Sage Journals: Discover world-class research

Abstract

To overcome the problems of structural parametric uncertainty and cable transmission model complexity, a nonlinear controller based on time-delay estimation and fuzzy self-tuning is proposed. The unknown dynamics and disturbances are estimated by time delaying the state of motion immediately before. The control gains are self-tuned by a fuzzy controller, which can reduce the errors caused by system’s uncertainties and external disturbances. Compared with the conventional Proportional-derivative (PD) and time-delay control, the result shows that the proposed control scheme based on time-delay estimation can improve the joint trajectory tracking accuracy of cable-driven robot by significantly reducing the control gains. With the PD gains self-tuned by fuzzy strategy, the mean square errors of trajectory tracking are decreased approximately by 5–20% more than the conventional time-delay control with constant gains. In addition, the experimental result shows that the proposed method has an effective inhibitory effect on dead zone in cable-driven joints. Experiment performed on position tracking control of a 2-degree-of-freedom cable-driven robot is presented to illustrate that the controller has the advantages of simple and reliable structure, model-free, strong robustness, and high tracking accuracy.

Keywords

Cable-driven robot time-delay estimation fuzzy control trajectory tracking

Introduction

Cable-driven robot usually mounts motor, reducer, and other transmission mechanisms on the base or away from joints, and the motion and force are transmitted by cables between actuator and joint.¹ Compared with the common robots, it has lightweight, low joint inertia, and large payload to mass. Cable-driven robots have been widely applied in various fields, such as industry, medical rehabilitation, bionic hands, and service.^2

–5 It has received great attention of researchers and institutions in recent years. Unlike the gear transmission, the cable itself is flexible. Due to some factors like tensioning or reversal, the tension is time varying during the transmission process, which makes cables more difficult to control.⁶ In addition, the winding of cables brings an unknown and complex frictional environment. Therefore, the precise modeling and control of kinematics/dynamics for cable-driven robot become a hot spot although many difficulties exist.⁷

Because the multi-degree-of-freedom (DOF) robot is a strongly coupled nonlinear multiple-input and multiple-output and complex system, other nonlinear factors, such as the cable flexibility and transmission friction, pose the obstacles to the precise modeling and parameter identification.⁸ To solve the problem of the uncertainties of system parameters and the difficulty of precisely acquiring dynamics model, the research of high-precision and high-speed control of the cable-driven robot has become a new focus.⁹ A method using Gaussian process regression and data cleaning to reduce position and angular errors achieved precise control to a cable-driven surgical robot, Raven Ⅱ. A H∞ methodology with cable tension distribution algorithm was proposed for a vision-based position control of 6-DOF cable-driven parallel robots (CDPRs).¹⁰ Although the methods show good trajectory tracking performance, the identification of kinematic and dynamic parameters is necessary and complex. Besides, Okur provided a full-state feedback nonlinear robust controller for tendon-driven robots in the presence of dynamical parametric uncertainty and additive external disturbances, while much information that the controller requires leads to much complexity.¹¹

To solve the complexity of dynamic modeling and external disturbances, time-delay estimation (TDE) technique provides a good control method.^12,13 The core of TDE is that the system dynamics at the current moment can be estimated by the last motion state of the closed-loop system, so as to compensate for the various uncertainties and disturbances.^14,15 It is a model-free control algorithm with simple structure and fast calculation to achieve good effects, and it has been widely applied in various fields.^16
–18 Many TDE-based sliding mode controllers have been proposed to perform high-precision and high-speed control with fast-tracking convergence and smooth motor inputs.^19

–22 To overcome slow data acquisition rate of an autonomous underwater vehicle, an integral sliding mode controller with conventional TDC was used to improve the performance. A continuous fractional-order nonsingular terminal sliding mode based on TDE ensures faster convergence and higher tracking precision under heavy lumped uncertainties.

Based on the abovementioned research studies, considering the structural characteristics and control difficulties of cable-driven robot, it is difficult to adopt a model-based method to control. Therefore, a model-free method based on TDE combined with fuzzy strategy should be developed to realize high-precision tracking control of cable-driven robot.

Dynamic model

The complete dynamical equations of cable-driven robot can be expressed as follows

J_{m} \ddot{θ} + B_{m} \dot{θ} = τ_{u} - τ_{d} (q, \dot{q}, θ, \dot{θ})

M (q) \ddot{q} + C (q, \dot{q}) \dot{q} + G (q) + F (q, \dot{q}) - τ_{s} = τ_{d} (q, \dot{q}, θ, \dot{θ})

τ_{d} (q, \dot{q}, θ, \dot{θ}) = K_{c} (θ - q) + D_{c} (\dot{θ} - \dot{q})

where $J_{m} \in R^{n \times n}$ and $B_{m} \in R^{n \times n}$ are effective moment of inertia matrix and damping matrix of motor and reducer, $M (q) \in R^{n \times n}$ is the mass matrix, $C (q, \dot{q}) \in R^{n \times 1}$ is the vector of centrifugal force and Coriolis force, $G (q) \in R^{n \times 1}$ is the gravity vector, $F (q, \dot{q})$ is the vector of Coulomb’s/viscous friction and other non-rigid effect, $τ_{u} \in R^{n \times 1}$ is the input torque vector, $τ_{d} (q, \dot{q}, θ, \dot{θ}) \in R^{n \times 1}$ is the control torque vector for joints, $τ_{s} \in R^{n \times 1}$ is the external disturbance, $K_{c}$ and $D_{c}$ are the impedance coefficient matrices, and $q$ and $\dot{q}$ are the angular displacement and velocity of joints, respectively. $θ$ and $\dot{θ}$ are those of motors, respectively.

The complete dynamical equation of cable-driven robot can be obtained by combining the equations (1), (2), and (3)

τ_{u} = J_{m} \ddot{θ} + B_{m} \dot{θ} + M (q) \ddot{q} + C (q, \dot{q}) \dot{q} + G (q) + F (q, \dot{q}) - τ_{s}

Design of controller

Design of proportional-derivative (PD) controller based on TDE

Expression (4) can be rewritten as

τ_{u} = \bar{M} \cdot \ddot{θ} + H

where $\bar{M} \in R^{n \times n}$ is a constant diagonal matrix. $H$ can be expressed as

H = (J_{m} - \bar{M} (t)) \ddot{θ} + D_{m} \dot{θ} + M (\ddot{q}) \ddot{q} + C (q, \dot{q}) \dot{q} + G (q) + F (q, \dot{q}) - τ_{s}

which shows that $H$ contains many system parameters in the robotic dynamics. For the cable-driven robot proposed in this article, model-based control algorithms have great difficulty in identifying dynamic parameters. However, once $H$ can be estimated reasonably with appropriate methods, the calculation of equation (5) will be simplified significantly, which can greatly improve the control performance. Introduce a constant gain $\bar{M}$ , and PD controller based on TDE is adopted for the cable-driven robot with the control law can be expressed by following equations

τ_{u} = \bar{M} \cdot u (t) + \hat{H} (t)

u (t) = {\ddot{θ}}_{d} + K_{D} \dot{e} + K_{P} e

where $\hat{H} (t)$ is the estimate for $H$ in equation (6), ${\ddot{θ}}_{d}$ is the expected acceleration vector of motors, $e = θ_{d} - θ$ and $\dot{e} = {\dot{θ}}_{d} - \dot{θ}$ are tracking error and its derivative of motors, respectively. $K_{P}$ and $K_{D}$ are both positive diagonal matrices containing PD parameters for each joint. It is worth to mention that constant gain $\bar{M}$ is not necessarily equal to $M (\ddot{q})$ since $\bar{M}$ is a constant gain of controller personally chosen.

By combining equations (7) and (8), the following equation can be written as

\bar{M} \cdot (\ddot{e} + K_{D} \dot{e} + K_{P} e) = H (t) - \hat{H} (t)

Assuming that $\hat{H} (t) = H$ , the closed-loop error equation of the system can be obtained with equations (8) and (9):

\ddot{e} + K_{D} \dot{e} + K_{P} e = 0

which shows that the closed-loop system is asymptotically stable. Estimated parameters for $H (t)$ of TDE

\hat{H} (t) = H (t - Δ t) = τ_{u} (t - Δ t) - \bar{M} \cdot \ddot{θ} (t - Δ t)

where $Δ t$ is sampling period of the system. If $Δ t$ given too small, $H (t - Δ t)$ of last sampling period could be used as the true $H (t)$

H (t) ≅ H (t - Δ t)

Considering equation (10), the law of TDC can be expressed as

τ_{u} = \bar{M} \cdot ({\ddot{θ}}_{d} + K_{D} \dot{e} + K_{P} e) + (τ_{u} (t - Δ t) - \bar{M} \cdot \ddot{θ} (t - Δ t))

which shows that precision of the dynamic model is mainly depending on $\hat{H} (t)$ and $Δ t$ . Ideally, if $Δ t \to 0$ , the equality of equation (12) holds.

Control of cable-driven robot based on TDE is illustrated in Figure 1 and the whole control system is divided into two parts: one is a conventional PD controller depending on the feedback error of system; another is a compensator for system based on linear estimation of control targets. In equation (13), $\bar{M}$ is a constant diagonal matrix and $K_{D}$ and $K_{P}$ are parameters of PD controller. By adjusting the above parameters, the calculation can be fast and simple enough for general high-speed robots. Moreover, the compensator is estimated in real time according to the change of actual external system parameters and guarantees better robustness even for systems with uncertainties.

Figure 1.

The control diagram of the TDE-based for cable-driven robot system. TDE: time-delay estimation.

Actually, the sampling period is unable to approach infinitesimal. Thus, in the case of limited sampling periods, estimation error of $H$ can be derived from equations (9) and (10)

H (t) - \hat{H} (t) = \bar{M} \cdot (u (t) - \ddot{θ} (t - Δ t))

TDE error is defined as

ε (t) = {\bar{M}}^{- 1} (H (t) - \hat{H} (t)) = u (t) - \ddot{θ} (t - Δ t)

By combining equation (14) into equation (15), the feedback error of the robot under closed-loop control can be expressed as

\ddot{e} (t) + K_{D} \dot{e} (t) + K_{P} e (t) = ε (t)

This equation is related to the control torque. And assume that $ε (t)$ is bounded, there must be existing a constant $a$

max_{t} (ε (t)) < a

As a result, the solution of the differential equation of system error $e_{0} (t)$ is bounded as well. Due to the property of the linear system with bounded-input, bounded-output (BIBO), the system can remain stable even estimated parameters are quite different from the real ones. To reduce the disturbance caused by estimation error and improve robustness of the system, a fuzzy PD controller is then designed.

Design of fuzzy PD controller based on TDE

Based on the PD controller with TDE, a fuzzy PD controller for parameter self-tuning is designed as shown in Figure 2.

Figure 2.

The diagram of the fuzzy PD controller with parameter self-tuning.

Fuzzy field of input and output variables and values of linguistic variables

Deviation e between encoder and expected joint angle and deviation rate $\dot{e}$ are used as inputs of the fuzzy controller, Kp and Kd are used as outputs. The basic domain of e, $\dot{e}$ , Kp, and Kd is [−3,+3]. By normalization, the basic domain of the input and output is discretized and described as fuzzy linguistic variables. According to this, e, $\dot{e}$ , Kp, and Kd can be described as {NB, NM, NS, ZO, PS PM, PB}.

e, \dot{e}, K p, K d \in {N B, N M, N S, Z O, P S, P M, P B}

Membership function

Membership function has a certain effect on control of the system. The steeper and higher resolution the membership function graph is, the higher control sensitivity it has; the milder and lower resolution it is, the better stability the system has. Considering the sensitivity and robustness of control, membership functions are composed of Gauss functions and triangle functions as shown in Figure 3.

Figure 3.

The member functions of $e$ and $\dot{e}$ .

Establishment of fuzzy rules

The fuzzy PD controller calculates $K p$ and $K d$ with input variables by fuzzy rules. The fuzzy rules of $K p$ and $K d$ are listed in Tables 1 and 2 and their surfaces are shown in Figure 4.

Table 1.

The fuzzy rules of Kp.

$Δ K p$		$\dot{e}$
$Δ K p$		NB	NM	NS	ZO	PS	PM	PB
e	NB	PB	PB	PM	PM	PS	ZO	NS
	NM	PB	PB	PM	PS	PS	ZO	NS
	NS	PM	PM	PM	PS	ZO	NS	NS
	ZO	PM	PM	PS	ZO	NS	NM	NM
	PS	PS	PS	ZO	NS	NS	NM	NM
	PM	PS	ZO	NS	NM	NM	NM	NB
	PB	ZO	ZO	NM	NM	NM	NB	NB

Table 2.

The fuzzy rules of Kd.

$Δ K d$		$\dot{e}$
$Δ K d$		NB	NM	NS	ZO	PS	PM	PB
e	NB	PS	NS	NB	NB	NB	NM	PS
	NM	PS	NS	NB	NM	NM	NS	ZO
	NS	ZO	NS	NM	NM	NS	NS	ZO
	ZO	ZO	NS	NS	NS	NS	NS	ZO
	PS	ZO	ZO	ZO	ZO	ZO	ZO	ZO
	PM	PB	NS	PS	PS	PS	PS	PB
	PB	PB	PM	PM	PM	PS	PS	PB

Figure 4.

The surface for adjusting Kp and Kd.

The fuzzy PD controller of cable-driven robot uses Mamdani-type fuzzy inference.^23

–26 The accurate values of output are calculated by area center algorithm. Convert the solution of $Δ K p$ obtained by fuzzy inference to output domain using quantification factor, and one new set of $K p$ and $K d$ is calculated which can be used as new control gains in next moment.

Simulation and experiment

Simulation

Control simulation of 4-DOF cable-driven robot by SimMechanics is presented. The parameters are listed in Table 3.

Table 3.

The parameters of control simulation for cable-driven robot.

Parameters	Values	Parameters	Values
Kp(PD)	[1200,1500,500,300]	Kd(PD)	[100,100,100,100]
Kp(TDE)	[15,15,15,15]	Kd(TDE)	[15,15,15,15]
ΔKp	[4,4,4,4]	ΔKd	[2,2,2,2]
Kp factor	[1500,1000,5000,3000]	Kd factor	[300,300,2000,800]
M	[0.2,0.2,0.2,0.04]	Δt	0.001s

Some parameters of the above PD controller are appropriately tuned. And for the convenience of comparison, parameters of the two controllers based on TDE are kept same. The results of joint tracking errors of the three control strategies are shown in Figure 5. The max absolute errors (MAE) and mean square errors (MSE) and quantization errors of joints are calculated in Table 4.^27

–30

Figure 5.

The trajectory tracking errors of three controllers.

Table 4.

The statistical table of simulation.

Quantified index	Joint 1			Joint 2
Quantified index	PD	TDE	Fuzzy TDE	PD	TDE	Fuzzy TDE
MAE (°)	0.191	0.059	0.060	0.328	0.128	0.118
Percent (%)	318.9	100	101.5	255.9	100	91.9
MSE (°)	0.117	0.035	0.317	0.227	0.066	0.061
Percent (%)	336.2	100	91.6	346.7	100	93.46
	Joint 3			Joint 4
Quantified index	PD	TDE	Fuzzy TDE	PD	TDE	Fuzzy TDE
MAE (°)	0.035	0.014	0.013	0.067	0.042	0.034
Percent (%)	258.1	100	91.9	158.3	100	81.4
MSE (°)	0.024	0.008	0.007	0.035	0.022	0.019
Percent (%)	316.1	100	93.4	158.4	100	87.1

According to the results listed in Table 4, TDC has greater advantage than the conventional PD controller. To meet trajectory tracking with high precision, higher gains are needed for the conventional PD controller with slower convergence of errors. However, TDC is able to significantly optimize the trajectory tracking with smaller PD gains and faster convergence. Moreover, its errors are optimized 3 times better than PD controller and can fulfill the requirement of high precision. Furthermore, besides advantages of TDC, a fuzzy controller with parameter self-tuning can adjust control gains in real time. The errors are reduced 5–20% and more.

Simulation above shows that TDE-based controllers have better trajectory tracking performance and reliability than the conventional PD controller with lower gains, higher precision and faster convergence. That parameter self-tuning fuzzy controller can improve the dynamic performance and robustness of the system and enhance the ability of anti-interference.

Experiment

Experiments are carried out with two-joint cable-driven robot. Trajectory tracking results of the two joints under three control strategies are shown in Figure 8, the tracking errors are shown in Figure 9 and the motor inputs are shown in Figure 10. The mean error, max error, and mean square error of the two joints are calculated in Table 5.

Figure 6.

Curves of the control gains adjusting: (a) self-tuning PD parameters curve of joint 3 and (b) self-tuning PD parameters curve of joint 4: (a) back slash curve of joint 3 and (b) back slash curve of joint 4.

Figure 7.

Curves of the trajectory tracking.

Figure 8.

Curves of the backhaul of the cable-driven joints.

Figure 9.

Curves of the tracking error.

Figure 10.

Curves of the motor input.

Table 5.

Statistical table of experiment.

Quantified index	Joint 1			Joint 2
Quantified index	PD	TDE	Fuzzy TDE	PD	TDE	Fuzzy TDE
Mean error (°)	2.435	2.989	0.279	0.315	0.254	0.269
Percent (%)	872.8	948.9	100	100	91.0	85.4
MSE (°)	2.652	3.913	0.404	0.436	0.377	0.397
Percent (%)	873.0	948.3	100	100	91.0	85.2
Max error (°)	5.129	8.191	2.782	3.123	2.736	1.738

According to the above data, TDC and fuzzy TDC with parameters self-tuning are better than the conventional PD controller. And benefiting from parameter self-tuning, the latter owns more advantages. Self-tuning PD parameters of joint 3 and joint 4 are shown in Figure 6.

Set the TDC as a reference (set to 100%), the relative percentages of MSE of joints 3 and 4 under conventional PD are 872.8% and 948.9%, respectively. Results of Fuzzy TDC are 91.0% and 85.4%, respectively. The relative percentages of mean error have the similar conclusion. As shown in Figure 7, error of the conventional PD controller has larger dead zone of backhaul when joints reverse their moving direction. The reason is, although tensioner has been adopted, the cables still get a slight stretch caused by its flexibility when the robot works. When cable-driven robot reverses its moving direction, the direction of internal tension in cables interchanges, and as a result, small dead zone exists. Backhaul dead zones of two joints are shown in Figure 8. Compared with the other two methods, TDE-based controllers are more robust with smaller angle error while backhaul dead zone appearing.

Conclusions

A nonlinear control technique based on TDE with fuzzy self-tuning is proposed, which can estimate the unknown dynamics parameters by the last motion state, and reduce the errors caused by system uncertainties and external disturbances. The method is simple and reliable in structure and has little dependence on the system model. The calculation of the complex dynamics for cable-driven robot can be avoided. Besides, it has the advantages of high trajectory tracking accuracy and strong robustness. The parameters self-tuned by fuzzy controller can adjust control gains dynamically to the change of the system model under estimation error and external disturbances. Compared with conventional TDC and PD controller, the simulation and experiment results show that the method based on TDE can improve the trajectory tracking accuracy of cable-driven robot while reducing the control gains significantly. Combined with the gains self-tuned by fuzzy controller, the MSEs of trajectory tracking are further reduced by 5–20%.

Furthermore, it can be found from experiments that the traditional PD controller has certain insufficiency of the dead zone brought by the cable driven when the joint rotation reversing, while the controller based on TDE has an effective inhibition effect. The result shows that the proposed method can meet the requirements of practical application, which further confirms the advantages of validity and robustness.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) discolsed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China [51705243, 51575256], the Natural Science Foundation of Jiangsu Province [BK20170789] and China Postdoctoral Science Foundation [2018M630552].

ORCID iD

Yaoyao Wang

Bai Chen

References

Townsend

Guertin

. Teleoperator slave -WAM design methodology. Industr Robot 1999; 26(3): 167–177.

Chalon

Baumann

. Dexhand: a space qualified multi-fingered robotic hand. In: 2011 IEEE international conference on robotics and automation, Shanghai, China, 9–13 May 2011, pp. 2204–2210. IEEE..

Hannaford

Rosen

Friedman

. Raven-II: An open platform for surgical robotics research. IEEE Trans Biomed Eng 2013; 60(4): 954–959.

Wang

Yan

Chen

. A new adaptive time-delay control scheme for cable-driven manipulators. IEEE Trans Indus Inform 1–1. DOI: 10.1109/TII.2018.2876605.

Wang

Yan

. Practical adaptive fractional-order nonsingular terminal sliding mode control for a cable-driven manipulator. Int J Robust Nonlinear Control 2019; 29(5): 1396–1417.

Palli

Borghesan

Melchiorri

. Modeling, identification, and control of tendon-based actuation systems. IEEE Trans Robot 2012; 28(2): 277–290.

Wang

Chen

Yan

. Adaptive super-twisting fractional-order nonsingular terminal sliding mode control of cable-driven manipulators. ISA Trans 2018. DOI: 10.1016/j.isatra.2018.11.009.

De Luca

Siciliano

Zollo

. PD control with on-line gravity compensation for robots with elastic joints: theory and experiments. Automatica 2005; 41(10): 1809–1819.

Reinecke

Chalon

Fried

. Guiding effects and friction modeling for tendon driven systems. In: 2014 IEEE international conference on robotics and automation (ICRA), Hong Kong, China, 31 May–7 June 2014, pp. 6726–6732. IEEE.

10.

Mahler

Krishnan

Laskey

. Learning accurate kinematic control of cable-driven surgical robots using data cleaning and Gaussian Process Regression. In: 2014 IEEE international conference on automation science and engineering (CASE), Taipei, Taiwan, 18–22 August 2014, pp. 532–539. IEEE.

11.

Chellal

Cuvillon

Laroche

. Model identification and vision-based H ∞ position control of 6-DoF cable-driven parallel robots. Int J Control 2017; 90(4): 684–701.

12.

Okur

Aksoy

Zergeroglu

. Nonlinear robust control of tendon–driven robot manipulators. J Int Robot Syst 2015; 80(1): 3–14.

13.

Youcef-Toumi

Ito

. A time delay controller for systems with unknown dynamics. J Dyn Syst Meas Control 1990; 112(1): 133.

14.

Hsia

Lasky

Guo

. Robust independent joint controller design for industrial robot manipulators. IEEE Trans Ind Electron 1991; 38(1): 21–25.

15.

Han

Chang

. Robust tracking of robot manipulator with nonlinear friction using time delay control with gradient estimator. J Mech Sci Technol 2010; 24(8): 1743–1752.

16.

Lee

Jin

Ahn

. Precise tracking control of shape memory alloy actuator systems using hyperbolic tangential sliding mode control with time delay estimation. Mechatronics 2013; 23(3): 310–317.

17.

Wang

Zhu

Yan

. Adaptive super-twisting nonsingular fast terminal sliding mode control for cable-driven manipulators using time-delay estimation. Adv Eng Software 2019; 128: 113–124.

18.

Wang

Jiang

Chen

. A new continuous fractional-order nonsingular terminal sliding mode control for cable- driven manipulators. Adv Eng Software. 2018; 119: 21–29.

19.

Jin

Kang

Chang

. Robust compliant motion control of robot with nonlinear friction using time-delay estimation. IEEE Trans Industr Elect 2008; 55(1): 258–269.

20.

Kim

Joe

. Time delay controller design for position control of autonomous underwater vehicle under disturbances. IEEE Trans Industr Elect 2015; 63(2): 1–1.

21.

Wang

Yan

Jiang

. Time delay control of cable-driven manipulators with adaptive fractional-order nonsingular terminal sliding mode. Adv Eng Software 2018; 121: 13–25.

22.

Wang

. Practical tracking control of robot manipulators with continuous fractional-order nonsingular terminal sliding mode. IEEE Trans Industr Elect 2016; 63(10): 6194–6204.

23.

de Jesús Rubio

Lughofer

Meda-Campaña

. Neural network updating via argument Kalman filter for modeling of Takagi-Sugeno fuzzy models. J Int Fuzzy Syst Preprint 2018; 2018: 1–12.

24.

Meng

Shi

Yao

. An inequality approach for evaluating decision making units with a fuzzy output. J Int Fuzzy Syst 2018; 34(1): 459–465.

25.

de Jesús Rubio

. SOFMLS: online self-organizing fuzzy modified least-squares network. IEEE Trans Fuzzy Syst 2009; 17(6): 1296–1309.

26.

Zhang

Han

. State estimation for static neural networks with time-varying delays based on an improved reciprocally convex inequality. IEEE Trans Neural Netwk Learn Syst 2018; 29(4): 1376–1381.

27.

de Jesús Rubio

. Discrete time control based in neural networks for pendulums. Appl Soft Comput 2018; 68: 821–832.

28.

Fourati

. Multiple neural control and stabilization. Neural Comput Appl 2018; 29(12): 1435–1442.

29.

Rubio

. Modified optimal control with a back propagation network for robotic arms. IET Control Theory Appl 2012; 6(14): 2216–2225.

30.

Yogambigai

Syed Ali

Zhu

. Exponential lagrange stability for markovian jump uncertain neural networks with leakage delay and mixed time-varying delays via impulsive control. Math Problem Eng 2018; 2018: 1–15.

A new fuzzy time-delay control for cable-driven robot

Abstract

Keywords

Introduction

Dynamic model

Design of controller

Design of proportional-derivative (PD) controller based on TDE

Design of fuzzy PD controller based on TDE

Fuzzy field of input and output variables and values of linguistic variables

Membership function

Establishment of fuzzy rules

Simulation and experiment

Simulation

Experiment

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References