-state accuracy. These advantages make it well-suited to the high-precision motion control requirements of this work.

Based on the dynamic characteristics of the Delta mechanism, the motion of the thermal nozzle can be approximated by the following multi-degree-of-freedom rigid-body dynamic equation:

where $\mathbf{q}\left(n\right)$ denotes the displacement vector, and $u\left(n\right)$ represents the control input. The matrices $M$, $C$, and $K$ correspond to the equivalent mass, damping, and stiffness of the system, while $B$ is the input distribution matrix. This equation provides the fundamental dynamic framework required for the PID-based motion control design. The simplified 3-DOF vibration model introduced in Section 2.2 is obtained by linearizing this general form under small-amplitude vibration assumptions.

Fig. 5Schematic diagram of the PID control system

In this system, independent PID control loops are designed for the $x$, $y$, and $z$ directions to accommodate the three degrees of freedom of the Delta mechanism’s spatial motion. This ensures that the motion trajectories along each axis can independently and synchronously follow the desired reference paths.

3.3. Vibration suppression strategy

Building upon the completed motion control of the thermal nozzle, this study further addresses the vibration issues induced during high-speed operations, which arise from structural flexibility and external micro-disturbances. Although the PID controller can improve the positioning accuracy to some extent, it is fundamentally designed for static error correction and thus lacks the capability to effectively suppress dynamic vibrations that compromise system stability and trajectory accuracy. This limitation is particularly significant in the application environment of Delta-type additive manufacturing systems, where the thermal nozzle structure is lightweight and exhibits limited stiffness. As a result, high-speed motion tends to excite small but persistent vibrations that can severely degrade printing quality.

Therefore, based on the PID motion control framework, this study introduces an adaptive vibration suppression controller using the FXLMS algorithm. By continuously monitoring the vibration error signals during motion, the FXLMS controller dynamically adjusts the compensation signal in real time to actively suppress vibrations at the thermal nozzle tip. This enhances both the stability and accuracy of trajectory tracking.

The motion control and vibration suppression modules are implemented in parallel within the system architecture. While the motion control module is responsible for tracking the desired trajectory, the vibration control module targets the suppression of dynamic micro-vibrations. These two modules work in a complementary manner to form a unified motion and vibration integrated control strategy.

FXLMS is a widely used adaptive filtering algorithm in applications such as active noise and vibration control, system identification, and signal processing. Compared with the FXLMS algorithm, FXLMS explicitly accounts for the dynamics between the actuator and the error sensor by filtering the reference signal through an estimate of the secondary path, which makes it particularly suitable for disturbance rejection in noisy environments.

In the proposed control framework, the adaptive filter coefficients are updated to minimize the measured vibration error, while the filtered reference signal effectively compensates for the influence of external disturbances and system dynamics on the adaptation process. This filtering mechanism significantly improves the robustness and convergence behavior of the adaptive algorithm when operating under broadband and time-varying disturbance conditions.

Fig. 6Schematic diagram of the adaptive filtering principle

A typical adaptive filtering structure is shown in Fig. 6. The notations are defined as follows.

Where $\mathbf{x}\left(n\right)$ represents the input signal at time $n$, $y\left(n\right)$ represents the output signal at time $n$, $d\left(n\right)$ represents the desired signal at time $n$, $e\left(n\right)$ represents the error signal, $v\left(n\right)$ represents the primary input disturbance signal. After receiving input signal $x\left(n\right)$, the output signal $y\left(n\right)$ after adaptive filtering is compared with the desired signal $d\left(n\right)$, to obtain the error signal $e\left(n\right)$, Furthermore, the error signal $e\left(n\right)$ is then used to automatically adjust the filter parameters $\mathbf{W}\left(n\right)$, In this way, the adaptive filter is gradually adjusted toward convergence, allowing the output $y(n+1)$ at the next time step to more closely approximate the desired signal.

From the principal diagram of the adaptive filter, it is clear that the system's output error is given by:

The output signal can be expressed as:

The weight coefficients of the $N$-th order adaptive filter at time $n$ can be expressed as:

where $\mu$ denotes the step size of the algorithm, and the convergence condition for the LMS algorithm is given by $1<\mu <\frac{1}{{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}}$, with ${\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ representing the maximum eigenvalue of the input signal’s autocorrelation matrix.

(1) VSSLMS-1 algorithm.

The NLMS algorithm is currently one of the most widely used Variable step-size LMS (VSS-LMS) algorithms[25], [26]. Its step-size update formula is given by:

In the NLMS algorithm, the normalized step size $\mu$ must satisfy 0 $<\mu <$ 2, although smaller values are generally preferred in practical applications to ensure stable adaptation. The algorithm exhibits robustness to variations in the energy of the reference signal $\mathbf{x}\left(n\right)$. To enhance numerical stability, particularly to prevent sudden step-size amplification when the reference signal amplitude approaches zero, which may otherwise destabilize or even cause divergence of the algorithm, a small positive constant $\epsilon$, much smaller than the reference signal power, is conventionally added to the denominator:

(2) VSSLMS-2 algorithm.

A VSS-LMS algorithm based on a nonlinear error function was proposed by F. Wang [27], and its step-size update rule is given by Eqs. (15-16):

The performance of this algorithm is largely influenced by the value of the damping factor $\upsilon$. The damping factor determines the duration for which the algorithm operates with a relatively large step size. A larger $\upsilon$ value results in a longer duration of large step-size operation. As $\upsilon$ approaches infinity, the algorithm degenerates into a fixed step-size LMS algorithm.

The computational complexity of an algorithm plays a critical role in its practical applicability, as it directly determines computational efficiency and real-time feasibility. In this section, we compare the computational complexity of several existing VSSLMS algorithms with that of the improved VSSLMS algorithm proposed in this study. The comparison focuses on key indicators such as the number of parameters that must be tuned before execution and the computational load required for step-size updates during each iteration.

In practice, many VSSLMS algorithms require upper and lower bounds on the step size to ensure stable operation. In response to this, we propose an improved variable step-size algorithm (ES-FXLMS), which is formulated as follows:

where $a\in \left[\mathrm{0.7,0.75}\right]$ and $b\in \left[\mathrm{1,1.5}\right]$.

Where $e\left(n\right)$ represents the vibration control error signal. The parameter $a$ specifies the upper bound of the step size, while $b$ determines the sensitivity of the step-size attenuation with respect to the error magnitude. The exponential term $\exp (-b\left|e\right(n\left)\right|)$ decreases monotonically as the error grows, ensuring that the adaptation becomes more conservative under large disturbances and preventing unstable gain escalation.

The sinusoidal component $\frac{\sin (e\left(n\right))+1}{2}$ provides a smooth, phase-dependent modulation of the step size. This term has been widely used in practical adaptive filtering to avoid abrupt variations in the update gain and to enhance robustness under oscillatory disturbances. Together, these two components form an empirically stable step-size function that provides fast initial convergence when the error is small while maintaining numerical stability during periods of larger or rapidly changing disturbances.

This formulation effectively balances convergence speed and stability, making it well-suited for the time-varying and multi-frequency disturbance environment encountered in space-based FDM systems.

4.1. Simulation system

To investigate and validate the vibration suppression performance of the thermal nozzle in Delta-type additive manufacturing equipment, a three-degree-of-freedom active vibration control simulation model was established using the dynamic modeling capabilities of ADAMS.

The model consists of two main subsystems: an excitation module and a suppression module. The excitation module simulates micro-vibrations caused by external disturbances, with its output serving as the input to the suppression module. The suppression module, driven by the control algorithm, generates real-time compensation forces to attenuate vibrations effectively.

This model enables simulation of micro-vibrations along the $x$, $y$, and $z$ axes. By integrating ADAMS with Simulink via a co-simulation platform and implementing an adaptive filtering-based control algorithm, the effectiveness of the ES-FXLMS Algorithm for vibration suppression was validated in co-simulation. The simulation results suggest a significant reduction in vibration amplitude and improved system stability, as illustrated in Fig. 7.

Fig. 7Co-simulation structure of ADAMS and Simulink

During co-simulation, a single-frequency sinusoidal excitation is introduced in Simulink to mimic realistic disturbances. This signal also serves as the reference input for the control algorithm, facilitating synchronization between excitation and compensation. The PID controller regulates the driving force to minimize trajectory tracking errors.

Meanwhile, ADAMS monitors the displacement at the center of the suppression module’s upper surface, which reflects the system’s vibrational response. This displacement is fed back to Simulink as an error signal, allowing the adaptive controller to compute a real-time compensation force. The calculated force is applied to the piezoelectric actuator, forming a closed-loop system for active vibration suppression.

The co-simulation environment is built in MATLAB/Simulink. The mechanical model is exported from ADAMS as a script and loaded in MATLAB using the admasses command. Within Simulink, the generated Adams sub-block is inserted into the control framework, as shown in Fig. 8.

To support flexible signal generation and online parameter tuning, a user-defined S-Function module is employed. This module enables custom signal input (e.g., step, sweep) and enhances the scalability of the control system by supporting real-time adjustments. Overall, the co-simulation framework facilitates accurate evaluation of the proposed control algorithm under realistic dynamic conditions.

Fig. 8Simplified ADAMS sub-block of co-simulation control system

4.2. Practical implementation of vibration detection

In the actual physical system, the $z$-direction vibration displacement of the thermal nozzle can be measured using high-precision non-contact laser displacement sensors or Micro-Electro-Mechanical Systems (MEMS) based accelerometers installed near the nozzle tip. The laser sensor offers micrometer-level accuracy and high response speed, making it suitable for capturing small-amplitude vibrations in real time. Alternatively, MEMS accelerometers can be used to detect acceleration signals, which are then integrated to obtain displacement information through a digital filter.

The measured signals are transmitted to the control system through a high-speed data acquisition module and processed in real time by the ES-FXLMS controller implemented in the Simulink environment. This configuration allows the same adaptive vibration suppression algorithm used in the simulation to be directly applied in practical HIL validation. The additional implementation details ensure that the proposed vibration detection and control strategy can be effectively transferred from simulation to an actual Delta-type additive manufacturing platform.

4.3. Integrated control system

Fig. 9 presents the Simulink block diagram of the integrated motion and vibration suppression control system based on the FXLMS algorithm. The system primarily consists of four modules: a trajectory input module, a PID control and inverse kinematics module, an FXLMS-based vibration control module, and a motor feedback module.

Fig. 9Simulink simulation model of the integrated motion and vibration suppression control system

Specifically, the trajectory input module receives the desired path signal and serves as the system input. The PID control and inverse kinematics module adjusts trajectory errors and converts them into motor drive parameters. The FXLMS vibration control module adaptively adjusts filter weights based on external disturbances and feedback signals, enabling active suppression of structural vibrations. The motor feedback module returns parameters such as displacement and current from the motion process back to the control system, enabling closed-loop correction and dynamic optimization.

By integrating motion control and vibration suppression into a unified control strategy, the system significantly enhances the operational accuracy and dynamic stability of the 3D printing device.

4.4. Simulation results and analysis

An integrated motion control model for the thermal nozzle system was developed based on a co-simulation platform combining MATLAB/Simulink and ADAMS. This model was designed to evaluate the response performance and tracking accuracy of a multi-axis control strategy across different motion directions. The overall simulation framework comprises a PID-based motion controller, an FXLMS adaptive filter module for active vibration suppression, a disturbance input module, and an ADAMS-based dynamic subsystem.

Fig. 10Comparison of motion trajectories along the x-axis with and without PID control

Fig. 11Comparison of motion trajectories along the y-axis with and without PID control

The control architecture utilizes a triple closed-loop feedback configuration to independently manage trajectory tracking along the $x$, $y$, and $z$ axis. The simulation duration was set to 5 seconds with a fixed time step of 0.0001 seconds. Controller parameters were iteratively tuned and optimized based on system response characteristics. To emulate external disturbances, a step function was applied as the initial input excitation.

Simulation results were analyzed by extracting the response trajectories along the $x$, $y$, and $z$ axis. These results were used to assess the system’s tracking stability under the proposed integrated control scheme. The application of the PID controller enabled the system to maintain close alignment with the desired trajectory, with negligible steady-state error and smooth transient response. These findings demonstrate the effectiveness of the PID strategy in improving trajectory precision and maintaining overall system stability.

Fig. 12Comparison of motion trajectories along the z-axis with and without PID control

The PID gains were tuned using a relay-based auto-tuning procedure in Simulink. Relay feedback was used to induce a stable limit cycle, from which the critical frequency characteristics were estimated for controller tuning. The tuning targeted a compromise between response speed and robustness. Frequency-domain verification of the tuned controller indicated a phase margin of approximately 65°, providing adequate stability margins against modeling uncertainties and external disturbances. After iterative validation, the final gains were selected as $K_p=$ 200, $K_i=$ 10, and $K_d=$ 9. Further increases in the gains amplified oscillations and degraded stability, confirming the suitability of the chosen parameter set.

Motion control simulations were performed along all three axes of the Delta-type additive manufacturing system. Figs. 10-12 present a representative case in which the actual trajectory under classical PID control (blue) is compared with the desired trajectory (red). The results reveal that the system achieves PID convergence and high short-term tracking accuracy during the initial phase. A magnified view in the top-left corner highlights the reduced dynamic deviation at the trajectory peak.

In the subsequent vibration phase, although residual oscillations persist, the overall deviation from the desired trajectory is significantly reduced, confirming the PID controller’s disturbance suppression capability and trajectory consistency. However, conventional PID control proves insufficient for maintaining long-term stability under periodic disturbances.

From a quantitative perspective, the deviation between the actual and desired trajectories under PID control remained within ±0.18 mm for most of the motion process, meeting the typical accuracy requirement of ±0.2 mm in high-precision additive manufacturing. The convergence time was approximately 0.45 seconds, indicating good responsiveness to short-term commands. Nevertheless, sustained disturbances continued to affect trajectory stability in later stages.

To address this, a reference trajectory modulation strategy was introduced to further enhance dynamic performance. This method maintained high tracking accuracy throughout the motion process and effectively suppressed the amplification of vibrations in the later stages. Simulations along the $x$, $y$, and $z$ axis validated the effectiveness of the proposed strategy in improving both motion quality and system robustness.

With this strategy applied, the peak deviation was further reduced to ±0.08 mm, significantly enhancing trajectory stability under dynamic loads. Compared to traditional PID control, the proposed method reduced the steady-state error by approximately 65 % and improved convergence speed by about 30 %. These findings demonstrate its adaptability and superior performance in practical additive manufacturing scenarios, particularly where vibration suppression is critical to ensuring part quality.

To further evaluate the effectiveness of the proposed active vibration control approach, a comparative analysis was conducted between the FXLMS based control strategy and conventional passive damping. Under identical disturbance conditions, the vibration responses of the thermal nozzle were recorded for both cases. We confirm that the active control strategy clearly outperformed passive damping in terms of suppression depth and convergence speed.

Moreover, passive damping devices are inherently limited by fixed parameters and structural constraints, making them less suitable for dynamic environments. In contrast, the proposed FXLMS-based control demonstrates real-time adaptability and effectively suppressed vibration energy within the 10-100 Hz range by over 40 dB in just 0.3 seconds, as verified by simulation data. This confirms its potential for application in space-constrained, dynamically varying environments such as orbital additive manufacturing platforms.

4.4.1. Single-frequency disturbance excitation

The vibration control algorithm was simulated in the printer's ADAMS environment, with the focus on vibration suppression along the $x$-axis. The choice to implement vibration control along the x-direction was based on the significant influence of disturbances and mechanical flexibilities in this particular direction. As the $x$-axis typically handles substantial dynamic loads and high-speed movements, it is more susceptible to vibration-induced deviations. Therefore, controlling vibration along the $x$-axis ensures more effective suppression of unwanted oscillations, leading to improved overall system stability and trajectory accuracy in the additive manufacturing process. Fig. 13 presents the vibration displacement curves of the thermal nozzle along a single degree of freedom under different control methods. When using passive damping, the nozzle exhibits relatively large vibrations and requires a longer time to stabilize. In contrast, the standard FXLMS algorithm significantly reduces vibration amplitude and ensures PID convergence. Furthermore, with the implementation of the ES-FXLMS algorithm proposed in this study, the suppression effect becomes more pronounced, achieving even better vibration mitigation performance.

Fig. 13Vibration suppression performance of three control strategies

Fig. 14 compares the power spectral density (PSD) of the thermal nozzle system under three control strategies: no control (green), FXLMS (blue), and the proposed ES-FXLMS algorithm (red). As shown, the ES-FXLMS algorithm achieves significantly deeper attenuation at dominant disturbance frequencies (e.g., 200 Hz, 300 Hz, 400 Hz). These harmonic peaks correspond to the system’s structural modal frequencies, which are prone to resonant amplification under periodic excitation. Suppressing these frequencies is therefore critical for maintaining structural stability and ensuring print quality in Delta-type additive manufacturing systems.

Fig. 14Comparison of power spectral density (PSD) under noise disturbances

Fig. 15MSE convergence curves of different algorithms under noise disturbances

The spectral peaks under the ES-FXLMS algorithm are markedly lower than those of the other two methods, indicating stronger attenuation across the target harmonic bands. The simulation was conducted over a 5-second duration with a sampling rate of 10 kHz and a time step of 0.0001 seconds. The external disturbance was modeled as a 10 Hz sinusoidal input with an amplitude of 1 mm.

These results validate the superior frequency-domain suppression capability of the proposed control strategy. While the FXLMS algorithm provides partial attenuation, it demonstrates limited suppression depth and narrower effective bandwidth. In contrast, the uncontrolled system exhibits pronounced harmonic amplification, with vibrational energy distributed across multiple frequency bands. This comparison highlights the ES-FXLMS algorithm’s effectiveness in addressing resonance-related issues and supports its engineering applicability in complex vibration scenarios.

In practical engineering environments, structural vibrations are often induced by complex and unpredictable external disturbances. To evaluate the robustness of the control algorithms under such conditions, broadband white noise was introduced in the simulation, superimposed with sinusoidal components corresponding to the system’s modal frequencies. This approach emulates real-world excitation and allows comprehensive performance evaluation.

To assess both dynamic response and steady-state behavior, the total mean square error (TMSE) – defined as the cumulative MSE over the full simulation time – was adopted as the evaluation metric. Fig. 15 presents the time-domain evolution of MSE under noise interference for the three algorithms. For clarity, the MSE was smoothed, and TMSE values were used to quantify the overall control performance.

From the curve trends, it is evident that the ES-FXLMS algorithm (red line) consistently maintains the lowest MSE throughout the simulation, outperforming both the FXLMS (blue) and the uncontrolled system (green). The ES-FXLMS algorithm achieves a TMSE of 1.18, compared to 26.92 and 249.79 for the FXLMS and no-control cases, respectively. This reflects a 95.6 % reduction relative to FXLMS, and over 99.5 % reduction compared to the uncontrolled case.

In the context of trajectory control, a TMSE below 2.0 typically corresponds to a peak displacement error within ±0.1 mm, which aligns with the accuracy requirements of high-precision additive manufacturing. The proposed algorithm not only meets this tolerance but significantly exceeds it, demonstrating its feasibility for deployment in precision-critical applications.

Although the uncontrolled case may exhibit a gradual reduction in error due to passive dissipation, it shows much larger residual oscillations and substantially slower stabilization than the controlled cases. In contrast, the ES-FXLMS algorithm achieves faster convergence and maintains smoother and more stable error suppression throughout the process.

All three algorithms show relatively fast initial convergence. However, only the ES-FXLMS algorithm continues to reduce error consistently and maintain long-term stability, confirming its adaptability and robustness under time-varying and noisy disturbance conditions. These observations demonstrate the practical value of the proposed control strategy for enhancing the reliability and precision of Delta-type additive manufacturing systems in real-world environments.

To quantitatively evaluate the vibration suppression performance of each control algorithm, the Total Mean Square Error (TMSE) and Convergence Time are adopted as the primary performance indicators. TMSE represents the average energy of the residual vibration error throughout the control process, where a smaller value indicates higher steady-state accuracy and better suppression stability. Convergence time is defined as the time required for the algorithm to reach the steady state after a disturbance, indicating the system’s dynamic response speed and adaptability. Therefore, an effective vibration suppression algorithm should achieve both a low TMSE and a short convergence time, ensuring rapid stabilization and minimal residual vibration. The combined comparison of these two metrics provides a comprehensive assessment of the algorithm’s suppression efficiency and robustness under complex space disturbance conditions.

Table 3Comparative performance of control strategies under disturbance conditions

Algorithm	Total mean square error (m2)	Convergence time (s)
No vibration suppression	249.79	4.75
FXLMS algorithm	26.92	2.7
ES-FXLMS algorithm	1.18	1.85

To provide a deeper understanding of the early-stage convergence behavior of the ES-FXLMS algorithm, an additional quantitative analysis was performed based on the time-domain response curves obtained in simulation. As shown in Fig. 13, although the full steady-state convergence requires approximately 1.85 s, the algorithm exhibits an extremely rapid initial reduction in residual vibration error. In particular, within the first 0.15-0.20 s, the error amplitude decreases by more than 70 %, effectively removing the dominant vibration components before entering the fine-tuning phase. This demonstrates that the ES-FXLMS algorithm achieves its essential suppression capability well ahead of full convergence.

This behavior is directly attributed to the nonlinear variable step-size mechanism, in which the exponentially decaying term provides a large initial adaptation gain, accelerating the suppression of strong vibration components, while the sinusoidal modulation further enhances responsiveness under slowly varying disturbances. The combination of these two mechanisms results in a convergence profile fundamentally different from the standard FXLMS family, where initial convergence is often much slower.

Furthermore, the early-stage suppression performance indicates that the algorithm’s practical effective convergence time is substantially shorter than the theoretical time required to reach full steady-state. Since the disturbances in space-based FDM are continuous and gradually varying rather than abruptly changing, the adaptive filter rapidly aligns itself with the disturbance characteristics during the initial phase and spends the remaining convergence period only refining the residual micro-vibrations. Such a convergence pattern ensures that the algorithm can provide reliable vibration suppression even when the layer deposition time is as short as 0.2-0.5 s.

4.4.2. Dual-frequency disturbance excitation

In the dual-frequency disturbance experiment, the No Control Algorithm was replaced with the NLMS Algorithm to assess the performance improvement offered by an adaptive Algorithm like NLMS compared to an uncontrolled system. Unlike the No Control scenario, where the system remains susceptible to disturbances without any correction, the NLMS Algorithm actively adapts to the changing disturbance conditions, resulting in better disturbance rejection and faster convergence. This makes the system more robust and capable of effectively suppressing vibration, even in complex disturbance environments, thus enhancing overall performance and stability.

To further evaluate the robustness of the proposed algorithm under such conditions and to better replicate realistic space vibration environments, a dual-tone sinusoidal excitation of 10 Hz and 25 Hz was selected. These frequencies were chosen based on modal analysis results and previously reported in-orbit vibration spectra, ensuring that both the primary structural resonance and representative higher-order disturbance components were covered. The vibration suppression performance of the three control algorithms was then systematically compared and analyzed under these excitation conditions.

Fig. 16Vibration suppression performance of three control strategies

Fig. 17Comparison of power spectral density (PSD) under noise disturbances

To provide a more scientific and comprehensive evaluation of the proposed ES-FXLMS algorithm, the NLMS algorithm was deliberately selected for comparison instead of the uncontrolled case. The rationale behind this choice is that NLMS employs a normalized step-size adjustment based on the instantaneous power of the reference signal, representing a classical and practically robust variant of the LMS family widely adopted in engineering applications. By comparing FXLMS, NLMS, and ES-FXLMS, the proposed method can be evaluated in terms of convergence rate, steady-state accuracy, and robustness under complex space disturbance conditions, thereby demonstrating its suitability for active vibration control under complex space disturbance environments.

Fig. 18MSE convergence curves of different algorithms under noise disturbances

Simulation results indicate that under dual-frequency excitation, the vibration suppression performance of all three algorithms declines compared to the single-tone or white-noise excitation cases. Specifically, under single-frequency excitation, the residual vibration amplitude can be suppressed below 0.02 mm, whereas under dual-frequency excitation, it remains below 0.05 mm. Regardless of the disturbance type, the ES-FXLMS algorithm consistently achieves the most significant reduction in vibration amplitude. Under dual-frequency excitation, the peak amplitudes of the FXLMS and NLMS algorithms are nearly identical, whereas ES-FXLMS maintains superior suppression.

A quantitative comparison of convergence time and steady-state error is summarized in Table 3. Even under complex dual-frequency disturbances, ES-FXLMS achieves faster convergence and higher steady-state accuracy than the other two algorithms, confirming its strong adaptability and robustness for active structural vibration control.

Table 4Comparative performance of control strategies under disturbance conditions

Algorithm	Total mean square error (m2)	Convergence time (s)
FXLMS algorithm	4742.35	47.2
NLMS algorithm	549.3	23.8
ES-FXLMS algorithm	69.71	10.9