A Controller for Delta Parallel Robot Based on Hedge Algebras Method

Abstract

Based on the Hedge Algebras (HA) methodology, this paper presents a method for managing the trajectory of a delta robot. This technique is easily applied to existing devices in the process control domain. In the proposed control method, the semantically quantitative mapping may replace the verbal values of the fundamental rule in an analogous manner (SQM). Consequently, the basic rule may be represented as a three-dimensional numerical grid with linear interpolations. Compared to the Fuzzy Logic controller, it reduces the controller’s complexity. In addition, when the control strategy is applied in reality, the system’s response time is decreased. The simulation results indicate that the proposed technique is more exact than previous approaches. The experimental results reveal that the proposed technique is effective for robot trajectory tracking difficulties.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In 1985, Clavel conceived the basic idea for the delta robot to stack chocolates in a factory. Today, the delta parallel robot is used in various industries owing to its advantages, including its rapid working speed, high accuracy, stability, and high load capacity [1–3]. However, delta robots often operate in a limited zone, including many singularities [4–6]. Throughout the past six decades, several researchers have investigated and proposed a paradigm for parallel robotics [7–14]. The references [15–18] explain the optimal designs and control of parallel robots. Since the parallel robot is a nonlinear system with a high number of parameters, the standard design method will be challenging to execute. Based on FLC (fuzzy logic controller), an optimal approach for the adaptive control of delta robots has been proposed [19]. Tho et al. [20] proposed an adaptive neuro-fuzzy inference system to handle the inverse kinematics of delta robots. This method optimizes the fuzzy controller’s input and output settings using a five-layer neural network. Numerous studies focus on creating control rules for delta robots to enhance the robot’s resistance to external forces. Some studies on delta robot trajectory control depend on sliding control [21–24] or active noise-removing control (ADRC) [25]. The Hedge Algebra (HA) theory was developed in 1990 and has been utilized for control since 2008. Studies using the hedge algebra [26–35] have shown favorable results, demonstrating the method’s merits. In recent years, HA theory has been used not only to control structural vibrations [36–40] but also to control serial robot trajectories [41–44]. The study findings demonstrate that using HAC in orbital control for serial robots is more successful than using PID or FLC.

There has been no previous study using the hedging method to control the trajectory of delta robots. This work proposes a delta robot trajectory control system based on the Hedge Algebra (HAC) approach. The necessity for mathematical formulas and understanding the delta robot’s kinematics complicates its control. Fuzzy control may be used to tackle this difficult situation. The traditional technique of fuzzy control may lose stability and accuracy during inference and optimization. Approximation inference is unaffected by the concept of fuzzy sets in hedge algebra. Consequently, the hedge algebra approach does not need to specify the kind of membership function, the number of membership functions, or the location of the fuzzy problem to resolve it. This technique eliminates design errors in controller development. Confounding effects and instability are also taken into account and handled throughout system development. The simulation outputs of the proposed approach are compared to those of a traditional fuzzy controller. In the instance of a fast-moving robot, the experimental results reveal that the proposed controller is helpful and error-free.

2. Delta Robot Dynamic

As illustrated in Figure 1, the structure of the delta robot model consists of five major components: a fixed base, a moveable base, three servomotors, three active arms, and three passive arms in parallelogram formation. The fixed base is mounted to the frame. Three motors and a reducer are installed at a 120-degree angle on the same plane. The reducer is attached to the actuating arm. The passive arm is a parallelogram that is attached to a moving platform. The platform is maintained perpendicular to the base. There are three identical drive chains connecting the moving platform to the base platform. Based on the publications [45, 46], the dynamic model is constructed.

[figure(s) omitted; refer to PDF]

The virtual work concept applied to the N-body system described by Arnol’d [46] can be written as follows:$$\begin{array}{}\text{(1)}& {\displaystyle \sum _{i=1}^N{m}_i{\ddot{x}}_i-F_i.\gamma x_i+I_i{\omega }_i{\dot{\omega }}_i+{\omega }_i\times I_i{\omega }_i-T_i.\gamma {\rho }_i}=0,\end{array}$$where $m_i$ is the body mass of body i; $I_i$ is the inertial of body i; ${\ddot{x}}_i$ is the acceleration of the body’s mass center; ${\omega }_i$ are the angular velocity ${\dot{\omega }}_i$ is the angular acceleration; $F_i,\hspace{0.17em}{T}_i$ are applied forces (gravity, for example) and moments, respectively; $\gamma {\rho }_i$ and $\gamma x_i$ represent the model of virtual displacements.

By utilizing the mean of the Jacobian matrices, $J_{n,i}$ and $J_{m,i}$, the virtual displacements can be expressed as a function of joint-variable displacements as follows:$$\begin{array}{}\text{(2)}& {\displaystyle \sum _{i=1}^N{J}_{n,i}^T{m}_i\ddot{x_i}-F_i+J_{m,i}^T{I}_i\dot{{\omega }_i}+{\omega }_i\times I_i{\omega }_i-T_i.\gamma {\rho }_i}=0.\end{array}$$

Figure 2 indicates the parameters of the delta robot model. Based on that, the delta robot dynamic derived from formula (2) is presented as follows:$$\begin{array}{}\text{(3)}& {\tau }_p+{\displaystyle \sum _{i=1}^3{\tau }_{u,i}}+{\displaystyle \sum _{i=1}^3{\tau }_{v,i}}=0,\end{array}$$where ${\tau }_p$ is the contribution of the forces operating on the moving plate; ${\tau }_{a,i}$ is the contribution of the force on the arm $i$; ${\tau }_{fa,i}$ is the contribution of the force on the forearm $i$.

[figure(s) omitted; refer to PDF]

It is noticeable that there are none of the contributions from the rotating terms in ${\tau }_n$ since the moving plate is always parallel to the base. The formulation of ${\tau }_n$ is rewritten as follows:$$\begin{array}{}\text{(4)}& {\tau }_p=J^T{{\displaystyle m}}_p\ddot{X_p}{{\displaystyle G}}_p,\end{array}$$where $J$ is the Jacobian matrix of the proposed model; ${\ddot{X}}_p$ is the end-effector acceleration; $m_p$ is the mass of traveling platform; and $G_p$ represents the relation between gravitational acceleration and the traveling platform. The equation of $G_p$ is indicated as follows:$$\begin{array}{}\text{(5)}& G_p=m_p{\begin{array}{ccc}0& 0& g\end{array}}^T,\end{array}$$where $g$ is the gravitational acceleration.

By arranging the contributions of the three arms into vector form and noticing that their motion consists only of rotations, we get$$\begin{array}{}\text{(6)}& {\tau }_u=I_u\ddot{q}-G_u-\tau ,\end{array}$$where $I_u$ is the actuating arm inertial; $q$ is the angle matrix of the driving arms; $\tau$ is the torque applied by each motor; and $G_u$ is the contribution of gravity.

The equation of the contribution of gravity $G_b$ is expressed as follows:$$\begin{array}{}\text{(7)}& G_u=m_u{r}_{u}g\begin{array}{ccc}\text{cos}{q}_1& \text{cos}{q}_2& \text{cos}{q}_3\end{array},\end{array}$$

with$$\begin{array}{c}\text{(8)}& m_u=m_a+m_e,r_u=L_A.\frac{1/2m_a+m_e}{m_a+m_e},\end{array}$$where $m_a$ and $m_e$ are the masses of the actuating arm and the elbow, respectively.

To compute the contribution of each forearm, ${\tau }_{v,i}$, can be written as follows:$$\begin{array}{}\text{(9)}& {\tau }_{v,i}=\frac{1}{3}{m}_{fa}{J}^T\ddot{X_n}+\frac{1}{2}{a}_{v,i}+J_{v,i}^T\frac{1}{2}\ddot{X_n}+a_{v,i}+-\frac{1}{2}{J_{v,i}+J}^T{m}_{fa}\begin{array}{c}0\\ 0\\ g\end{array},\end{array}$$where $m_{fa}$ is the forearm mass; $J_{v,i}$ is the forearm Jacobian matrix; and $a_{v,i}$ is the gravity has been added and where the acceleration of the upper part of the forearm$$\begin{array}{}\text{(10)}& a_{v,i}=-{}_i{}^{R}R\begin{array}{c}{l}_a\text{sin}{q}_i\\ 0\\ l_a\text{cos}{q}_i\end{array}\ddot{q_i}+\begin{array}{c}{l}_a\text{cos}{q}_i\\ 0\\ -l_a\text{sin}{q}_i\end{array}\dot{q_i}.\end{array}$$

In general, the dynamics of the delta robot can be calculated as follows:$$\begin{array}{}\text{(11)}& \tau =I_u\ddot{q}+J^T\ddot{X_n}-G_u-J^T{G}_p+{\displaystyle \sum _{i=1}^3{\tau }_{v,i}}.\end{array}$$

3. HA Controller

According to the definitions, theorems, and propositions of Bui et al. [26], Ho and Wechler [27] recount, summarize, and provide the idea and essential formulas of HA theory. Following is an illustration of the particular semantically quantifying mapping (SQM) values of a HA structure. The linguistic variable X can be represented by a standard Hedge Algebras structure named $AX$. The formulation group of $\hspace{0.17em}AX$ is defined as follows:$$\begin{array}{}\text{(12)}& \begin{array}{c}AX=X,G,C,H,\le ,\\ G=\text{Negative,Positive}=c^{-},c^{+},\\ C=0;W;1,\\ H=\text{Very,Little}=H^{+}\cup H^{-},\end{array}\end{array}$$where $G$ are the primary terms of X, which is represented by the negative value $c^{-}$ and the positive value $c^{+}$; $C$ is the set of constants included: 0-absolutely negative, 1-absolutely positive, and W-neutral; $H$ is the set of hedges, in which $H^{+}$ represent all negative ones and $H^{-}$ represents all negative ones; the symbol “$\le$” indicates a partly ordering relationship for X.$$\begin{array}{}\text{(13)}& \begin{array}{c}{H}^{-}=h_{-1},\mathrm{...},h_{-q}\text{with\hspace{0.17em}}{h}_{-1}<h_{-2}<\dots <h_{-q},\\ H^{+}=h_1,\mathrm{...},h_p\text{with\hspace{0.17em}}{h}_1<h_2<\dots <h_p.\end{array}\end{array}$$

In this case, $p=q=1$, and as a result, SQM value $\phi$ for all linguistic values of a linguistic variable with its term-set $X$ are determined through only independent fuzziness parameters, which are $0<fm{c}^{-}$ and $\mu h^{-}<1$, where $fm{c}^{-}$ is fuzziness measure of $h^{-}$.

In previous research and development of HA theory, all possible semantically quantifying mappings (SQMs) of linguistic values of the linguistic variable X were determined explicitly. Based on the fuzziness measurements, $fm$ and $\mu$, of the principal words and hedges, respectively, these mappings translate linguistic values of linguistic variables in the linguistic domain to semantically quantified ones in the interval [0, 1]. All SQMs of linguistic values of AX could be determined through [24] based on the two given independent parameters, $fm{c}^{-}$ and $\mu h^{-}$, as shown above. In fact, a finite number of linguistic values of a linguistic variable will be often used in practical problems in general and in structural fuzzy control problems in particular. SQM-values of typical linguistic values of AX for cases of $fm{c}^{-}=fm{c}^{+}=0.5$ and $\mu h$, are presented in Table 1. Table 2 displays typical linguistic values with SQMs of X, where Ne, Po, V, and L represent negative, positive, very, and little, respectively.

Table 1

Parameter for variables $e$, $\dot{e}$, and $u$.

		Input 1 $e$Input 2 $\dot{e}$		Output $u$
		μ (h)		μ (h)
$H^{-}$	Little (L)	μ(h−)	0.35	μ(h−)	0.5
$H^{+}$	Very (V)	μ(h+)	0.65	μ(h+)	0.5

Table 2

The variable parameters.

Symbols	Ranges	Linguistic value with SQMs
		VN	LN	W	LP	VP
$e$	$-a,a$	0.2113	0.4387	0.5	0.5613	0.7888
$\dot{e}$	$-b,b$	0.2113	0.4387	0.5	0.5613	0.7888
U	$-c,c$	0.125	0.375	0.5	0.625	0.875

The framework of an HA-based controller [23] is clearly shown in Figure 3. To be more detailed, the signal u can take part as the control variable after the denormalization process, $e$ and $\dot{e}$ are the proportional-derivative state variables used for the normalization.

[figure(s) omitted; refer to PDF]

Figure 4 depicts the mapping between the reference ranges (in the real domain) and the SQM ranges of the state variables throughout the normalization process. The $e_S$ and ${\dot{e}}_S$ are the SQM values of $e$ and $\dot{e}$, respectively.

[figure(s) omitted; refer to PDF]

In this study, a typical HA-rule base is made up of 25 control rules, each of which has its SQM of linguistic values. Table 3 illustrates the arrangement of all rules, with the orders put in brackets.

Table 3

Rule base of the HAC with SQMs.

$\dot{e}$	VN = 0.2113	LN = 0.4387	W = 0.5	LP = 0.5613	VP = 0.7888
$e$
VN = 0.2113	VN = 0.125	VN = 0.125	VN = 0.125	LN = 0.375	W = 0.5
LN = 0.4387	VN = 0.125	VN = 0.125	LN = 0.375	W = 0.5	LP = 0.625
W = 0.5	VN = 0.125	LN = 0.375	W = 0.5	LP = 0.625	VP = 0.875
LP = 0.5613	LN = 0.375	W = 0.5	LP = 0.625	VP = 0.875	VP = 0.875
VP = 0.7888	W = 0.5	LP = 0.625	VP = 0.875	VP = 0.875	VP = 0.875

As shown in Figure 4, the denormalization method steps are established to extract the actual value of the control variable u from its SQM value $u_s$.

The 3D surface in Figure 5 geometrically represents the mechanism of the HA-rule base. In addition, it demonstrates the HA inference component for calculating the SQM value $u_S$ from $e_s$ and ${\dot{e}}_S$.

[figure(s) omitted; refer to PDF]

The denormalization step to bring out the real value of the control variable u from its SQM value $u_S$ is shown in Figure 6.

[figure(s) omitted; refer to PDF]

Three identical HACs were explored to operate three servo motors for the trajectory control of the delta robot. Figure 7 depicts a schematic depiction of the HAC coupled to the delta robot platform. By calculating the inverse kinematic equations, the desired trajectory of the traveling platform in the workspace is converted into joint space. The error is determined as the difference between the actual location of the joint and the planned position. Two inputs were used by the Hedge-algebras controller: the input signal error $et$ and its temporal derivative ${\dot{e}}_{i}t$. This controller also features a voltage $u_{i}t$ output signal for controlling servo motors.

[figure(s) omitted; refer to PDF]

4. Simulation and Experimental Results

4.1. Simulation Results

In this part, three simulations are conducted to validate the HAC’s performance. The reference trajectory of the delta robot is a spiral using the following equation:$$\begin{array}{}\text{(14)}& \begin{array}{c}x=100\text{cos\hspace{0.17em}}\pi t,\\ y=100\text{sin\hspace{0.17em}}\pi t,\\ z=-280.69+12t.\end{array}\end{array}$$

In the first case, the system response following a set trajectory without interference is shown in Figures 8 and 9.

[figure(s) omitted; refer to PDF]

Figures 9(a)–9(c) illustrate the x, y, and z position errors, respectively. The torque supplied to the actuators is compared in Figures 9(d)–9(f) The HAC response oscillates within 0.5 mm in the x-axis, has a range of up to 2.3 mm, stabilizes at 0.5 mm after 1 s in the y-axis, and has an oscillation amplitude of 0.25 mm in the z-axis. The amplitude of FLC’s oscillations is about 9 percent more than that of HAC, and its response time is similarly slower than the average of 0.4 seconds. PID stabilizes after 1.2 seconds with the greatest amplitude of oscillation compared to the other two controllers.

Figures 10 and 11 illustrate the system’s response to the second situation (a 1 kg weight is placed on the delta robot’s moveable platform during operation). HAC’s constant amplitude of oscillation along the x-axis is around 0.7 mm, whereas FLC’s is over 9% and PID’s is over 20%. The amplitude of y-axis variations for HAC and FLC is 2.9 mm at 0.1 s and stabilizes within 1 mm afterward. The HAC has the smallest oscillation amplitude of the three and is stable at 1 mm along the z-axis. Due to the load’s effects, the average error across all three controllers is now 10% more than in the first situation.

[figure(s) omitted; refer to PDF]

The last instance is tested when the system is exposed to a load disturbance, as shown in Figures 12 and 13. The noise input is a sine function with the following equation:$$\begin{array}{}\text{(15)}& {\partial }_{\text{dis}}=30\text{sin}40t.\end{array}$$

[figure(s) omitted; refer to PDF]

Figures 13(a)–13(c) show the results of the position error on the x, y, and z axes, respectively. Similar to the previous two cases, the HAC’s error response is continually changing due to noise, but it remains steady within 0.5 mm. FLC oscillates with a larger amplitude than HAC, although PID is the most unstable. Additionally, HAC reacts faster than FLC.

Various profiles are simulated to evaluate the performance and robustness of the controllers. Figures 14–25 depict the system’s reaction outcomes. The findings demonstrate that the suggested controller can function consistently throughout the working process. Hedge Algebra outperforms PID and FLC in terms of accuracy and reaction time.

[figure(s) omitted; refer to PDF]

4.2. Experimental Results

Figure 26 depicts the experimental system’s configuration. The system consists of a power source, a fixed platform, a mobile platform, a robot frame, three sets of servomotors, and a controller. The two controllers, HAC and FLC, are tested following the trajectory in the formula to demonstrate the performance of the suggested controller.

[figure(s) omitted; refer to PDF]

Figures 27 and 28 show the trajectory error made by the controls. It can be seen that the proposed controller has a smaller error than FLC and PID. HAC might be reduced by an average of 7% and 18% when compared to FLC and PID, respectively. Additionally, HAC reacts more quickly than conventional FLC. Due to problems with the system installation during testing, the HAC displays a 3.2 mm y-axis overshoot and fluctuates greatly in motion. The results, however, demonstrate that HAC’s control performance performs well in experimental environments.

[figure(s) omitted; refer to PDF]

To demonstrate the robustness of the control law against parametric disturbances, extra weights are added to the delta robot’s traveling plate during operation. Figure 29 shows the results of further 500 g load tests.

[figure(s) omitted; refer to PDF]

The systematic response achieved, which is implemented with the second and third desired orbitals, is shown in Figures 30–33, respectively.

[figure(s) omitted; refer to PDF]

The benefits of utilizing a controller based on hedge algebra were shown throughout the system design phase. In controller design, HAC may generate an algebraic structure in the form of a functional connection, enabling an arbitrarily large collection of linguistic values to be formed to express input-output interactions. As a result, the quality of the produced control system may be superior to that of fuzzy control. The disadvantage of HAC is that it takes expert knowledge or skill to design the control rule and the range of variables. As a consequence, the design will become more problematic or an automated design solution based on predefined quality requirements will be required. This is a challenge for the next research direction.

5. Conclusions

The recommended hedge algebra controller proved successful in guiding the delta parallel robot’s direction. Additionally, FLC and PID controllers are used in experimental models and comparative simulations. Data from simulations and experiments show that HAC performs effectively in terms of reaction time and setting error even when there are load disturbances. When compared to FLC and PID controllers, the controller’s efficiency has also been found to be higher. In trials, HAC has proven to be quite useful for orbital robot control of delta robots. By merging HAC with optimization techniques in the future, setup time and error might be decreased. For assembly or grading systems that need great precision, delta systems with quick reaction times and few errors are acceptable.

Acknowledgments

This research was funded by the Hanoi University of Industry (HAUI) (grant number 02-2022-RD/HĐ-ĐHCN).

References

[1] L. Rey, R. Clavel, "The delta parallel robot," Parallel Kinematic Machines, pp. 401-417, 1999.

[2] I. Bonev, "Delta parallel robot-the story of success," Newsletter, 2001. , 2001, https://www.parallelmic.org

[3] M. Mahmoodi, M. G. Tabrizi, K. Alipour, "A new approach for kinematics-based design of 3-rrr delta robots with a specified workspace," Proceedings of the 2015 AI & Robotics (IRANOPEN), .

[4] F. Pierrot, C. Reynaud, A. Fournier, "DELTA: a simple and efficient parallel robot," Robotica, vol. 8 no. 2, pp. 105-109, DOI: 10.1017/s0263574700007669, 1990.

[5] V. Wijk, J. L. Herder, "Dynamic balancing of clavel’s delta robot," Computational kinematics, pp. 315-322, 2009.

[6] S. Yang, Y. Li, "Kinematic analysis of deployable parallel mechanisms," Proceedings of the Institution of Mechanical Engineers - Part C: Journal of Mechanical Engineering Science, vol. 234 no. 1, pp. 263-272, DOI: 10.1177/0954406218825325, 2020.

[7] S. Yang, Y. Li, "Classification and analysis of constraint singularities for parallel mechanisms using differential manifolds," Applied Mathematical Modelling, vol. 77, pp. 469-477, DOI: 10.1016/j.apm.2019.07.040, 2020.

[8] H. Zhang, H. Fang, D. Zhang, Q. Zou, X. Luo, "Trajectory tracking control study of a new parallel mechanism with redundant actuation," International Journal of Aerospace Engineering, vol. 2020,DOI: 10.1155/2020/7178103, 2020.

[9] X. J. Liu, J. Wang, H. Zheng, "Workspace atlases for the computer aided design of the Delta robot," Proceedings of the Institution of Mechanical Engineers - Part C: Journal of Mechanical Engineering Science, vol. 217 no. 8, pp. 861-869, DOI: 10.1243/095440603322310413, 2003.

[10] X. J. Liu, J. Wang, K. K. Oh, J. Kim, "A new approach to the design of a DELTA robot with a desired workspace," Journal of Intelligent and Robotic Systems, vol. 39 no. 2, pp. 209-225, DOI: 10.1023/b:jint.0000015403.67717.68, 2004.

[11] H. Zhang, H. Fang, Q. Zou, D. Zhang, "Dynamic modeling and adaptive robust synchronous control of parallel robotic manipulator for industrial application," Complexity, vol. 2020,DOI: 10.1155/2020/5640246, 2020.

[12] Y. Li, D. Shang, Y. Liu, "Kinematic modeling and error analysis of Delta robot considering parallelism error," International Journal of Advanced Robotic Systems, vol. 16 no. 5,DOI: 10.1177/1729881419878927, 2019.

[13] H. Shen, Q. Meng, J. Li, J. Deng, G. Wu, "Kinematic sensitivity, parameter identification and calibration of a non-fully symmetric parallel Delta robot," Mechanism and Machine Theory, vol. 161,DOI: 10.1016/j.mechmachtheory.2021.104311, 2021.

[14] D. Shang, Y. Li, Y. Liu, S. Cui, "Research on the motion error analysis and compensation strategy of the delta robot," Mathematics, vol. 7 no. 5,DOI: 10.3390/math7050411, 2019.

[15] J. P. Barreto, B. Corves, "Resonant delta robot for pick-and-place operations," IFToMM World Congress on Mechanism and Machine Science, vol. 73, pp. 2309-2318, 2019.

[16] K. Miller, "Optimal design and modeling of spatial parallel manipulators," The International Journal of Robotics Research, vol. 23 no. 2, pp. 127-140, DOI: 10.1177/0278364904041322, 2004.

[17] A. M. Hay, J. A. Snyman, "Methodologies for the optimal design of parallel manipulators," International Journal for Numerical Methods in Engineering, vol. 59 no. 1, pp. 131-152, DOI: 10.1002/nme.871, 2004.

[18] T. Sun, S. Yang, "An approach to formulate the hessian matrix for dynamic control of parallel robots," IEEE, vol. 24 no. 1, pp. 271-281, DOI: 10.1109/tmech.2019.2891297, 2019.

[19] J. Zhang, C. Lian, R. Gao, "3-Degree-of-freedom parallel robot control based fuzzy theory," Proceedings of the 2010 Second International Conference on Intelligent Human-Machine Systems and Cybernetics, pp. 221-224, .

[20] T. P. Tho, N. T. Thinh, N. T. Tuan, M. N. T. Nhan, "Solving inverse kinematics of delta robot using ANFIS," Proceedings of the 2015 15th International Conference on Control, Automation and Systems (ICCAS), pp. 790-795, .

[21] C. E. Boudjedir, D. Boukhetala, M. Bouri, "Nonlinear PD plus sliding mode control with application to a parallel delta robot," Journal of Electrical Engineering, vol. 69 no. 5, pp. 329-336, DOI: 10.2478/jee-2018-0048, 2018.

[22] T. Su, X. Liang, G. He, Q. Zhao, "Robust trajectory tracking of delta parallel robot using sliding mode control," Proceedings of the 2019 IEEE Symposium Series on Computational Intelligence (SSCI), pp. 508-512, .

[23] I. Hashlamon, Sliding Mode Based Virtual Sensor for Feedback Linearization in delta Robots with Unknown Dynamic Model, 2020.

[24] W. Yang, W. Liu, "Delta robot trajectory tracking based on fuzzy adaptive sliding mode controller," Proceedings of the 2021 China Automation Congress (CAC), pp. 7041-7046, .

[25] L. A. Castañeda, A. Luviano-Juárez, I. Chairez, "Robust trajectory tracking of a delta robot through adaptive active disturbance rejection control," IEEE Transactions on Control Systems Technology, vol. 23 no. 4, pp. 1387-1398, DOI: 10.1109/tcst.2014.2367313, 2015.

[26] H. L. Bui, C. H. Nguyen, N. L. Vu, C. H. Nguyen, "General design method of hedge-algebras-based fuzzy controllers and an application for structural active control," Applied Intelligence, vol. 43 no. 2, pp. 251-275, DOI: 10.1007/s10489-014-0638-6, 2015.

[27] N. C. Ho, W. Wechler, "Extended hedge algebras and their application to fuzzy logic," Fuzzy Sets and Systems, vol. 52 no. 3, pp. 259-281, DOI: 10.1016/0165-0114(92)90237-x, 1992.

[28] D. Vukadinović, M. Bašić, C. H. Nguyen, N. L. Vu, T. D. Nguyen, "Hedge-algebra-based voltage controller for a self-excited induction generator," Control Engineering Practice, vol. 30, pp. 78-90, DOI: 10.1016/j.conengprac.2014.05.006, 2014.

[29] N. D. Anh, H. L. Bui, N. L. Vu, D. T. Tran, "Application of hedge algebra based fuzzy controller to active control of a structure against earthquake," Structural Control and Health Monitoring, vol. 20 no. 4, pp. 483-495, DOI: 10.1002/stc.508, 2013.

[30] D. Vukadinović, M. Bašić, C. H. Nguyen, N. L. Vu, "Hedge-algebra speed controller–experimental implementation in a sensorless stand-alone WECS," Proceedings of the 2020 International Symposium on Industrial Electronics and Applications (INDEL), .

[31] T. M. Hoang, T. Nguyen, N. L. Vu, N. Hong-Duc, "A novel fuzzy inference system based on hedge algebras to enhance energy efficiency in wireless sensor networks," Proceedings of the 2018 IEEE 3rd International Conference on Communication and Information Systems (ICCIS), pp. 73-78, .

[32] N. T. Duy, N. Van Tinh, Approximating the bell-shaped Function Based on Combining Hedge Algebras and Particle Swarm Optimization, vol. 2 no. 4, pp. 136-142, 2019.

[33] D. Vukadinović, T. D. Nguyen, C. H. Nguyen, N. L. Vu, M. Basic, I. Grgic, "Hedge-algebra-based phase-locked loop for distorted utility conditions," Journal of Control Science and Engineering, vol. 2019,DOI: 10.1155/2019/3590527, 2019.

[34] D. Vukadinović, M. Bašić, C. H. Nguyen, N. L. Vu, "Optimization of a hedge-algebra-based speed controller in a stand-alone WECS," Proceedings of the 2020 5th International Conference on Smart and Sustainable Technologies (SpliTech), .

[35] N. C. Ho, N. V. Long, "Fuzziness measure on complete hedge algebras and quantifying semantics of terms in linear hedge algebras," Fuzzy Sets and Systems, vol. 158 no. 4, pp. 452-471, DOI: 10.1016/j.fss.2006.10.023, 2007.

[36] H. L. Bui, N. An Tran, H. Q. Cao, "Active control based on hedge-algebras theory of seismic-excited buildings with upgraded tuned liquid column damper," Journal of Engineering Mechanics, vol. 149 no. 1,DOI: 10.1061/jenmdt.emeng-6821, 2023.

[37] N. Tien-Duy, H. L. Bui, General Optimization Procedure of the Hedge Algebras Controller for Controlling Dynamic Systems, 2022.

[38] H. L. Bui, M. N. Pham, T. T. H. Nguyen, "Swing-up control of an inverted pendulum cart system using the approach of Hedge-algebras theory," Soft Computing, vol. 26 no. 10, pp. 4613-4627, DOI: 10.1007/s00500-022-06968-2, 2022.

[39] B. Hai-Le, T. N. An, "Multi-objective optimal design of TMDs for increasing critical flutter wind speed of bridges," Journal of Wind Engineering and Industrial Aerodynamics, vol. 225, 2022.

[40] H. L. Bui, Q. C. Tran, "A new approach for tuning control rule based on hedge algebras theory and application in structural vibration control," Journal of Vibration and Control, vol. 27 no. 23-24, pp. 2686-2700, DOI: 10.1177/1077546320964307, 2021.

[41] T. D. Nguyen, D. D. Vu, "Designing hedge algebraic controller and optimizing by genetics algorithm for sserial robots adhering trajectories," Journal off Computer Science and Cybernetics, vol. 36, pp. 265-283, 2020.

[42] T. Nguyen Van, S. Y. Yi, P. Bui Khoi, "Hedge algebras-based admittance controller for safe natural human–robot interaction," Advanced Robotics, vol. 34 no. 24, pp. 1546-1558, DOI: 10.1080/01691864.2020.1852958, 2020.

[43] P. B. Khoi, N. V. Toan, "Hedge-algebras-based controller for mechanisms of relative manipulation," International Journal of Precision Engineering and Manufacturing, vol. 19 no. 3, pp. 377-385, DOI: 10.1007/s12541-018-0045-8, 2018.

[44] K. Lai, N. H. Vu, T. H. Lai, "The hedge-algebras-based controller for robotic arm," Proceedings of the International Conference on Engineering Research and Applications, pp. 572-579, .

[45] A. Codourey, "Dynamic modeling of parallel robots for computed-torque control implementation," The International Journal of Robotics Research, vol. 17 no. 12, pp. 1325-1336, DOI: 10.1177/027836499801701205, 1998.

[46] V. I. Arnol'd, Mathematical methods of classical mechanics, vol. 60, 2013.