Elliptical vibration cutting (EVC), as a precision machining technology, is used in many applications. In precision machining, control accuracy plays an essential role in improving the machinability of difficult-to-machine materials. To improve the control accuracy, dynamic and static characteristics of the system need to be tuned to obtain the optimal parameters. In this paper, we use a glowworm algorithm with an improved adaptive step size to tune the parameters of a robust adaptive fuzzy controller. We then obtain the optimal controller parameters through simulation. The optimal solution of the controller parameters is then applied to a 3D EVC system model for simulation and closed-loop testing experiments. The results indicate that a good agreement between the ideal curve and the tracking signal curve verifies the optimality of the controller parameters. Finally, under certain cutting conditions, the workpieces of three different materials are cut with two different cutting methods. The study revealed that the surface roughness value is reduced by 20 %–32 %, which further verifies the effectiveness of the optimal controller's parameters.

Elliptical vibration cutting (EVC) technology was first proposed by Shamoto in 1984 (Ma et al., 2004). For decades, EVC devices evolved from 2D to 3D, and many experiments verified their superiority in the machining of difficult-to-machine materials. Since the 1990s, in-depth research works have been conducted on elliptical vibration cutting and designing EVC devices (Tan et al., 2018; Kim and Loh, 2007; Zhang and Song, 2019). The machinability of difficult-to-process material was also examined in Lu et al. (2017) and Dong et al. (2020), and path planning was reported on in Kim and Loh (2008), Jieqiong et al. (2013) and so on. However, the improvement of control accuracy in the process of 3D EVC is a challenging task, and both the dynamic and static characteristics of the system need to be tuned to obtain the optimal parameters. Precision control also plays a significant role in the machining performance of 3D EVC. Therefore, we use a glowworm algorithm with an improved adaptive step size to tune the parameters for a robust adaptive fuzzy controller.

To meet the control requirements and achieve effective precision machining, suitable controller parameters must be carefully selected. Parameter tuning improves the dynamic and static characteristics of the system by changing the parameters of the control unit to achieve a more efficient control performance. In engineering, different methods, such as the critical proportionality method, response curve method, attenuation curve method (Yang et al., 2012; Verboven et al., 2005; Petkov, 2018), etc., are often used for parameter tuning. However, for complex control systems, the optimal parameters for the tuning of the controller need intelligent optimization algorithms. Instances of such intelligent algorithms are particle group algorithms (Karakuzu, 2010), genetic algorithms (Ortiz et al., 2018), neural network algorithms (Sheng et al., 2017) and various new bionic intelligence algorithms, e.g., bat algorithms (Li et al., 2018; Xue et al.,2015), cuckoo search algorithms (Yildiz, 2012; Gandomi et al., 2011), etc.

Extended research has been done on tuning controller parameters in recent
years. Soma et al. (2004) proposed a fictitious reference iterative tuning
(FRIT), where only one shot of experimental data is required to perform the
offline nonlinear optimization and obtain the optimal parameter of the
controller in the real closed loop. The result of the control experiments
confirmed the validity of FRIT. García-Gutiérrez et al. (2019)
proposed an optimization program based on the cuckoo search (CS) algorithm
to optimize all parameters of a fuzzy logic control (FLC) and applied their
proposed algorithm to a nonlinear magnetic levitation system. Comparative
simulation results were also provided to validate the featured improvement
of such an approach, which can be extended to other FLC-based control
systems. Nie et al. (2017) proposed an adaptive chaotic particle swarm
optimization (ACPSO) to optimize the parameters of the proportional integral
derivative (PID) controller. The results of the performance test for their
algorithm showed that ACPSO was efficient when used to find the best
parameters of the PID controller. Furthermore, Zhu and Liu (2020) conducted
extensive studies on machining chatter and presented a novel approach to
detect the milling chatter based on variational mode decomposition (VMD) and
energy entropy. It was found that the parameters, such as the number of modes
(

Although the above article has tuned the controller parameters and further verified the optimality of the controller parameters through simulations and experiments, it has not been applied in the machining experiment, and the effectiveness of the optimal parameters of the controller cannot be obtained more intuitively. In this paper, based on the nonlinear Wiener model of 3D EVC, an improved adaptive step size glowworm swarm optimization algorithm (IASGSO) is used to tune the parameters of the robust adaptive fuzzy controller and obtain the optimal solution of the controller parameters. The optimal solution of the controller parameters is also applied to the 3D EVC device system model for simulation and closed-loop test experiments. Our simulation results verify the optimality of the robust adaptive fuzzy controller parameters. Finally, a copper rod, an aluminum rod and a titanium alloy (Ti6A14V) are machined by 3D EVC and 3D EVC with a robust adaptive fuzzy controller improved by IASGSO. The effectiveness of the control system is also verified by analyzing and comparing the surface morphology and surface roughness values of the produced samples.

In this paper, the 3D EVC device is a hybrid drive three-dimensional
elliptical vibration-assisted cutting device (3D EVC), as shown in Fig. 1. The device
of the developed apparatus is mainly composed of two parts, i.e, the upper part and
the lower part. The lower part has a compliant mechanism that can be fixed on a lathe
by the connecting block. The piezoelectric stack 1 and the piezoelectric
stack 2 are also fixed on the lower compliant mechanism, and the
piezoelectric stack 3 is fixed on the upper compliant mechanism. For easy
analysis, three axes are defined, e.g., the axis along piezoelectric stack 1
is defined as

The parallel series hybrid driving system is driven by three ballast stacks
of two parallels and one vertical with the nonlinear resonance, and the
position of the space 3D ellipse is achieved by adjusting the signal
parameters of the piezoelectric stack drive. There are three piezoelectric stacks positioned to drive the flexible hinges along three directions,
respectively, and there is no coupling between them. At the same time, the

The 3D EVC system.

The principle of the 3D EVC is shown in Fig. 2. The concrete cutting principle is the continuous progression of the tool in the process of material removal. Therefore, the movement trajectory of the tool is a three-dimensional elliptical motion. As shown in Fig. 2, points 1–5 represent the position of the tool at different times, which constitutes a complete cutting cycle. It can be divided into two stages, i.e., effective cutting and tool–workpiece separation.

The principle of the 3D EVC.

The objective is to achieve the best machining performance and precision control in the precision machining of elliptical vibration cutting. Therefore, a perfect control system is required to be developed to achieve such a precision control. This section focuses on system modeling of the 3D EVC device and identification of the system model.

The nonlinear Wiener system model of the single-input, single-output (SISO)
process of 3D EVC can be represented, as in Lu et al. (2019), as follows:

Identification experiments for Eq. (1) use the improved adaptive step size
glowworm swarm optimization algorithm (IASGSO). According to Lu et al. (2019), the identification results of the

In this section, we propose an improved design of a robust adaptive fuzzy
controller for SISO-uncertain nonlinear systems. Consider a class of
nonlinearly indeterminate SISO systems that are subject to external
disturbances as follows:

The IASGSO optimized solution versus the number of iterations for

Here we define the tracking error,

The setting of the system control parameters.

Value of each parameter (test parameters

According to the control algorithm in Lin et al. (2020), the control rate,
the intermediate calming function and the parameter adaptive rate are
selected as follows:

System response curve (test

System response curve (test

Before optimizing the controller parameters, an appropriate performance
indicator function should be selected as the fitness function of the
algorithm. The error-integral criterion is one of the most commonly used
performance indicators for measuring the performance of control systems. There are the following four error integrals: integral of the squared error criterion (ISE), integral of time multiplied by squared error criterion (ITSE), integral of the absolute value of error criterion (IAE) and integral of time multiplied by the absolute value of error criterion (ITAE). Among these, ISE is preferred for suppressing large errors, and IAE is better at suppressing smaller errors. ITAE can also suppress long-term errors effectively, making the adjustment time shorter, and the performance indicators of ITSE can control large deviations and shorten the adjustment time. Therefore, ITES is the most appropriate fitness function for this algorithm, and it is defined as follows:

Value of each parameter (test parameters

Using the robust adaptive fuzzy controller presented in Sect. 2.3, it can
be seen that, if the precision control is to be achieved, we need to achieve
the desired tracking accuracy. Hence, the tracking error (

Parameters of the robust adaptive fuzzy controller are determined by the
IASGSO algorithm (Lu et al., 2019), as described in the following:

Initializing the IASGSO parameters means the population size is 20, the maximum number of iterations is 200, the neighborhood threshold (the number of neighborhood fireflies) is 5 and the perception radius and decision radius are both 2.048. Furthermore, the concentration of fluorescein is 5, and the step length is 0.03. It is also necessary to set the value range of the controller parameters that need to be optimized. These ranges are defined as

According to the value interval of the parameters, 20 groups of different control parameter points are selected as the parameter set which needs to be optimized. The ITSE is then used as the adaptability function of the algorithm, as shown in Table 1.

Calculating the adaptability function values of each parameter set updates the glowworm luciferin and completes the change phase of the renewal of fluorescein.

Updating the neighbors within the decision domain and the glowworm's movement location and completing the update phase of glowworm position.

Updating the adaptive step and dynamic decision domain range to complete the update phase of the dynamic perceptive range.

Deciding whether the algorithm meets the target according to the maximum number of iterations. If it is satisfied, then the optimal solution is achieved.

The 3D EVC closed-loop control system.

We use the above process to tune the parameter based on the IASGSO
algorithm. The simulation in this paper was implemented in MATLAB, and the
optimization trajectory of control parameters (

A step response performance test was used to verify the optimal degree of
control parameters which concluded from the previous results. Here we change
the value of the parameters according to the response curve of the system
with different values of parameters. We then obtain the impact of variations
in the controller parameter changes on the system response curve; hence, the
influence of adjusting parameters on the control characteristics could be
determined. The following are the results:

For constant values of

For constant values of

Tracking curve-fitting diagram.

The overall system of cutting experiments.

As it is seen above, the following conclusions can be drawn: by increasing
the values of

Surface roughness values for work parts under both machining methods.

The 3D EVC closed-loop control system is shown in Fig. 6. The system comprises a 3D EVC device, an NI cRIO-9033 controller capacitive displacement sensor, a power amplifier and a Lenovo notebook computer. This section investigates the displacement tracking and velocity tracking of the sinusoidal signal of the 3D EVC device.

The robust adaptive fuzzy controller was programmed by LabVIEW software, and
the NI cRIO-9033 controller was used to collect data. The display control of the
LabVIEW front panel is used to process the collected data and output the
fitting curve. The external interference in the experiment is also taken as

Running the program in the system design software LabVIEW and using and NI cRIO-9033 controller to transfer the collected data to the Wiener system model of the 3D EVC device, the displacement tracking curve-fitting diagram is obtained, as in Fig. 7a. The speed tracking curve-fitting diagram is also shown in Fig. 7b. As it is seen in Fig. 7, the results showed a good agreement between the ideal speed signal curve and the tracking speed signal curve. Similarly, the ideal speed signal curve is consistent with the tracking speed signal curve. Both displacement tracking curve and velocity tracking curve have a high fitting degree, which indicated that the robust adaptive fuzzy controller under the optimal parameters provides a better tracking effect.

The 3D EVC device-cutting experiment was conducted on the Nanoform 250 ultra-machine made by AMETEK (USA). The main instruments included the independently developed 3D EVC device, a NI cRIO-9033 controller, a C series analog input module NI 9222, a C series analog output module NI 9263, a DE-5300-013 displacement sensor, a power amplifier, a Nanoform 250 ultra-machine tool, a workstation and so on. The overall system of the cutting experiment can be seen in Fig. 8.

Here we examine the machining precision of 3D EVC under the controlling of the robust adaptive fuzzy controller with optimal parameters. A total of three workpieces with different materials (a copper rod, an aluminum rod and a titanium alloy, Ti6A14V) are considered for comparative verification experiments with two methods, including 3D EVC and 3D EVC under the control of robust adaptive fuzzy controller improved by IASGSO.

As shown in Fig. 9, the surface appearance of the copper rods, aluminum rods and titanium alloys (Ti6A14V) are compared under the control of 3D EVC and 3D EVC controlled by robust adaptive fuzzy controller, respectively.

The results of the surface roughness under two different cutting methods are shown in Table 4. As shown in Fig. 10 and Table 4, the copper rod, aluminum rod or titanium alloy (Ti6A14V) all indicate the comparative advantage in the surface roughness of 3D EVC processing under the control of robust adaptive fuzzy controller improved by IASGSO. The surface roughness value reduced to 20 %–32 %. Under the process of 3D EVC, obvious scratches and pits appeared on the surface morphology of the workpiece, but this phenomenon is avoided during the processing of 3D EVC under the control of the robust adaptive fuzzy controller improved by IASGSO. It confirms that the robust adaptive fuzzy controller under optimal participation has a comparative advantage in the processing accuracy of 3D EVC and, thus, verifies the validity of the optimal parameters of the controller.

To improve dynamic and static characteristics of the system and achieve an
ideal control accuracy in the process, this paper tunes the parameters of
the 3D EVC robust adaptive fuzzy controller and finds and verifies the optimal
solution through simulation and experiment. Finally, the surface roughness
was compared through cutting experiments, which proved the effectiveness
and the following conclusions are formed:

The controller parameters were tuned by the IASGSO algorithm, and the optimal solution of controller parameters was obtained by MATLAB simulation. Analyzing the system response curve under different parameters under the step response performance test, it is known that increasing the value of

The controller was applied to the 3D EVC device system model, and the displacement tracking signal, speed tracking signal and tracking error were analyzed through simulation. The control object had a little jitter at the beginning, but it could quickly stabilize and smoothly move towards the direction of the ideal displacement signal and the speed signal. This verifies the effectiveness of the optimal parameters of the controller and the robustness of the controller.

A closed-loop test experiment was performed on the 3D EVC control system, and the displacement tracking curve and the speed tracking curve were collected with the help of LabVIEW and the NI cRIO-9033 controller. The result showed that the ideal curve and the tracking signal curve were approximately coincident. This verifies the rationality of the parameter adjustment of the robust adaptive fuzzy device.

Under certain cutting conditions, copper rods, aluminum rods and titanium alloys were machined by 3D EVC and 3D EVC under the control of a robust adaptive fuzzy controller improved by IASGSO. The surface topography of the three sets of workpieces was then analyzed and compared. The surface roughness values were reduced by 20 %–32 %, which verifies the rationality of the controller parameter setting and the effectiveness of the control system.

No data sets were used in this article.

YSD conceptualized and visualized the project and wrote, reviewed and edited the paper with the help of MML. MML and JKZ did the investigation. JKZ also visualized the project and wrote the original draft with HW. The methodology, data curation and formal analysis was conducted by HW, while JQL supervised the project.

The authors declare that they have no conflict of interest.

The authors thank the reviewers for their valuable comments and Copernicus Publications for their copy-editing and typesetting services.

This research has been supported by the National Natural Science Foundation of China (grant no. U19A20104), Science and Technology Development Projects of Jilin Province (grant no. 20190201303JC) and Micro-Nano and Ultra-Precision Key Laboratory of Jilin Province (grant no. 20140622008JC).

This paper was edited by Jeong Hoon Ko and reviewed by Rui Huang and two anonymous referees.