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Mechanical Systems and Signal Processing
journal homepage: www.elsevier.com/locate/ymssp
Actuator saturation during active vibration control of milling
Muhammet Ozsoy a,b,∗, Neil D. Simsa, Erdem Ozturk c
aDepartment of Mechanical Engineering, The University of Sheffield, S1 3JD, Sheffield, UK
bDepartment of Mechanical Engineering, Eskisehir Technical University, 26555, Eskisehir, Turkey
cAdvanced Manufacturing Research Center, The University of Sheffield, Wallis Way, Rotherham, S1 3JD, Sheffield, UK
ARTICLE INFO
Communicated by X. Si
Keywords:
Robotic assisted milling
Active chatter control
Saturation model
Chatter stability
Inertial actuator
ABSTRACT
Machining chatter is a common problem in the manufacturing industry that can lead to reduced
productivity, poor surface quality, and accelerated tool wear. Various methods have been
proposed to suppress chatter, including passive, active, and hybrid techniques. Active control
methods, in particular, have gained increasing attention due to their potential for achieving
higher suppression effectiveness and adaptability to different machining conditions. However,
one of the main challenges of active control is the occurrence of actuator saturation, which
happens when the actuator reaches its maximum output and cannot provide any further control
action. This can lead to instability and deterioration of suppression performance. Despite its
significance, the issue of actuator saturation in machining chatter suppression has not received
much attention in the literature. Therefore, this paper aims to fill this gap by providing
a detailed investigation of the effects of actuator saturation on the performance of active
control methods for chatter suppression. The paper presents a comprehensive review of existing
literature on machining chatter suppression methods, with a specific focus on active control
techniques and their associated problems, such as saturation. An experimental scenario is
presented that illustrates the problem of actuator saturation in the context of robotically assisted
milling. The paper then proposes a novel actuator saturation model in the frequency domain
that can significantly inform the selection of cutting parameters, potentially enhancing material
removal rates and operational productivity. By addressing this research problem, this paper aims
to make a significant contribution to the field of machining chatter suppression and stimulate
further research in this direction.
1. Introduction
Regenerative chatter remains a significant impediment in machining processes, leading to detrimental effects such as self-excited
vibrations, elevated cutting forces, compromised surface quality, and accelerated tool wear [1]. Particularly impacting on thin-walled
and flexible structures due to their low dynamic stiffness, chatter necessitates robust mitigation strategies.
In pursuit of establishing precise mathematical models, extensive studies have been conducted on the chatter mechanism [2,3].
Altintas and Budak [4–7] expanded the theory of regenerative chatter by considering both the tool and workpiece as multi-degree-of-
freedom structures. Their approach involved the development of a stability model by analysing the time-periodic equation of motion
using a Fourier series expansion. This method accurately predicts the boundaries of chatter; meanwhile Insperger and Stepan [8–10]
introduced the semi-discretisation method, to efficiently account for the time period coefficients.
∗Corresponding author at: Department of Mechanical Engineering, Eskisehir Technical University, 26555, Eskisehir, Turkey.
E-mail address: muhammet.ozsoy@eskisehir.edu.tr (M. Ozsoy).
https://doi.org/10.1016/j.ymssp.2024.111942
Received 31 May 2024; Received in revised form 3 September 2024; Accepted 9 September 2024
Mechanical Systems and Signal Processing 224 (2025) 111942
Available online 11 October 2024
0888-3270/© 2024 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license
( http://creativecommons.org/licenses/by/4.0/ ).
M. Ozsoy et al.
Additional damping and structural alterations have proven effective in enhancing chatter stability [11,12], or newly designed
cutting tools to improve vibration suppression performance [13]. However, passive methods like tuned mass dampers (TMDs) require
manual intervention and lack adaptability to changing system configurations. This issue can be avoided by implementing a semi-
active chatter reduction method [14] or an active damping system [15] and also higher performance can be obtained via active
control methods.
Early research efforts in active chatter control focused on improving boring operations. Klein and Nachtigal [16,17] proposed
an active control scheme utilising an electrohydraulic servo system to enhance boring bar performance. Experimental findings
demonstrated a significant improvement in the critical limiting depth of cut. However, they noted that the performance of active
vibration control depended on understanding the principal modes and cutting force angle. Similarly, Glaser and Nachtigal [18]
investigated a special boring bar with two hydraulic chambers for actuation. However, control system limitations hindered optimal
chatter control performance.
Active regenerative chatter control techniques have also been applied in milling operations. For instance, Munoa et al. [19]
developed a biaxial active actuator, comparing one and two-axis actuators in experiments. Two-axis actuators demonstrated
enhanced machining stability, resulting in more accurate operations. Monnin et al. [20,21] introduced an active control system
integrated into a spindle unit, utilising four piezoelectric stack actuators with equal spacing. Various control methods, including
adaptive control [22], model-predictive control [23], 𝐻∞[24], and a robust control [25,26], were investigated for integrating
piezoelectric stack actuators into spindle units. Zhang and Sims [27] explored a piezoelectric patch actuator for active damping in
milling operations, reporting an increase in the critical depth of cut through positive position feedback control strategy. Using the
piezoelectric patch actuator can pose challenges regarding mounting. Additionally, in numerous tests, controller performance was
degraded by actuator saturation.
Alternative methods have been proposed to enhance chatter stability. Huyanan and Sims [28] presented three control methods,
the skyhook controller (DVF), the virtual passive absorber (VPA), and the virtual active tuned mass damper (VATMD). A proof-mass
actuator was attached to the workpiece to reduce the vibrations. The cutting tests were presented using the virtual passive absorber
method tuned by Sims method [29]. Due to the material removal effect, the dynamic properties of the structure were changed,
which caused stability degradation. To overcome this problem, automatically tuned actuators were examined. Beudaert et al. [30]
developed a portable inertial actuator capable of automatic controller parameter tuning for flexible structures, demonstrating
improved dynamic properties in experimental and industrial settings. Also, Zaeh et al. [31] introduced an automatically tuned
inertial actuator for machining vibration suppression, employing direct velocity feedback (DVF) and 𝐻∞control methods to enhance
stability. Moreover, Kleinwort et al. [32] proposed a particle swarm optimisation-based automatic tuning method for active vibration
control, comparing its performance with existing control methods in cutting tests. Validation of controller performance were carried
out for DVF, 𝐻∞and a novel adaptive FxLMS control [33]. Significantly improved chatter stability was achieved by implementing
the active control methods.
Additionally, alternative active control solutions such as the new machine tool feed drive system [34,35] and designing active
workpiece holders [36] have been explored to mitigate chatter. Active magnetic bearing (AMB) systems [37], despite promising
advantages such as increased system stiffness, face technical challenges including low specific load, bandwidth limitations, and
spindle dynamics complexity, which must be addressed for future applications.
Active damping systems have gained traction in recent years [19,38], utilising inertial actuators and control strategies such as
direct velocity feedback (DVF) [39,40]. Model-based approaches such as Linear Quadratic Regulator (LQR) and 𝐻∞control exhibit
promise, yet necessitate prior knowledge of structural dynamics for effective parameter tuning [31,41]. The 𝜇synthesis method
offers advantages in stability and robustness, particularly in the face of uncertainties [42,43].
Recently, Ozsoy et al. [44] proposed a virtual inerter based dynamic vibration absorber (IDVA) by utilising an inertial actuator,
building upon the passive chatter suppression method developed by Dogan et al. [45]. The numerical analysis revealed that an IDVA
could enhance absolute chatter stability by over 20% compared to the effectiveness of a TMD. However, this improvement was
constrained by the maximum actuator force utilised in the experimental setup. Subsequently, the numerical findings were validated
through cutting trials [46]. They [47] also analysed the uncertainty and robustness of IDVA considering various layouts for the
active control systems. The potential benefits and limitations of IDVAs, considering actuator saturation and parameter uncertainty,
were demonstrated.
More broadly, robotic machining and robotic-assisted machining has earned attention, due to their potential to offer more
versatility and to increase productivity [48]. Robotic-assisted machining, as explored by [49], demonstrates promise in improving
form accuracy by utilising robotically-supported workpieces. This is particularly relevant when the workpiece being machined is
flexible or difficult to clamp using traditional fixture designs.
Ozsoy et al. [50] proposed a novel mobile active vibration control method employing a robotic arm specifically adapted for
milling operations. Later work [51] extended this to explore the experimental performance of a basic prototype from a dynamics
perspective. By combining automation capabilities with vibration control, the approach aimed to overcome the challenges faced by
active vibration control methods in practical production environments. They defined a simplified dynamic scenario, and introduced
time-domain and frequency-domain models of the system for numerical study. Several control systems were designed and their
efficiency in chatter stability prediction was evaluated through simulations validated by experimental frequency response function
(FRF) tests. However, this previous work did not demonstrate the performance of the approach in a machining environment.
Based upon this literature, it is clear that active vibration control offers the potential to greatly improve machining productivity
through the reduction of regenerative chatter. However, a key obstacle is the onset of actuator saturation effects, which have been
shown to exist for a wide range of control strategies and implementation scenarios. Meanwhile, the practical implementation of
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Fig. 1. Self-excited vibrations in milling.
active vibration control could be eased by combining the method with robotic and automation approaches, but this approach has
not yet been tested in a machining environment.
The present contribution aims to tackle these two shortfalls. First, machining experiments are presented to verify the robotic-
assisted concept in real cutting conditions. These experiments also serve to illustrate the existence of actuator saturation issues
when considering the complex forced vibrations that occur during milling. Next, a novel actuator saturation model is developed to
predict the interdependency between saturation effects and machining process parameters. Experimental validation demonstrates
the efficacy of the saturation model in a machining context. The effect of different machining parameters and active vibration control
strategies are then discussed, before drawing conclusions.
2. Background theory
To begin, it is helpful to briefly summarise regenerative chatter mechanisms and how such chatter is mitigated by active
vibration control. Extensive and accurate models exist elsewhere in the literature [52,53], so this section provides only a simplified
formulation.
Consider the rigid cutting tool and flexible workpiece, as illustrated in Fig. 1; here (to simplify and maintain consistency with
subsequent analyses) a single coordinate 𝑥is employed. The interaction involves the rigid cutting tool inducing waviness on the
flexible workpiece’s surface with each cut. The phase relationship between successive waviness imprints results in variations in
the instantaneous chip thickness, denoted as ℎ(𝑡). These variations lead to instability in the cutting process, triggering exponential
growth in vibrations and cutting forces.
Considering the cutting force’s proportionality to the removal chip area, and simplifying the time-periodic coefficients by using
the first term of a Fourier series expansion [5], the predicted stability limit in a single degree-of-freedom (SDOF) system becomes:
𝑎𝑐𝑟 =−1
𝑁𝑡
2𝜋𝑏𝑥𝑥𝐾𝑐Re(𝐺(𝑗 𝜔𝑐))
.(1)
Here, 𝑎𝑐𝑟 is the critical limiting depth of cut above which chatter occurs. Meanwhile, 𝐾𝑐is cutting stiffness, 𝑁𝑡is the number
of flutes, and 𝑏𝑥𝑥 is a directional coefficient which is related to the cutting conditions and arises from the time-averaging that is
introduced by the Fourier series expansion and truncation [5].
Tobias and Fishwick [2] explained the relationship between the critical limiting depth of cut and the time delay inherent in
the machining process. This relationship gave rise to stability pockets, known as the stability lobe diagram (SLD). The diagram
illustrates the stability boundary as a function of spindle speed and axial depth of cut. Additionally, Merritt [54] represented the
chatter theory as a feedback loop system, neglecting the process damping phenomena introduced by Tobias at lower spindle speeds,
and achieved stability using the Nyquist stability criterion.
From the perspective of active vibration control, the key term in (1) is Re(𝐺(𝑗𝜔𝑐)). Here, 𝜔𝑐is the chatter frequency, and 𝐺
is the frequency response function (FRF) of the flexible structure. Therefore, chatter stability is directly related to the negative
real component of the FRF. Many active vibration control approaches can directly modify this value [55], by drawing upon
classical and optimal control strategies such as Direct Velocity Feedback, Linear Quadratic Regulator, and 𝐻∞control. Alternatively,
bespoke mechanisms of feedback control can be employed, such as delayed feedback [2,54] that accounts for the special nature of
regenerative chatter as a time-delayed differential equation.
3. Robotic assisted milling: experimental proof of concept
This section extends earlier work [51] by experimentally demonstrating how active vibration control could be applied as part
of a robotic assisted milling operation, and to illustrate the impact of actuator saturation effects.
Mechanical Systems and Signal Processing 224 (2025) 111942
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Fig. 2. Conceptual illustration of robotic assisted vibration control during milling. (a) side view; (b) top view, showing the cutting force 𝑓𝑐, total support force
𝑓𝑡, total proof-mass actuator force 𝑓𝑎𝑐𝑡 and robot support force 𝑓𝑟.
Fig. 3. Experimental setup for the milling tests.
3.1. Experimental configuration and method
The ultimate application of robotic assisted milling with active vibration control is illustrated schematically in Fig. 2. Employing a
six-axis industrial robot, an end effector is positioned against the machining workpiece, which remains secured through conventional
fixturing techniques. The articulated robot is strategically manoeuvred to establish a beneficial contact pressure between the end
effector and the workpiece. In contrast to previous approaches [49] the primary function of the end effector is to apply dynamic
forces to the workpiece through a proof-mass (or inertial) actuator. This intentional imposition of forces aims to enhance chatter
stability during the machining process.
Fig. 2b illustrates the forces at play at the interface among the robot, workpiece, and tool. Notably, the force exerted by the
proof-mass actuator 𝑓𝑎𝑐𝑡 influences the overall support force 𝑓𝑡. However, it is crucial to acknowledge that this support force
remains exclusively positive due to the absence of rigid attachment between the robot and the workpiece through the robot-
workpiece interface. Additionally, the support force undergoes influence from forces transmitted through the robot’s structure itself,
necessitating consideration of both the robot’s structural dynamics and those of the workpiece in the design of the inertial actuator’s
control system.
In order to assess the efficacy of this concept in a controlled laboratory environment, previous work has developed a prototype
configuration where the robotic manipulator is represented by a simplified structure. The present section validates the findings of
that work in a machining environment, using the experimental setup depicted in Fig. 3.
Details concerning the control system design and the control system parameter optimisation are provided in [51] and the relevant
system parameters are summarised in Table 1, along with the full parameters of the system.
The transfer function of the actuator can be defined as:
𝑓𝑎𝑐𝑡 (𝑠)
𝑉𝑖𝑛(𝑠)= 3 𝑠2
𝑠2+ 15.834𝑠+ 2785.6(2)
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Table 1
Tuning parameters for machining experiments.
Control method Controller parameters
Direct Velocity Feedback (DVF) 𝑔𝑑𝑣𝑓 = 253
Virtual Passive Absorber (VPA) 𝜇𝑐= 0.0061
Proportional Integrated Derivative (PID) 𝑔𝑎= 9.97×10−4,𝑔𝑣= 256,𝑔𝑝= 2.88 × 104
Linear Quadratic Regulator (LQR) 𝑄= 8262.4,𝑅= 0.028
𝐻Infinity (𝐻∞)𝐺𝑠𝑡 = 9.92 × 105,𝑓1= 59.12,𝑓2= 9.95
𝜇Synthesis 𝐺𝑠𝑡 = 2.2 × 107,𝑓1= 4553.84,𝑓2= 4.29
Table 2
Structural, machining and actuator parameters.
Preloaded structural parameters Machining parameters
Natural frequency 129.3 Hz Tool diameter 16 mm
Damping ratio 1.34% Number of teeth 4
Stiffness 1.34×107N m−1 Tool helix angle 45◦
Flexible Robot Parameters Material Al-7075-T6
Natural frequency 23 Hz, 47 Hz Cutting stiffness 𝐾𝑟180×106N m−2
Damping ratio 4.3%, 2.9% Cutting stiffness 𝐾𝑡660×106N m−2
Stiffness 0.79×106N m−1, 2×106N m−1 Milling type Down milling
Actuator Parameters Radial depth of cut Half immersion
Natural frequency 8.4 Hz Feed per tooth 0.05 mm
Damping ratio 0.15
The experimental setup involved securing the flexible structure onto the CNC table, illustrated in Fig. 3. An Aluminium alloy block
(Al 7075-T6, dimensions 100 ×100x300 mm) was used as a workpiece, and the clamping method of the workpiece and the structure
of the fixture involve a rigid connection using bolts on a steel box section so as to achieve repeatable dynamic characteristics that
could be represented as a single-degree-of-freedom system.
The proof mass actuator was clamped to a steel beam-like structure, which was shaped such that it possessed a low number of
dominant modes of vibration and was representative of a serial arm robot. The beam and actuator structure was pushed against the
workpiece to ensure positive contact, representing the robot-workpiece interface depicted in Fig. 1.
During milling, measurements of the workpiece and beam accelerations were obtained using an accelerometer (model PCB
353B18). To monitor the spindle speed, a hall-effect sensor was strategically positioned near the tool holder. This sensor detected
one revolution of the spindle by capturing changes in voltage caused by the two slots on the tool holder as the spindle rotated. Data
acquisition from these experiments was facilitated by a data logger (model NI DAQ USB-4431), enabling comprehensive recording
and analysis of experimental parameters and responses.
The milling parameters and tool properties are summarised in Table 2. The cutting operations were conducted on the side
opposite to the location of the beam. For the cutting operations, a cutting length of 100 mm was used along with an 8 mm (half)
radial immersion for the specified axial depth of cuts. Additionally, the feed direction of the cutting tool was oriented out of the
page.
3.1.1. Chatter detection methods
The detection of chatter during machining operations often relies on well-established methods such as once-per-revolution
sampling and Fast Fourier Transform (FFT) spectrum analysis, as highlighted in studies like Schmitz et al. [56,57]. These methods
capitalise on the periodicity observed in signals during machining, particularly at the tooth passing frequency (𝑓𝑡𝑝 =𝑁𝑡𝑁
60 , where 𝑁
is the spindle speed in rpm and 𝑁𝑡is the total number of tool teeth) and their harmonics, especially when non-zero runout occurs.
Runout during cutting operations introduces the spindle rotation frequency (𝑓𝑠=𝑁
60 ) into the FFT spectrum [58]. Frequencies other
than these fundamental components in the FFT spectrum often indicate the presence of chatter [59], making FFT analysis a valuable
tool for chatter detection.
Another method, once-per-revolution sampling, involves synchronised data collection during cutting operations [57,60]. Davies
et al. [60] presented the Poincaré plot, depicting the tool motion in both directions for each revolution. In stable cutting, this plot
should demonstrate consistent tool positions across each revolution. However, in the case of chatter, variations in tool position
occur from one revolution to another. This disparity is evident in the plot, indicating chatter occurrence. Alternatively, plotting
the tool positions in both directions against each other, known as the Poincaré map, can reveal differences. A stable cut exhibits a
concentrated cluster of data points, while chatter leads to a dispersed cluster.
In this study, both the once-per-revolution sampling and FFT spectrum methods are employed to detect chatter phenomena. After
each cutting operation, these techniques are utilised using acceleration data from the flexible structure. The once-per-revolution
sampling method is employed to create Poincaré maps and time-domain plots for each cut, enabling the observation and analysis
of tool motion patterns, aiding in the identification of chatter occurrences.
Mechanical Systems and Signal Processing 224 (2025) 111942
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3.2. Machining results
The results obtained from one typical controller – Direct Velocity Feedback control – are first presented in detail, before
summarising the performance of the other controllers.
To begin, Fig. 4 provides a visual illustration of the controller performance, by illustrating the surface finish. In the presence of
active control (control on), no chatter marks are evident, indicative of a stable cut. However, upon deactivating the control (control
off), chatter marks become visible on the cut surface. Fig. 5 presents once-per-revolution samples, Poincaré plot, and FFT spectrum
for the corresponding machining conditions. With control, the Poincaré plot shows concentrated data points that are indicative of
stable machining. When the controller is switched off, then chatter ensues and this is indicated by the changes in the FFT and the
Poincaré plot.
Fig. 6 shows the predicted stability of the system, both with (solid line) and without (dashed line) control. It can be seen that the
addition of the DVF controller substantially increases the predicted chatter stability, in line with previously published work [51]. The
stability lobe diagram can be predicted using various methods [61,62]. In the present study, stability was predicted using the method
developed by Budak and Altintas [4]. While the specifics of this approach are beyond the scope of the current contribution, the theory
and methodology are widely reported elsewhere [4,52]. The controlled SLD is obtained using the FRF results after implementing
the control method. The present contribution provides rigorous experimental validation of this result through machining trials, as
indicated by the markers on Fig. 6. In general, there is reasonably accurate agreement between the experimental data points and
the stability predictions. In fact, the results presented in 4and 5are one of the unexpected scenarios where the controller was
more stable then expected. This could be attributed to structural variations in clamping conditions, leading to small changes in the
stability boundaries.
Results at 2800 rpm are presented in Fig. 7 for the scenarios labelled A–D on Fig. 6. Scenarios A (2800 rpm, 0.5 mm depth of cut
(doc)) and Point B (2800 rpm, 1.5 mm doc) represent uncontrolled machining. In Scenario A, the Poincaré plot shows concentrated
data points that are indicative of stable machining. In Scenario B, an elliptical pattern in the Poincaré plot and the presence of an
additional vibration frequency, alongside tooth-pass and run-out frequencies in the FFT spectrum, confirmed an unstable (chatter)
cut. These experimental observations align closely with predictions, considering a critical limiting depth of cut of 1.2 mm for the
uncontrolled flexible structure.
In the controlled scenario, stability was predicted at Scenario C (2800 rpm, 2.5 mm doc) in accordance with the controlled
SLD. However, unexpected stability was observed even at increased depths of cut (Scenario D). Here, the Poincaré plot and the FFT
spectrum indicate broadband vibrations in addition to the expected quasi-periodic motion; this was subsequently attributed to the
loss-of-contact between the beam-actuator assembly (representing the robotically assisted system) and the workpiece. This result
highlights a potential challenge with the implementation of robotic-assisted active vibration control, in that the pre-load applied
between the robot and the workpiece must be sufficient to avoid such nonlinear behaviour.
Fig. 8 provides further example data sets under DVF control, for Scenarios E-H on Fig. 6. Scenario E shows an unstable case
that matches the predicted stability boundary, and Scenario F is stable as predicted. However, at Scenario G (2100 rpm, 5 mm doc)
represents a marginally stable case, where the onset of chatter becomes evident in the FFT spectrum. Finally, Scenario H shows an
unstable result, which is particularly important for the present study. Here, the irregularity in the FFT and in the Poincaré section
were attributed to actuator saturation effects, with the actuator control signal exceeding the linear range for the proof-mass-actuator
and amplifier components.
The experimental outcomes for six control methods are depicted in Fig. 9. Notably, a consistent actuator loss-of-contact issue
was observed across all control methods at a spindle speed of 2800 rpm. This observation suggests heightened forced vibrations
and structural nonlinearity at this particular spindle speed. Remarkably, the milling experiments imply that the loss-of-contact
issue transcends the variation in controller types, highlighting its correlation with the dynamic characteristics of the structure and
the specific cutting parameters employed. Therefore, further work may be needed to fully understand the onset of this actuator
loss-of-contact phenomenon.
However alongside this, the issue of actuator saturation requires further investigation. To further explore this, additional
machining experiments were conducted for DVF control in the region 1700–2250 rpm. It should be noted that the preloaded
structural parameters are slightly different than the previous application due to the clamping conditions. The control gains are
optimised considering this modification for the saturation predictions. The flexible workpiece, represented by a flexure and an Al
7075-T6 block, exhibits distinct dynamic properties with a natural frequency of 128.1 Hz, a damping ratio of 1.48%, and a stiffness
value of 1.13 ×107N m−1. The results are summarised in Fig. 10. These actuator forces are plotted against the chosen cutting
parameters on the SLD, with the saturation islands encircled with red dashed lines. Remarkably, the experimental actuator force
data aligns closely with the predicted location of saturation islands, confirming the accuracy of the predicted saturation regions.
Here, numerical values indicate the amplitude of the control (command or reference) signal below saturation which is 27 N. It can
be seen that the saturation effect appears to exist in isolated islands of the stability diagram and that in some cases the nonlinearity
that this introduces means that chatter occurs below the predicted stability boundary. Therefore, this demonstrates that for active
vibration control implementations, the issue of actuator saturation must be carefully considered.
4. A new actuator saturation model
This section introduces a new actuator saturation model in the frequency-domain, drawing upon Schmitz’s surface location error
model [53,63].
Mechanical Systems and Signal Processing 224 (2025) 111942
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Fig. 4. The chatter marks when the control is on and off with the parameters 2900 rpm, 5 mm doc for DVF controlled structure.
Fig. 5. The chatter marks when the control is on (a) and off (b) with the parameters 2900 rpm, 5 mm doc for DVF controlled structure. Spindle frequency
(runout) ( ), tooth passing frequency (■), chatter frequency (⧫).
4.1. Frequency-domain solution
The methodology introduced by Budak and Altintas [5,7] for SLD prediction was discussed in the previous sections. Their ap-
proach involved an analysis of the milling process, expanding the time-varying force coefficients into Fourier series representations.
Building upon this framework, this section introduces an actuator force model in the frequency domain, applying Fourier series
expansions. The objective of this model is to elucidate the impact and relevance of actuator saturation islands in active vibration
control during milling.
To predict these actuator saturation islands using a frequency-domain model, two fundamental assumptions have been consid-
ered. Firstly, the actuator force exclusively operates along the 𝑦axis while the cutting feed direction remains along the 𝑥axis.
Secondly, for the scope of stable machining scenarios, any regeneration effect has been omitted. These assumptions serve as the
foundation for the subsequent procedural steps:
1. Compute the cutting force in the 𝑦direction within the frequency domain, denoted as 𝐹𝑦(𝑤), by employing a Fourier series
representation.
2. Determine the displacement 𝑌(𝑤)along the 𝑦axis within the frequency domain. Achieve this by multiplying 𝐹𝑦(𝑤)by the
FRF of the structure in the 𝑦direction.
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Fig. 6. Stability lobe diagram with experimental results for DVF control method. DVF controlled SLD ( ), uncontrolled SLD ( ), uncontrolled stable
cut (+), uncontrolled chatter (◊), controlled stable cut (◦), controlled marginal cut (⊳), controlled chatter (×), controlled loss-of-contact (⋆), onset of actuator
saturation in time-domain simulation (□).
3. Employ an inverse Fourier transform on 𝑌(𝑤)to transform it back to the time-domain, resulting in the vibration 𝑦(𝑡).
4. Derive the voltage value employed by the control method utilising the vibration signal 𝑦(𝑡). For example, compute the
structure’s velocity using 𝑦(𝑡), then multiply this velocity by the corresponding gain factor, such as 𝑔𝑑𝑣𝑓 , to obtain the voltage
for DVF control.
5. Calculate the actuator force using the determined voltage value.
4.1.1. Fourier force model
The tangential 𝐹𝑡and normal 𝐹𝑛cutting forces are described as:
𝐹𝑡(𝜃) =𝐾𝑡𝑏ℎ(𝜃) + 𝐾𝑡𝑒𝑏
𝐹𝑛(𝜃) =𝐾𝑛𝑏ℎ(𝜃) + 𝐾𝑛𝑒𝑏(3)
where 𝐾𝑡,𝐾𝑡𝑒,𝐾𝑛, and 𝐾𝑛𝑒 are the tangential, tangential edge, normal, and normal edge cutting coefficients, respectively. 𝑏and ℎ
denote the axial depth of cut and chip thickness.
The cutting force in 𝑦direction, 𝐹𝑦(𝜃)are described as:
𝐹𝑦(𝜃) = −𝑏−𝐾𝑡𝑓𝑡
2
𝑁𝑡
𝑗=1
𝑔(𝜃𝑗)(1 − cos(2𝜃𝑗)) + 𝐾𝑛𝑓𝑡
2
𝑁𝑡
𝑗=1
𝑔(𝜃𝑗)(sin(2𝜃𝑗))
−𝐾𝑡𝑒
𝑁𝑡
𝑗=1
𝑔(𝜃𝑗) sin(𝜃𝑗) + 𝐾𝑛𝑒
𝑁𝑡
𝑗=1
𝑔(𝜃𝑗) cos(𝜃𝑗)(4)
where 𝑓𝑡and 𝑁𝑡are the feed per tooth and total number of teeth, respectively. 𝑔(𝜃𝑗)is the switching function which is defined in
Eq. (5). The angle of each tooth, 𝑗, at any instant in time is 𝜃𝑗=𝜔𝑡 +2𝜋
𝑁𝑡
(𝑗− 1), (rad), where 𝜔is the spindle rotation frequency
(rad/s).
𝑔(𝜃𝑗) = 1, 𝜃𝑠≤𝜃𝑗< 𝜃𝑒
0, 𝜃𝑗< 𝜃𝑠, 𝜃𝑗> 𝜃𝑒
(5)
where 𝜃𝑠and 𝜃𝑒are the start and exit tool immersion angles, respectively.
The cutting force in 𝑦direction, denoted as 𝐹𝑦, is expressed using the equivalent Fourier series once the Fourier coefficients, 𝑎𝑛
and 𝑏𝑛, are computed.
𝐹𝑦(𝜃) =
𝑁𝑡
𝑗=1𝑎0+
∞
𝑗=1
(𝑎𝑛cos(𝑛𝜃𝑗) + 𝑏𝑛sin(𝑛𝜃𝑗))(6)
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Fig. 7. Once per revolution samples, Poincaré map and the FFT spectrum for (a) stable cut A (2800 rpm, 0.5 mm), (b) chatter cut B (2800 rpm, 1.5 mm), (c)
stable cut C (2800 rpm, 2.5 mm), (d) loss-of-contact cut D (2800 rpm, 5.5 mm). Spindle frequency (runout) ( ), tooth passing frequency (■), chatter frequency
(⧫).
The 𝑎0term can be defined utilising Eq. (7), where the full revolution of the cutting tool is divided into three parts. 𝜃1denotes
the entry angle in down milling or exit angle in up milling.
𝑎0=1
2𝜋∫2𝜋
0
𝐹𝑦(𝜃)𝑑𝜃 =1
2𝜋∫𝜃1
0
𝐹𝑦(𝜃)𝑑𝜃 +∫𝜋
𝜃1
𝐹𝑦(𝜃)𝑑𝜃 +∫2𝜋
𝜋
𝐹𝑦(𝜃)𝑑𝜃(7)
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Fig. 8. Once per revolution samples, Poincaré map and the FFT spectrum for (a) chatter cut E (2700 rpm, 4 mm), (b) stable cut F (2000 rpm, 5 mm), (c)
marginal cut G (2100 rpm, 5 mm), (d) chatter cut H (2100 rpm, 6 mm). Spindle frequency (runout) ( ), tooth passing frequency (■), chatter frequency (⧫).
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Fig. 9. Stability lobe diagrams with experimental results for (a) DVF control, (b) VPA control, (c) PID control, (d) LQR control, (e) H∞control, (f) 𝜇-synthesis
control. Controlled SLD ( ), uncontrolled SLD ( ), controlled stable cut (◦), controlled marginal cut (⊳), controlled chatter (𝑥), controlled loss-of-contact
(⋆), onset of actuator saturation in time-domain simulation (□).
Fig. 10. The predicted saturation islands (encircled with red dashed lines) and experimental actuator forces for DVF controlled system. Controlled SLD ( ),
uncontrolled SLD ( ).
If a down milling operation is considered, only the middle integral term is nonzero in Eq. (7) according to the tool engagement,
and after the integration process is performed, Eq. (8) is obtained. Otherwise, in up milling, only the first integral term is nonzero,
and the integration limits become zero to 𝜃1in Eq. (8).
In the case of a down milling operation, considering the tool engagement, only the middle integral term is non-zero in Eq. (7).
Upon integration, this leads to the expression given in Eq. (8). Conversely, for up milling, solely the first integral term is non-zero,
and the integration limits are adjusted from zero to 𝜃1in Eq. (8) (see Fig. 11).
𝑎0= −𝑏𝑁𝑡
2𝜋−𝐾𝑡𝑓𝑡𝜃
2+𝐾𝑡𝑓𝑡
4sin(2𝜃) − 𝐾𝑛𝑓𝑡
4cos(2𝜃) + 𝐾𝑡𝑒 cos(𝜃) + 𝐾𝑛𝑒 sin(𝜃)𝜋
𝜃1
(8)
The coefficients 𝑎𝑛and 𝑏𝑛can be defined as:
𝑎𝑛=1
𝜋∫2𝜋
0
𝐹𝑦(𝜃) cos(𝑛𝜃)𝑑𝜃 (9)
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Fig. 11. Angles for the integrals considering the milling type.
𝑏𝑛=1
𝜋∫2𝜋
0
𝐹𝑦(𝜃) sin(𝑛𝜃)𝑑𝜃 (10)
The integrals, similar to Eq. (7), can be segmented for the 𝜃and 𝜋angles. Following the same recursive patterns of the integration,
the equations can be calculated for 𝑛= 3,4,5,…. The Fourier series coefficients 𝑎1, 𝑎2, 𝑎3,…and 𝑏1, 𝑏2, 𝑏3,…are given in Appendix.
To accommodate the helix angle 𝛾in depicting the milling forces, the tool can be divided into 𝐴axial segments. Each segment
is considered to possess a zero helix angle and is rotated relative to another segment by an angle 𝜒=2𝑑𝑏 t an(𝛾)
𝑑, (rad), where 𝑑𝑏 and
𝑑are the segment height and the tool diameter, respectively. Subsequently, the Fourier series can be reformulated as follows:
𝐹𝑦(𝜃) =
𝐴
𝑖=1
𝑁𝑡
𝑗=1𝑎0+
∞
𝑛=1
(𝑎𝑛cos(𝑛𝜃𝑗) + 𝑏𝑛sin(𝑛𝜃𝑗))(11)
where 𝜃𝑗=𝜔𝑡 +2𝜋
𝑁𝑡
(𝑗− 1) − 𝜒(𝑖− 1). Increased segmentation into a larger number of slices offers a more precise determination of
the milling force.
Once the cutting force is calculated, the inverse Fourier transform is applied to 𝑌(𝑤)to obtain the vibration of the structure 𝑦(𝑡).
In order to implement DVF control, the velocity of the structure is derived from the vibration of the structure 𝑦(𝑡). The corresponding
voltage value for the controller is then calculated by multiplying this velocity by the gain factor 𝑔𝑑𝑣𝑓 . Finally, the actuator force is
determined using the calculated voltage value, employing the transfer function of the actuator as described in Eq. (2).
4.2. Verification of actuator saturation islands
This subsection aims to experimentally validate the proposed actuator saturation model. In Fig. 12, the predicted actuator
saturation is compared to the experimental data that was previously presented in Fig. 10. It can be seen that there is good agreement
between the model and experimental data. In particular, a saturation-free region was observed between the predicted saturation
islands, confirming the accuracy of the predictive models. Moreover, the second saturation island, projected to occur at higher depths
of cut (around 20 mm), was validated by the experimental results, aligning closely with predictions. However, in the region between
two SLDs, instances of chatter and marginal cuts with actuator saturation were observed. These arose due to the chosen depth of cuts
over the predicted saturation points, exceeding the actuator’s capacity to deliver sufficient force. Some disparity emerged between
the frequency-domain predictions and experimental results occurred, particularly in the 2050–2200 rpm range. This could be related
to actuator loss of contact effects or other nonlinearities.
The comparison between predicted saturation islands in the frequency domain and the experimental outcomes reveals a
significant alignment, validating the reliability and accuracy of the frequency-domain predictions.
5. Discussion
Within this section, an analysis is conducted on the actuator saturation model alongside the practical implementation of the
concept of robotic-assisted milling.
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Fig. 12. The predicted saturation islands and experimental actuator forces for DVF controlled system. Controlled SLD ( ), uncontrolled SLD ( ).
Fig. 13. Actuator saturation islands in frequency-domain for each control method. (a) DVF control, (b) VPA control, (c) PID control, (d) LQR control, (e) H∞
control, (f) 𝜇-synthesis control.
5.1. Saturation for different control strategies
The detailed procedure in Section 4.1 is employed to identify actuator saturation islands across various controller types. Given
the observed actuator saturation occurrence across all control methods near the spindle speed associated with the highest depth of
cut, an assessment is conducted within the spindle speed range of 1700 to 2200 rpm. This assessment entails scanning the spindle
speed range at 50 rpm intervals, while the axial depth of cut is incremented in intervals of 0.5 mm. This investigation spans from
zero to the stability boundary.
The visual representation of saturation islands in the frequency-domain for each control method is presented in Fig. 13. It can
be seen that the saturation phenomena is not strongly impacted by the choice of controller. Furthermore, in terms of computational
efficiency, the frequency domain method notably outperforms the traditional time domain approaches. Specifically, using 5 and 10
terms of the Fourier force model in the frequency domain method reduces computation time considerably. However, it is worth
noting that employing higher terms in the Fourier force model increases the computation time for the frequency domain method.
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Fig. 14. Surface location errors in freq-domain for each control method. (a) DVF control, (b) VPA control, (c) PID control, (d) LQR control, (e) H∞control, (f)
𝜇-synthesis control.
5.2. Relation to surface location error
Fig. 14 displays surface location errors, following the work of [53]. Interestingly, these errors do not mirror the saturation
island pattern. They are distinct since surface location errors are determined by tool vibrations at an exact moment, while actuator
saturation islands are specifically linked to the vibration difference over time. This comprehension of saturation islands can
significantly inform the selection of cutting parameters within the surface location error figures, potentially enhancing material
removal rates and operational productivity.
5.3. Influence of helix angle
The influence of the helix angle on the cutting force profile is assessed by examining zero and 45-degree helix angles concerning
the DVF control method. The comparison of saturation islands is conducted utilising identical control parameters.
Actuator saturation islands in the frequency-domain are depicted in Figs. 15 for the DVF controlled structure using 45-degree and
zero helix angles, respectively. The helix angle significantly influences the number of saturation islands, their respective locations,
and actuator forces by affecting the delay in the cutting process. When employing the tool with a 45-degree helix angle, two
saturation islands are observed. This is attributed to the helix angle’s ability to reduce forced vibrations in the controlled direction,
impacting the cutting process delay. In contrast, despite the same number of islands occurring with 30 and 45-degree helix angles,
their positions differ.
Conversely, using a tool with a zero helix angle leads to higher forced vibrations in the controlled direction, resulting in a single
large saturation island. In this scenario, once saturation begins, it persists until the stability boundary without any intervening
saturation-free zones. This observation underlines how varying helix angles influence forced vibrations and subsequent actuator
saturation islands during machining.
5.4. Robotic assisted active vibration control milling
The previous discussion has focussed on the broader issue of how actuator saturation can be predicted for active vibration control
in milling. However, the scenario presented in the present study has also explored the suitability of this in a robotic-assisted milling
scenario. Whilst the proof-of-concept has been experimentally demonstrated in Section 3, some other issues merit further work:
Material removal during cutting might influence the dynamic response of the structure. Consequently, retuning of controller
parameters may be needed to ensure optimised controller performance. The incorporation of 𝜇synthesis control, accounting for
associated uncertainties, presents a promising area for the robotic-assisted milling concept. In the present study the change in the
workpiece dynamics (due to material removal) was negligible.
Potential loss-of-contact issues between the robot and the structure have been observed in the experiments. This requires further
investigation, for example by developing analytical predictions and using them to avoid the loss-of-contact occurring. The loss-of-
contact could also be addressed by adjusting the robot’s position to ensure continuous contact during operation. This can be achieved
by maintaining a static preload force higher than the actuator’s control force. However, force vibrations or structural nonlinearities
could still contribute to loss-of-contact occurrences, potentially preventing the control system’s performance.
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Fig. 15. Actuator saturation islands in frequency-domain, DVF controlled, (a) zero helix angle, (b) 45 helix angle.
Different forms of contact between the robot and the workpiece represent an alternative approach for robotic-assisted milling.
While this study utilised rigid metallic contact, employing rolling contact using soft or hard rubber could yield varied system
responses due to altered actuator force transmission mechanisms.
Additional modes may emerge during milling, potentially requiring further optimisation of the actuator to achieve optimal
performance. Simulation results indicate that the actuator effectively managed multiple vibration modes, including bending and
torsion, demonstrating its versatility and reliability when tuned to address these modes.
When chatter occurs during the machining process, selecting appropriate parameters according to the proposed method involves
understanding and using saturation islands. By identifying these saturation-free zones, particularly between uncontrolled and
controlled stability lobe diagrams (SLDs), optimal cutting parameters can be chosen. This approach helps to:
•Avoid Saturation Zones: Select cutting parameters that fall within regions identified as free from saturation to maintain effective
actuator performance.
•Enhance Stability: Choose parameters that align with stable regions of the SLDs, improving overall stability and reducing the
likelihood of chatter.
•Increase Productivity: Optimise material removal rates and operational productivity by avoiding conditions that lead to
actuator saturation and unstable machining.
6. Conclusion
The conclusions and main contributions from this research that can be drawn are:
•Machining experiments are demonstrated to verify the robotic-assisted milling concept using various control methods (DVF,
VPA, PID, LQG, 𝐻∞, and 𝜇synthesis).
•The critical limiting depth of cut (𝑏𝑐𝑟𝑖𝑡 ) demonstrates a significant enhancement, increasing by a factor of 2.9 compared to
systems without active control and by 4.4 compared to systems lacking robotic-assisted support.
Mechanical Systems and Signal Processing 224 (2025) 111942
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M. Ozsoy et al.
•The practical effectiveness of each controller is assessed via a simplified model and validation experiments, demonstrating
strong alignment between predicted stability boundaries and experimental observations.
•A newly proposed predictive model for actuator saturation, based on frequency-domain analysis, is introduced. This model
provides insights into how actuator performance can be optimised across different control methods.
•The actuator saturation model with stability lobe diagrams are integrated to identify and avoid saturation-free zones. This
integration enhances the accuracy of parameter selection and improves machining stability.
•The predicted saturation islands in the frequency model match with the experimental actuator forces, which demonstrates the
model accuracy.
•Additionally, the influence of the helix angle on the location of saturation islands and actuator force is explored, highlighting
the impact of cutting tool characteristics on the milling process.
In conclusion, the presented approach demonstrates feasibility in enhancing milling stability via robotic-assisted milling concepts.
The experimental verification of a novel actuator saturation model has been conducted. It is shown that practical considerations of
actuator saturation, using this model, could offer deeper insights into the performance of active control.
CRediT authorship contribution statement
Muhammet Ozsoy: Writing – original draft, Validation, Software, Methodology, Investigation, Formal analysis, Conceptualiza-
tion. Neil D. Sims: Writing – review & editing, Supervision. Erdem Ozturk: Writing – review & editing, Supervision.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared
to influence the work reported in this paper.
Data availability
Data will be made available on request.
Acknowledgements
The first author would like to acknowledge the support from the Turkish Ministry of National Education by providing the
scholarship.
Appendix
The Fourier series coefficients
The Fourier coefficients, 𝑎𝑛and 𝑏𝑛, for 𝑦direction force series:
𝐹𝑦(𝜃) =
𝐴
𝑖=1
𝑁𝑡
𝑗=1𝑎0+
∞
𝑛=1
(𝑎𝑛cos(𝑛𝜃𝑗) + 𝑏𝑛sin(𝑛𝜃𝑗))(12)
where 𝜃𝑗=𝜔𝑡 +2𝜋
𝑁𝑡
(𝑗− 1) − 𝜒(𝑖− 1). The force can be accurately determined by the higher number of slices.
The terms in Eqs. (13) through (18) are determined using Eqs. (9) and (10). Considering the down milling operation, the
integration limits become from 𝜃1to 𝜋, otherwise zero to 𝜃1.
𝑎1= − 𝑏𝑁𝑡
𝜋𝐾𝑡𝑓𝑡−1
4sin(𝜃) + 1
12 sin(3𝜃)+𝐾𝑛𝑓𝑡−1
4cos(𝜃) − 1
12 cos(3𝜃)
+𝐾𝑡𝑒1
4cos(2𝜃)+𝐾𝑛𝑒1
2(𝜃) + 1
4sin(2𝜃)𝜋
𝜃1
(13)
𝑎1= − 𝑏𝑁𝑡
𝜋𝐾𝑡𝑓𝑡1
4(𝜃) − 1
4sin(2𝜃) + 1
16 sin(4𝜃)+𝐾𝑛𝑓𝑡−1
16 cos(4𝜃)
+𝐾𝑡𝑒−1
2cos(𝜃) + 1
6cos(3𝜃)+𝐾𝑛𝑒1
2sin(𝜃) + 1
6sin(3𝜃)𝜋
𝜃1
(14)
𝑎𝑛= − 𝑏𝑁𝑡
𝜋𝐾𝑡𝑓𝑡−1
2𝑛sin(𝑛𝜃) + 1
4(𝑛− 2) sin((𝑛− 2)𝜃) + 1
4(𝑛+ 2) sin((𝑛+ 2)𝜃)
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M. Ozsoy et al.
+𝐾𝑛𝑓𝑡1
4(𝑛− 2) cos((𝑛− 2)𝜃) − 1
4(𝑛+ 2) cos((𝑛+ 2)𝜃)
+𝐾𝑡𝑒−1
2(𝑛− 1) cos((𝑛− 1)𝜃) + 1
2(𝑛+ 1) cos((𝑛+ 1)𝜃)
+𝐾𝑛𝑒−1
2(𝑛− 1) sin((𝑛− 1)𝜃) + 1
2(𝑛+ 1) sin((𝑛+ 1)𝜃)𝜋
𝜃1
, 𝑛 = 3,4, .. (15)
𝑏1= − 𝑏𝑁𝑡
𝜋𝐾𝑡𝑓𝑡3
4cos(𝜃) − 1
12 cos(3𝜃)+𝐾𝑛𝑓𝑡1
4sin(𝜃) − 1
12 sin(3𝜃)
+𝐾𝑡𝑒−1
2(𝜃) + 1
4sin(2𝜃)+𝐾𝑛𝑒−1
4cos(2𝜃)𝜋
𝜃1
(16)
𝑏2= − 𝑏𝑁𝑡
𝜋𝐾𝑡𝑓𝑡1
4cos(2𝜃) − 1
16 cos(4𝜃)+𝐾𝑛𝑓𝑡1
4(𝜃) − 1
16 sin(4𝜃)
+𝐾𝑡𝑒−1
2sin(𝜃) + 1
6sin(3𝜃)+𝐾𝑛𝑒−1
2cos(𝜃) − 1
6cos(3𝜃)𝜋
𝜃1
(17)
𝑏𝑛= − 𝑏𝑁𝑡
𝜋𝐾𝑡𝑓𝑡1
2𝑛cos(𝑛𝜃) − 1
4(𝑛− 2) cos((𝑛− 2)𝜃) − 1
4(𝑛+ 2) cos((𝑛+ 2)𝜃)
+𝐾𝑛𝑓𝑡1
4(𝑛− 2) sin((𝑛− 2)𝜃) − 1
4(𝑛+ 2) sin((𝑛+ 2)𝜃)
+𝐾𝑡𝑒𝑓𝑡−1
2(𝑛− 1) sin((𝑛− 1)𝜃) + 1
2(𝑛+ 1) sin((𝑛+ 1)𝜃)
+𝐾𝑛𝑒𝑓𝑡−1
2(𝑛− 1) cos((𝑛− 1)𝜃) − 1
2(𝑛+ 1) cos((𝑛+ 1)𝜃)𝜋
𝜃1
, 𝑛 = 3,4, .. (18)
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