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Research ArticleResearch on a LADRC Strategy for Trajectory Tracking Control ofDelta High-Speed Parallel Robots

Cheng Liu1 Yanming Cheng 1 Dejun Liu1 Guohua Cao2 and Ilkyoo Lee3

1College of Electrical and Information Engineering Beihua University Jilin China2College of Mechanical and Electric Engineering Changchun University of Science and Technology Changchun China3Division of Electrical Electronics amp Control Engineering Kongju National University Cheonan Republic of Korea

Correspondence should be addressed to Yanming Cheng mychengkongjuackr

Received 19 June 2020 Revised 9 August 2020 Accepted 17 August 2020 Published 30 August 2020

Academic Editor Kalyana C Veluvolu

Copyright copy 2020 Cheng Liu et al )is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In order to better track the planned trajectory of Delta high-speed parallel robot this paper proposes a dynamics control strategyfor Delta high-speed parallel robots based on the linear active disturbance rejection control (LADRC) strategy which realizesdecoupling control through observing and compensating the coupling and internal and external disturbances between the threejoints Firstly the structure and dynamics model of the Delta high-speed parallel robot are analyzed respectively Secondly thecontrol scheme of the Delta high-speed parallel robot dynamic LADRC strategy is constructed and then the system stability isanalyzed Taking a representative 8-shaped space helical variance trajectory as a given input of the system and a triangular wave asan external disturbance as given disturbance input of the system simulations are carried out to demonstrate the effectiveness ofthe proposed LADRC strategy results indicate that the system with the LADRC strategy has a good quick and precise real-timetrajectory tracking and strong robustness

1 Introduction

)e parallel robot has the advantages of high speed light-weight and strong flexibility which is complementary to theserial robot )ereby an increasing number of researcheswere carried out by numerous research institutions at homeand abroad [1 2] At present the parallel robot has not beenwidely used inasmuch as it has a difficult design of theparallel mechanism difficult kinematics solution complextrajectory planning difficult trajectory tracking control andother problems [2ndash6] In this paper Delta parallel robot ischosen as a study object the control of Delta parallel robotcan be divided into two types one is kinematics controlwhich ignores the centripetal force Coriolis force and allkinds of disturbances of the robot It directly controls therotation angle of the servo motor through the given rotationangle of trajectory planning which is mainly used in the low-speed parallel robot the other is dynamics control Underthe condition of high-speed motion if the centripetal forceCoriolis force and all kinds of disturbances are ignored the

operation accuracy of the robot will decrease and the jointsof the robot will vibrate )erefore the dynamics controlmethod is designed to improve the dynamic response of therobot system and effectively control all kinds of forces anddisturbances in high-speed motion which realize that thecontrol of the robot in high-speed motion accuracy is ofgreat importance )e controls of three joints of Deltaparallel robot are coupled with each other and the controlobject has nonlinear characteristics )erefore its controlhas always been a difficult issue in the field of parallel robotsthe quality of control strategy directly affects the quality oftrajectory tracking the speed and operation accuracy of theparallel robot

)e existing control strategies that have been applied tothe parallel robot include PID control [7ndash11] calculatedtorque control [12ndash17] and sliding mode variable structurecontrol [18ndash20] )ese methods have higher requirementson the model of the control object and the operatingconditions and operating environment are deterministic Inorder to improve the control performance some scholars

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 9681530 12 pageshttpsdoiorg10115520209681530

combined the intelligent control strategy with the traditionalcontrol strategy on control of the Delta parallel robot[21ndash28]

It is generally accepted that classical PID control hasgood robustness and reliability and it is easy to be realized Itplays an important role in single input and single outputapplications )e outputs of three joints of Delta parallelrobot are coupled with each other and the control isnonlinear )e classical PID control is difficult to ensurehigh-precision and high-speed trajectory tracking and haspoor robustness )erefore scholars at home and abroad arecommitted to researching the improved PID controlmethod

In literature [7ndash11] starting from the robot joint or jointdrive servo system PD control PD+ speed feed-forwardPD+position feed-forward a nonlinear combination of PIDparameters online optimization of PID parameters fuzzyself-tuning PID and other controllers as well as variousdynamic compensation controllers are used to enable therobot joints to track the given trajectory curve )e methodof calculating torque control is to simplify the dynamicmodel of each joint and carry out decoupling control whichcan achieve linear control of each joint In literature [12ndash17]the dynamics control of the robot is realized based on thecomputational torque control method which achieves a goodtrajectory tracking control In order to improve the trackingperformance of the system various improvements andcompensations are made to predict or calculate the torque inthe literature and the robustness of the system is analyzedIn the nonlinear control system the sliding mode variablestructure control strategy is very popular with researchersbut the controller itself has the problems of chattering andeasy to be disturbed In literature [18ndash20] a new adaptivevariable structure controller a new smooth sliding modecontrol algorithm and a combination of REF neural net-work and sliding mode control algorithm are used to solvethe chattering problem of the traditional sliding modecontrol with strong robustness Since the advent of artificialintelligence control technology in the 1960s an increasingnumber of control systems have been added to the elementsof artificial intelligence )ere are many active figures ofartificial intelligence control technology in control of robottrajectory tracking of the robot Experts and scholars haveapplied various neural networks predictive control fuzzyalgorithm expert system and other technologies to controlof robot trajectory tracking of the robot which achieve goodresults Moreover the performance of the controller isimproved by combining the artificial intelligence methodwith PID control calculation torque method and slidingmode variable structure control such as the achievements inliterature [11 20 27ndash29] However the aforementionedmethods excessively depend on the model and are difficult tobe realized due to the complexity of calculation Conse-quently these methods cannot meet the requirements ofhigh-precision and high-speed control of high-speed parallelrobot

In literature [30] the method of combining the ActiveDisturbance Rejection Controller (ADRC) with General-ized Proportional Integral (GPI) was adopted ADRC has

the advantage of being model independent and General-ized Proportional Integral Observer (GPIO) was used in-stead of Extended State Observer (ESO) which has a betterperformance comparing with the calculated torqueHowever the entire system has many parameters to betuned which can achieve good results at a specific loadtime Nevertheless the performance of the system de-creases when the load changes and the parameters cannotbe optimized

In the last few years Professor Gao Ziqiang proposed thelinear active disturbance rejection control method [31](LADRC) which mainly deals with the linearization of theExtended State Observer and the error feedback combinedcontroller which is convenient for stability analysis andparameter tuning by using the frequency domain method)e LADRC has been widely used in a wider range owing toits advantages of simple parameter tuning and strong anti-interference ability and more achievements have beenachieved in some literature [32ndash38] Motivated by theaforementioned analysis this paper proposes an LADRCstrategy to apply to the dynamic stability control of Deltahigh-speed parallel robot for the first time Simulation re-sults demonstrate that the system with LADRC strategy hasthe performance of a good trajectory tracking and strongrobustness

2 Structure and Dynamics Modeling of DeltaHigh-Speed Parallel Robots

21 Structure of Delta High-Speed Parallel Robots In thispaper the structure sketch of Delta high-speed parallel robotis depicted in Figure 1 A1A2A3 is a fixed platform B1B2B3is a mobile platform Ai is a rotating joint Bi and Ci arespherical joints AiCi is an active arm and CiBi is a drivenarm i 1 2 3 O is the center of the fixed platform and P isthe center of the mobile platform R is the outer circle radiusof the fixed platform and r is the outer circle radius of themobile platform Both A1A2A3 and B1B2B3 are regulartriangles

22 Dynamics Modeling of Delta High-Speed Parallel RobotsAccording to the literature [30] the dynamics equation ofdelta high-speed parallel robot can be expressed by

M(θ)euroθ + C(θ _θ) _θ + G(θ) τ (1)

Add friction force and disturbance to formula (1) andformula (2) can be obtained as

M(θ)euroθ + C(θ _θ)euroθ + G(θ) + F( _θ) + τ d τ (2)

where θ isin Rn is the rotation angle of the three joints of Deltahigh-speed parallel robot M(θ) isin Rnxn is inertia matrix ofthe three joints of Delta high-speed parallel robot C(θ _θ) iscentripetal force and Coriolis force G(θ) isin Rn is the gravityterm F( _θ) isin Rn is friction τ d is disturbance term and τ isservo input

Formula (2) is transformed into formula (3) as

2 Mathematical Problems in Engineering

euroθ minusMminus1

C(θ _θ) _θ minus Mminus1

G(θ) minus Mminus1

F( _θ) minus Mminus1

(θ)τ d + Mminus1τ

(3)

3 Study on LADRC Strategy

31 Scheme of LADRC )e mechanical structure of theDelta high-speed parallel robot is complex and the threejoints are coupled with each other )e control structure isshown in Figure 2 X1(s)X2(s) and X3(s) are inputs ofDelta high-speed parallel robot Y1(s) Y2(s) and Y3(s) areoutputs of the Delta high-speed parallel robot When oneXi(s) is changed at the same time the other two Xi(s) willfollow it and three Yi(s) will change )ereby in order torealize the trajectory tracking control of Delta high-speedparallel robot three joints need to be input at the same timeand decoupled each other

When the parallel robot is running the decomposed loadon each joint changes nonlinearly with the change of poseand angular acceleration which cannot be expressed line-arly)erefore PID control cannotmeet the requirements ofthe high-speed and high-precision control system of Deltahigh-speed parallel robot According to the motion re-quirements and nonlinear characteristics of Delta high-speed parallel robot this paper adopts the LADRC strategyto control Delta high-speed parallel robot An 8-shapedspace helical variance trajectory as the input signal is used toverify the trajectory tracking of the control strategy

Figure 3 shows the LADRC strategy)e LADRC strategyhas strong robustness in that many uncertain factors areomitted in dynamics modeling In practical control these areuncertain factors of control while the LADRC strategy doesnot depend on the establishment of the dynamics modelwhich circumvents the error problem of the dynamics modelof the parallel robot Control idea is to control each jointseparately the coupling parameters between the joints theomitted factors in modeling and the disturbance in actualoperation are all treated as the disturbance to compensate

Finally the high-precision trajectory tracking control of Deltahigh-speed parallel robot is realized

32Designof SystemController )edynamics of Delta high-speed parallel robot are expressed by formula (3) which canbe converted into

euroθ minusa1_θ minus a2θ + f0(θ _θω t) + bμ (4)

where μ is the input of the system θ is the output of thesystem minusa1

_θ minus a2θ is known modeling dynamics of objectf0(θ _θω t) is unknown modeling dynamics of objects andsum of external disturbances b is the uncertain control gainand b0 is the approximate value of uncertain control gain bletf(θ _θω t) f(middot) minus a1

_θ minus a2θ + f0(θ _θω t) + (b minus b0)μwhich is called ldquototal disturbancerdquo

Let x1 θ x2 _θ and x3 f(middot) Formula (4) expressesstate equation of the second-order objects which can bewritten as

_x1 x2

_x2 x3 + b0μ

_x3 h + _f

(5)

)e state equation of the LESO (Linear Extended StateObserver) designed for formula (3) is expressed as follows

e z1 minus θ

_z1 z2 minus β1f1(e)

_z2 z3 minus β2f2(e) + b0μ

_z3 minusβ3f3(e)

(6)

μ minusz3 + μ0

b0 (7)

where L is the gain of LESO L β1 β2 β31113858 1113859T

fi(e) z1 minus θ (i 1 2 3) and β1 β2 and β3 are selected

B1

B2

B3

C3 C1

C2

A3

A2

A1

LBC

O R

r

Mobile platform

Driven arm

Active arm

Fixed platform

P

Figure 1 Mechanism sketch of three-degree-of-freedom Deltahigh-speed parallel robots

+

+

+ +

+

+

+

++

+ +

X1 (s)

X2 (s)

X3 (s)

Y1 (s)

Y2 (s)

Y3 (s)

G11 (s)

G12 (s)

G13 (s)

G21 (s)

G22 (s)

G33 (s)

G31 (s)

G32 (s)

G23 (s)

Figure 2 Control system structure of Delta high-speed parallelrobot

Mathematical Problems in Engineering 3

appropriate gains of LESO respectively LESO can track allvariables including z1 θ z2 _θ and z3 f(middot) in thesystem expressed through formula (6) in real time

Linear control law can be expressed by

u0 kp v minus z1( 1113857 + k d _v minus z2( 1113857 (8)

According to formula (6)

z1 β1s

2+ β2s + β3

s3

+ β1s2

+ β2s + β3θ +

b0s

s3

+ β1s2

+ β2s + β3μ

z2 β2s + β3( 1113857s

s3

+ β1s2

+ β2s + β3θ +

b0 s + β1( 1113857s

s3

+ β1s2

+ β2s + β3μ

z3 β3s

2

s3

+ β1s2

+ β2s + β3θ minus

b0β3s3

+ β1s2

+ β2s + β3μ

(9)


μ0 1b0

kp v minus z1( 1113857 + k d _v minus z2( 1113857 minus z31113960 1113961 (10)

Substitute z1z2 and z3into formula (11) as follows

μ 1b0

s3

+ β1s2

+ β2s + β3s3

+ β1 + k d( 1113857s2

+ β2 + kp + k dβ11113872 1113873skp + k ds1113872 1113873v

kpβ1 + k dβ2 + β31113872 1113873s2

+ kpβ2 + k dβ31113872 1113873s + kpβ3s3 + β1s

2+ β2s + β3

⎡⎣ ⎤⎦ (11)

321 Stability Analysis From equation (13) the singleclosed-loop structure diagram is shown in Figure 4

One has

G1(s) kp + k ds (12)

G2(s) s3

+ β1s2

+ β2s + β3s3

+ β1 + k d( 1113857s2

+ β2 + kp + k dβ11113872 1113873s (13)

H(s) kpβ1 + k dβ2 + β31113872 1113873s

2+ kpβ2 + k dβ31113872 1113873s + kpβ3

s3

+ β1s2

+ β2s + β3

(14)

From the structure diagram in Figure 4 the closed-looptransfer function can be obtained as

Gb(s) G1(s)G2(s)G(s)( 1113857b0( 1113857

1 + G2(s)G(s)( 1113857b0)H(s)((15)

(1) 6e Scenario of Precisely Known Controlled Object It canbe seen from the above analysis that dynamics model for-mula (3) of the parallel robot is established by omitting themoment of inertia and the friction between joints therebythe model is considered to be known

Theorem 1 When the model of the controlled object isknown accurately the differential tracker does not affect thestability of the system but only affects the zero point of thesystem Reasonable selection of b0 kp k d β1 β2 and β3 canmake the system stable

Proof If the object model is known precisely its transferfunction is

G(s) k0

s2

+ a1s + a2 (16)

Substituting formulae (14)ndash(18) into formula (17) oneobtains

Trajectory planning LADRC

LADRC

LADRC

Deltaparallel robot

Xd Xd

θd θd

θ θ

θd2 θd2

θ2 θ2

θd3 θd3

θ3 θ3

θd1 θd1

θ1 θ1

Figure 3 Delta high-speed parallel robot based on LADRC

G1 (s) G2 (s) G (s)1b0

H (s)

f0 ()yv

ndash

ndash

Figure 4 )e single closed-loop structure diagram


Gb(s) kp + k ds1113872 1113873 s

3+ β1s

2+ β2s + β31113960 1113961k0

b0 s3

+ β1 + k d( 1113857s2

+ β2 + kp + k dβ11113872 1113873s1113960 1113961 s2

+ a1s + a21113960 11139611113966 1113967 + k0 kpβ1 + k dβ2 + β31113872 1113873s2

+ kpβ2 + k dβ31113872 1113873s + kpβ31113960 1113961

(17)

Closed-loop characteristic equation is presented as

D(s) b0s5

+ b0 β1 + k d + a1( 1113857s4

+ b0 β2 + kp + k dβ1 + a1 β1 + k d( 1113857 + a21113872 1113873s3

+ b0a1 β2 + kp + k dβ11113872 1113873 + b0a2 β1 + k d( 1113857 + k0 kpβ1 + k dβ2 + β31113872 11138731113960 1113961s2

+ b0a2 β2 + kp + k dβ11113872 11138731113960

+ k0 kpβ2 + k dβ31113872 11138731113961s + k0kpβ3

(18)

)e following constraint can be specified

D0 b0

D1 b0 β1 + k d + a1( 1113857

D2 b0 β2 + kp + k dβ1 + a1β1 + a1k d + a21113872 1113873

D3 b0a1 β2 + kp + k dβ11113872 1113873 + b0a2 β1 + k d( 11138571113960

+ k0 kpβ1 + k dβ2 + β31113872 11138731113961

D4 b0a2 β2 + kp + k dβ11113872 1113873 + k0 kpβ2 + k dβ31113872 11138731113960 1113961

D5 k0kpβ3

(19)

From Routh criterion the stability of the system can beobtained as

s5

D0 D2 D4

s4

D1 D3 D5

s3

B31 B32

s2

B41 B42

s1

B51

s0

B61

B31 D1D2 minus D0D3

D1

B32 D1D4 minus D0D5

D1

B41 B31D3 minus D1B32

B31

B42 D5

B51 B41B32 minus B31D5

B41

B61 D5

(20)

(2) 6e Scenario of Unknown Parameters of the ControlledObject When Delta high-speed parallel robot runs at a highspeed the coupling relationships between joints are sig-nificantly relevant to the effects of the moment of inertiaand friction between joints )ey will result in the systemchatter in the process of operation and ultimately destroythe stability of the system and decrease trajectory trackingperformance if these impact factors were not taken intoaccount )erefore in the design of the LADRC strategythese factors are taken into consideration as unknownmodel parameters of the system when carrying outmodeling

)e stability of the systemmodel is proved as follows Letthe nominal model of the object be Gn and then the actualobject is G(s) Gn(1 + δG(s)) δG is the perturbation ofnominal model and meets |δG(jω)|le δ(Gω)δG(ω) isbounded the uncertainty of multiplicative norm

Owing to the closed-loop characteristic equation1 + G2(s)(1b0)G(s)H(s) 0 the following expression isobtained as

1 + G2(s)1b0

Gn(s)(1 + δG)H(s) 0 (21)

According to robust stability criterion for arbitraryωthe inequality satisfying formula (21) is as follows

b0 + G2(s)Gn(s)H(s) + G2(s)Gn(s)H(s)δG(s) 0

δG(ω)ltΔG(ω) b0 + G2(s)Gn(s)H(s)

G2(s)Gn(s)H(s)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(22)

)e conditions given by formula (22) can stabilize thesystem

It can be seen that selecting b0 kp k d β1 β2 and β3properly can make the system stable which can guaranteethat the system has the stability and certain robustness

It can be concluded from the above stability proof thatthe LADRC strategy not only realizes the dynamics controlof the trajectory tracking of Delta high-speed parallel robotbut also guarantees that the uncertain factors of the modelwill not affect the stability of the system at high-speedoperation


322 Design of Delta High-Speed Parallel Robots )econtroller structure of the LADRC strategy is shown inFigure 5

As shown in Figure 5 the LADRC strategy only needs tobe set by setting up stability conditions of LESO which areβ1 gt 0 β2 gt 0 and β3 gt 0 and meet β1β2 gt β3

As depicted in Figure 2 Delta high-speed parallel robotis composed of three inputs X1(s) X2(s) and X3(s) andthree outputs Y1(s) Y2(s) and Y3(s) which are coupledwith each other and can be constructed as shown in thefollowing formula

Y1(s) G11(s)X1(s) + G12(s)X2(s) + G13(s)X3(s)

Y2(s) G21(s)X1(s) + G22(s)X2(s) + G23(s)X3(s)

Y3(s) G31(s)X1(s) + G32(s)X2(s) + G33(s)X3(s)

⎧⎪⎪⎨

⎪⎪⎩

(23)

According to formulas (6) and (23) the system equationof Delta high-speed parallel robot can be obtained asexpressed in

euroθ1 f1 θ1 _θ1 μ1(t) t1113872 1113873 + k1μ1

euroθ2 f2 θ2 _θ2 μ2(t) t1113872 1113873 + k2μ2

euroθ3 f1 θ3 _θ3 μ3(t) t1113872 1113873 + k3μ3

(24)

where f1(θ1 _θ1 μ1(t) t) f2(θ2 _θ2 μ2(t) t) and(f3(θ3 _θ3 μ3(t) t)) are equivalent comprehensive distur-bance which are the comprehensive functions of the cou-pling term between the three axes the uncertainty term inhigh-speed motion and various disturbance terms in thefield )e control structure block diagram is shown inFigure 6

4 Simulation and Results Analysis

In this section to verify the performance of the controllerbased on the proposed LADRC strategy the primary pa-rameters of Delta high-speed parallel robot are summarizedin Table 1

An 8-shaped trajectory that is a representative spacehelical variance trajectory is taken as inputs of the systemwhich is expressed as

x1 t

x2 800 sin 03t +1360

1113874 1113875lowast π1113874 1113875

x3 800 sin(06tlowast π)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)

)e pictures of different views corresponding to formula(25) are depicted in Figure 7

For obtaining trajectories under some conditions thefirst step is to verify the performance of the system whenthe disturbance is not introduced and then verify theperformance of the system when the disturbance is in-troduced To better estimate the control effect the angle

limit time (speed) limit and space limit of the robot areremoved

Case 1 )e case of 8-shaped trajectory input withoutdisturbance

According to zi(i 1 2 3) in formula fd9(9) stabilityconditions of LESO must meetβ1 gt 0 β2 gt 0 β3 gt 0 andβ1β2 gt β3 On this basis it is concluded that coef-ficientsβ1 1 β2 65000 and β3 600 through a largenumber of simulation experiments )e trajectory of Deltahigh-speed parallel robot is set as the 8-shaped trajectory)e given trajectories of joint 1 joint 2 and joint 3 are setaccording to formula (25) According to Figure 6 thesimulation curve of trajectory tracking control is obtained inFigure 8 the blue circle denotes the given circular trajectoryand the red curve denotes the trajectory controlled by theLADRC strategy

In order to verify the robustness of the controller againstdisturbance a triangular wave external disturbance with anamplitude of 50mm and a period of 02Hz is added after theDelta high-speed parallel robot runs to 5 s )e disturbancesignal is shown in Figure 9

Figure 10 shows the tracking trajectories of three jointsin the scenario of only PID control without the LADRCstrategy

)e tracking trajectory of only using PID control underthe condition of without the LADRC strategy is obtained inFigure 11

From Figures 10 and 11 it is very obvious that only usingPID control completely cannot control the tracking tra-jectory )erefore it is very necessary to introduce theLADRC strategy to resolve the antidisturbance of PIDcontrol Figures 12 and 13 show the effects of trackingtrajectory control when the LADRC strategy is used

In Figure 13 the fluctuation of the three-dimensionaltrajectory is not obvious and the 8-shaped trajectory isslightly vibrated It is verified that the LADRC strategy canrealize the decoupling control of three joints and has strongrobustness under the disturbance of the triangular wave for arepresentative space helical variance trajectory of 8-shapedtrajectory input

Simulation results show that the LADRC strategy canrealize the decoupling control of three joints and realize thequick and precise real-time tracking of the given trajectoryand has good robustness

TDControl

lawb0

G (s)v

z1

z2

3z

y

minus

minusminus e

err

LESO

minus

1s

1b

1s

1s

μ0 μ

β1

β2

β3

Figure 5 Structure diagram of LADRC for the second-ordersystem


TD

Delt

a hig

h-sp

eed

para

llel r

obot

minus

minusminus minus

LESO 2

TD Control law 1

minus

minusminus e1

e1

minus

LESO 1

v1

z11 z21z31

z32

z33

z22

z23z13

z12

v2

v3 TD Control law 2

Control law 2

minus

minusminus minus

LESO 3

μ2

μ1

3micro

μ01

μ02

μ03

1b01

1b02

1b03

b01

b02

r2r2

r3

e2

e3

e3

r3

e2

b03

Figure 6 Simulation structure block diagram

Table 1 Primary parameters of Delta high-speed parallel robot for simulation

Parameters ValueRated output power of servo motor 750WRated speed of servo motor 3000 rpmRated torque of servo motor 238M middot mMoment of inertia of servo motor 159 times 10minus4 (kgm2)

Reduction ratio 20 1Mass of active arm 235 kgMass of driven arm 09 kgLength of active arm 400mmLength of driven arm 1000mm

1000500 1000

500Position y (mm)

0

Position x (mm)0ndash500 ndash500ndash1000 ndash1000

05

101520253035

Posit

ion z (

mm

)

(a)

ndash800 ndash600 ndash400 ndash200 0 200 400 600 800 1000ndash1000Position x (mm)

ndash1000

ndash800

ndash600

ndash400

ndash200

0

200

400

600

800

1000

Posit

ion y (

mm

)

(b)

Figure 7 Continued


ndash5

0

5

10

15

20

25

30

35Po

sitio

n z (

mm

)

ndash800 ndash600 ndash400 ndash200 0 200 400 600 800 1000ndash1000Position y (mm)

(c)

ndash5

0

5

10

15

20

25

30

35

Posit

ion z (

mm

)


(d)

Figure 7 8-shaped trajectory (a))e space trajectory in 3D coordination system (b))e projection on the x-y plane (c))e projection onthe y-z plane (d) )e projection on the x-z plane

1000

500

0

ndash500

Posit

ion y

(mm

)

ndash1000

0

5

10

15

20

25

30

35

Posit

ion z (

mm

)

1000

Position x (mm)

800 600 400 200 0 ndash200 ndash400 ndash600 ndash800ndash1000

Given trajectoryTracking trajectory

Figure 8 Trajectories without disturbance

ndash60

ndash40

ndash20

0

20

40

60

Am

plitu

de (m

m)

5 10 15 20 25 300Time (s)

Figure 9 Disturbance signal


Joint 1

010203040

Posit

ion

(mm

)5 10 15 20 25 300

Time (s)


(a)

Joint 2

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(b)

Joint 3

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(c)

Figure 10 Trajectories of 3 joints without the LADRC strategy (a) Trajectories of joint 1 (b) Trajectories of joint 2 (c) Trajectories of joint 3

1000500 1000

500Position y (mm)

0


05

101520253035

Posit

ion z (

mm

)


Figure 11 )e tracking trajectory of without the LADRC strategy


Joint 1

010203040

Posit

ion

(mm

)5 10 15 20 25 300

Time (s)Given trajectoryTracking trajectory

(a)

Joint 2

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(b)

Joint 3

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(c)

Figure 12 Trajectories of 3 joints with the LADRC strategy (a) Trajectories of joint 1 (b) Trajectories of joint 2 (c) Trajectories of joint 3

1000500 1000

500Position y (mm)

0



05

101520253035

Posit

ion z (

mm

)

Figure 13 )e tracking trajectory with the LADRC strategy


5 Conclusion

Aiming at the motion requirements nonlinear character-istics and the omission of the relevant factors for modelingDelta high-speed parallel robot this paper proposes anLADRC strategy to be applied to the dynamic control ofDelta high-speed parallel robot )rough the stabilityanalysis the LADRC strategy can realize the dynamiccontrol of the trajectory tracking of Delta high-speed parallelrobot and when the robot is running at high speed thestability of the model will not be affected by the uncertainfactors of the system

In order to verify the proposed LADRC strategy thecircular trajectory a representative 8-shaped space helicalvariance trajectory and a triangular wave external distur-bance with large amplitude are input respectively )e PIDmethod and the linear ADRC control strategy are used tosimulate respectively )e simulation and comparativeanalysis show that the LADRC strategy is applied to theDelta high-speed parallel robot control which has a goodquick and precise real-time trajectory tracking and strongrobustness

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was supported by the Education Department ofJilin Province (Grant JJKH20200044KJ) the Jilin ProvincialDevelopment and Reform Commission (Grant 2018C035-1) Jilin Provincial Science and Technology Department(Grants 20160101276JC and 20150312040ZG) and Project ofBeihua University (Grant 201901012)

References

[1] Z X Cai Fundamentals of Robotics Mechanical IndustryPress Beijing China 2009

[2] J P Merlet Parallel Robots Springer Press DordrechtNetherlands 2006

[3] S Cong and W W Shang Parallel Robot Modeling ControlOptimization and Application Electronic Industry PressBeijing China 2009

[4] X S Chen Z L Chen and M X Kong ldquo)e developmentand current situation of parallel robot researchrdquo Robotvol 24 no 5 pp 464ndash470 2002

[5] B J Yang G Q Cai J M Luo et al ldquoResearch status of lowDOF parallel robotsrdquo Machine Tool and Hydraulic vol 34no 5 pp 202ndash205 2006

[6] Q L Ai S J Zu and F Xu ldquoResearch progress on kinematicsand singularity of parallel mechanismrdquo Journal of ZhejiangUniversity Engineering Edition vol 46 no 8 pp 1345ndash13592012

[7] U )akar V Joshi and V V Yawahare ldquoFractional-order PIcontroller design for PMSM a model-based comparativestudyrdquo in Proceedings of the International Conference onAutomatic Control amp Dynamic Optimization TechniquesIEEE Pune India September 2017

[8] Q Zhao PWang and J Mei ldquoController parameter tuning ofdelta robot based on servo identificationrdquo Chinese Journal ofMechanical Engineering vol 28 no 2 pp 267ndash275 2015

[9] Y QWang and T Huang ldquoResearch on the parameter tuningmethod of Delta mechanism controllerrdquo Mechanical Designand Manufacturing vol 08 pp 20ndash22 2008

[10] Y Su D Sun L Ren and J K Mills ldquoIntegration of saturatedPI synchronous control and PD feedback for control ofparallel manipulatorsrdquo IEEE Transactions on Robotics vol 22no 1 pp 202ndash207 2006

[11] X-G Lu M Liu and J-X Liu ldquoDesign and optimization ofinterval type-2 fuzzy logic controller for delta parallel robottrajectory controlrdquo International Journal of Fuzzy Systemsvol 19 no 1 pp 190ndash206 2017

[12] F Pierrot V Nabat O Company S Krut and P PoignetldquoOptimal design of a 4-DOF parallel manipulator from ac-ademia to industryrdquo IEEE Transactions on Robotics vol 25no 2 pp 213ndash224 2009

[13] H Z Huang Research on Structural Parameter Optimizationand Motion Control of Delta Parallel Robot Harbin Instituteof Technology Harbin China 2013

[14] M Rachedi M Bouri and B Hemici ldquoApplication of an Hinfincontrol strategy to the parallel deltardquo in Proceedings of the2012 2nd International Conference Communications Com-puting and Control Applications (CCCA) December 2012

[15] M Rachedi B Hemici and M Bouri ldquoApplication of themixed sensitivity problem Hinfin and H2 to the parallel deltardquoin Proceedings of the 3rd International Conference on Systemsamp Control October 2013

[16] M Rachedi M Bouri and B Hemici ldquoHinfin feedback controlfor parallel mechanism and application to delta robotrdquo inProceedings of the 22nd Mediterranean Conference on Controland Automation (MED) pp 1476ndash1481 University ofPalermo Palermo Italy IEEE June 2014

[17] X B Guo Kinematic Planning and Dynamic Control of DeltaParallel Robot Technology of Guangdong UniversityGuangdong China 2015

[18] G Q Gao Q Q Ding and W Wang ldquoApplication of RBFneural network sliding mode variable structure control inparallel robotrdquo Industrial Instrumentation and Automationvol 02 pp 35ndash39 2012

[19] K Y Bi Research on the Control of a New 6ree TranslationParallel Robot Mechanism Jiangsu University Jiangsu China2006

[20] G Q Gao Z M Fang X M Niu et al ldquoResearch on thesmooth sliding mode control of a new three translationparallel robot mechanismrdquo Modular Machine Tool and Au-tomatic Machining Technology vol 11 pp 81ndash83 2004

[21] C Y Lian J J Zhang L H Shi et al ldquoJoint simulation of 3-DOF parallel robot based on fuzzy controlrdquoMachine Tool andHydraulic vol 39 no 09 pp 122ndash125 2011

[22] O Linda and M Manic ldquoUncertainty-Robust design of in-terval type-2 fuzzy logic controller for Delta parallel robotrdquoIEEE Transactions on Industrial Informatics vol 7 no 4pp 661ndash670 2011

[23] Y C Huang and Z L Huang ldquoNeural network based dy-namic trajectory tracking of delta parallel robotrdquo in Pro-ceedings of International Conference on Mechatronics andAutomation pp 1938ndash1941 Beijing China IEEE 2015


[24] K Zheng and C Wang ForcePosition Hybrid Control of6PUS-UPU Redundant Driven Parallel Manipulator Based on2-DOF Internal Model Control Hindawi Publishing CorpLondon UK 2014

[25] J Z Hui and Y K Yang ldquoResearch on nonlinear dynamiccontrol method of 2-DOF redundant drive parallel robotrdquoMechanical Engineering and Technology vol 5 no 2pp 150ndash164 2016

[26] A Vivas and P Poignet ldquoPredictive functional control of aparallel robotrdquo Control Engineering Practice vol 13 no 7pp 863ndash874 2005

[27] M A Mirza S Li and L Jin ldquoSimultaneous learning andcontrol of parallel stewart platforms with unknown param-etersrdquo Neurocomputing vol 266 pp 114ndash122 2017

[28] J K Liu MATLAB Simulation of Advanced PID ControlElectronic Industry Press Beijing China 3rd edition 2011

[29] Y C Fan Y S Li Q Yu et al ldquoFuzzy adaptive PID controlsimulation of Stewart parallel robot based on ADAMS andMATLABrdquo Instruments and Equipment vol 3 no 3pp 63ndash71 2015

[30] Mario Ramırez-NeriaHebertt et al ldquoActive disturbance re-jection control applied to a Delta parallel robot in trajectorytracking tasksrdquo Asian Journal of Control vol 17 no 2pp 636ndash647 2015

[31] Z Gao ldquoScaling and bandwidth-parameterization basedcontroller tuningrdquo in Proceedings of the American ControlConference pp 4989ndash4997 Denver CO USA June 2003

[32] H Y Jin L L Liu and W Y Lan ldquoStability conditions oflinear ADRC for second order systemsrdquo Journal of Auto-mation vol 44 no 09 pp 1725ndash1728 2018

[33] Z Q Chen Y Cheng M W Sun et al ldquoSome progress oflinear ADRC theory and engineering applicationrdquo Informa-tion and Control vol 46 no 03 pp 257ndash266 2017

[34] D Yuan X J Ma Q H Zeng et al ldquoStudy on the frequencycharacteristics and parameter configuration of the linearADRC for the second order systemrdquo Control 6eory andApplication vol 30 no 12 pp 1630ndash1640 2013

[35] R Zhang ldquoParameter identification based on series extendedstate observerrdquo Journal of System Simulation vol 06pp 793ndash795 2002

[36] P W Li Application of Advanced Control 6eory in ControlSystem of Cold Mill Science and Technology of BeijingUniversity Beijing China 2019

[37] B Liu J Hong and L Wang ldquoLinear inverted pendulumcontrol based on improved ADRCrdquo Systems Science amp ControlEngineering vol 7 no 3 pp 1ndash12 2019

[38] Y Liu H Liu and Y Meng ldquoActive disturbance rejectioncontrol for a multiple-flexible-link manipulatorrdquo Journal ofHarbin Institute of Technology vol 25 no 1 pp 18ndash28 2018


combined the intelligent control strategy with the traditionalcontrol strategy on control of the Delta parallel robot[21ndash28]

It is generally accepted that classical PID control hasgood robustness and reliability and it is easy to be realized Itplays an important role in single input and single outputapplications )e outputs of three joints of Delta parallelrobot are coupled with each other and the control isnonlinear )e classical PID control is difficult to ensurehigh-precision and high-speed trajectory tracking and haspoor robustness )erefore scholars at home and abroad arecommitted to researching the improved PID controlmethod

In literature [7ndash11] starting from the robot joint or jointdrive servo system PD control PD+ speed feed-forwardPD+position feed-forward a nonlinear combination of PIDparameters online optimization of PID parameters fuzzyself-tuning PID and other controllers as well as variousdynamic compensation controllers are used to enable therobot joints to track the given trajectory curve )e methodof calculating torque control is to simplify the dynamicmodel of each joint and carry out decoupling control whichcan achieve linear control of each joint In literature [12ndash17]the dynamics control of the robot is realized based on thecomputational torque control method which achieves a goodtrajectory tracking control In order to improve the trackingperformance of the system various improvements andcompensations are made to predict or calculate the torque inthe literature and the robustness of the system is analyzedIn the nonlinear control system the sliding mode variablestructure control strategy is very popular with researchersbut the controller itself has the problems of chattering andeasy to be disturbed In literature [18ndash20] a new adaptivevariable structure controller a new smooth sliding modecontrol algorithm and a combination of REF neural net-work and sliding mode control algorithm are used to solvethe chattering problem of the traditional sliding modecontrol with strong robustness Since the advent of artificialintelligence control technology in the 1960s an increasingnumber of control systems have been added to the elementsof artificial intelligence )ere are many active figures ofartificial intelligence control technology in control of robottrajectory tracking of the robot Experts and scholars haveapplied various neural networks predictive control fuzzyalgorithm expert system and other technologies to controlof robot trajectory tracking of the robot which achieve goodresults Moreover the performance of the controller isimproved by combining the artificial intelligence methodwith PID control calculation torque method and slidingmode variable structure control such as the achievements inliterature [11 20 27ndash29] However the aforementionedmethods excessively depend on the model and are difficult tobe realized due to the complexity of calculation Conse-quently these methods cannot meet the requirements ofhigh-precision and high-speed control of high-speed parallelrobot

In literature [30] the method of combining the ActiveDisturbance Rejection Controller (ADRC) with General-ized Proportional Integral (GPI) was adopted ADRC has

the advantage of being model independent and General-ized Proportional Integral Observer (GPIO) was used in-stead of Extended State Observer (ESO) which has a betterperformance comparing with the calculated torqueHowever the entire system has many parameters to betuned which can achieve good results at a specific loadtime Nevertheless the performance of the system de-creases when the load changes and the parameters cannotbe optimized

In the last few years Professor Gao Ziqiang proposed thelinear active disturbance rejection control method [31](LADRC) which mainly deals with the linearization of theExtended State Observer and the error feedback combinedcontroller which is convenient for stability analysis andparameter tuning by using the frequency domain method)e LADRC has been widely used in a wider range owing toits advantages of simple parameter tuning and strong anti-interference ability and more achievements have beenachieved in some literature [32ndash38] Motivated by theaforementioned analysis this paper proposes an LADRCstrategy to apply to the dynamic stability control of Deltahigh-speed parallel robot for the first time Simulation re-sults demonstrate that the system with LADRC strategy hasthe performance of a good trajectory tracking and strongrobustness

2 Structure and Dynamics Modeling of DeltaHigh-Speed Parallel Robots

21 Structure of Delta High-Speed Parallel Robots In thispaper the structure sketch of Delta high-speed parallel robotis depicted in Figure 1 A1A2A3 is a fixed platform B1B2B3is a mobile platform Ai is a rotating joint Bi and Ci arespherical joints AiCi is an active arm and CiBi is a drivenarm i 1 2 3 O is the center of the fixed platform and P isthe center of the mobile platform R is the outer circle radiusof the fixed platform and r is the outer circle radius of themobile platform Both A1A2A3 and B1B2B3 are regulartriangles

22 Dynamics Modeling of Delta High-Speed Parallel RobotsAccording to the literature [30] the dynamics equation ofdelta high-speed parallel robot can be expressed by

M(θ)euroθ + C(θ _θ) _θ + G(θ) τ (1)

Add friction force and disturbance to formula (1) andformula (2) can be obtained as

M(θ)euroθ + C(θ _θ)euroθ + G(θ) + F( _θ) + τ d τ (2)

where θ isin Rn is the rotation angle of the three joints of Deltahigh-speed parallel robot M(θ) isin Rnxn is inertia matrix ofthe three joints of Delta high-speed parallel robot C(θ _θ) iscentripetal force and Coriolis force G(θ) isin Rn is the gravityterm F( _θ) isin Rn is friction τ d is disturbance term and τ isservo input

Formula (2) is transformed into formula (3) as


euroθ minusMminus1


G(θ) minus Mminus1



(3)












_x1 x2

_x2 x3 + b0μ

_x3 h + _f

(5)


e z1 minus θ



_z3 minusβ3f3(e)

(6)

μ minusz3 + μ0

b0 (7)



B1

B2

B3

C3 C1

C2

A3

A2

A1

LBC

O R

r

Mobile platform

Driven arm

Active arm

Fixed platform

P


+

+

+ +

+

+

+

++

+ +

X1 (s)

X2 (s)

X3 (s)

Y1 (s)

Y2 (s)

Y3 (s)

G11 (s)

G12 (s)

G13 (s)

G21 (s)

G22 (s)

G33 (s)

G31 (s)

G32 (s)

G23 (s)







z1 β1s

2+ β2s + β3

s3

+ β1s2

+ β2s + β3θ +

b0s

s3

+ β1s2

+ β2s + β3μ

z2 β2s + β3( 1113857s

s3

+ β1s2

+ β2s + β3θ +

b0 s + β1( 1113857s

s3

+ β1s2

+ β2s + β3μ

z3 β3s

2

s3

+ β1s2


b0β3s3

+ β1s2

+ β2s + β3μ

(9)


μ0 1b0



μ 1b0

s3

+ β1s2

+ β2s + β3s3

+ β1 + k d( 1113857s2

+ β2 + kp + k dβ11113872 1113873skp + k ds1113872 1113873v

kpβ1 + k dβ2 + β31113872 1113873s2

+ kpβ2 + k dβ31113872 1113873s + kpβ3s3 + β1s

2+ β2s + β3

⎡⎣ ⎤⎦ (11)


One has

G1(s) kp + k ds (12)

G2(s) s3

+ β1s2

+ β2s + β3s3

+ β1 + k d( 1113857s2

+ β2 + kp + k dβ11113872 1113873s (13)

H(s) kpβ1 + k dβ2 + β31113872 1113873s

2+ kpβ2 + k dβ31113872 1113873s + kpβ3

s3

+ β1s2

+ β2s + β3

(14)


Gb(s) G1(s)G2(s)G(s)( 1113857b0( 1113857

1 + G2(s)G(s)( 1113857b0)H(s)((15)




G(s) k0

s2

+ a1s + a2 (16)



LADRC

LADRC

Deltaparallel robot

Xd Xd

θd θd

θ θ

θd2 θd2

θ2 θ2

θd3 θd3

θ3 θ3

θd1 θd1

θ1 θ1


G1 (s) G2 (s) G (s)1b0

H (s)

f0 ()yv

ndash

ndash



Gb(s) kp + k ds1113872 1113873 s

3+ β1s

2+ β2s + β31113960 1113961k0

b0 s3

+ β1 + k d( 1113857s2

+ β2 + kp + k dβ11113872 1113873s1113960 1113961 s2

+ a1s + a21113960 11139611113966 1113967 + k0 kpβ1 + k dβ2 + β31113872 1113873s2

+ kpβ2 + k dβ31113872 1113873s + kpβ31113960 1113961

(17)


D(s) b0s5

+ b0 β1 + k d + a1( 1113857s4

+ b0 β2 + kp + k dβ1 + a1 β1 + k d( 1113857 + a21113872 1113873s3


+ b0a2 β2 + kp + k dβ11113872 11138731113960

+ k0 kpβ2 + k dβ31113872 11138731113961s + k0kpβ3

(18)


D0 b0

D1 b0 β1 + k d + a1( 1113857

D2 b0 β2 + kp + k dβ1 + a1β1 + a1k d + a21113872 1113873

D3 b0a1 β2 + kp + k dβ11113872 1113873 + b0a2 β1 + k d( 11138571113960

+ k0 kpβ1 + k dβ2 + β31113872 11138731113961


D5 k0kpβ3

(19)


s5

D0 D2 D4

s4

D1 D3 D5

s3

B31 B32

s2

B41 B42

s1

B51

s0

B61

B31 D1D2 minus D0D3

D1

B32 D1D4 minus D0D5

D1


B31

B42 D5


B41

B61 D5

(20)




1 + G2(s)1b0

Gn(s)(1 + δG)H(s) 0 (21)




G2(s)Gn(s)H(s)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(22)








Y1(s) G11(s)X1(s) + G12(s)X2(s) + G13(s)X3(s)

Y2(s) G21(s)X1(s) + G22(s)X2(s) + G23(s)X3(s)

Y3(s) G31(s)X1(s) + G32(s)X2(s) + G33(s)X3(s)

⎧⎪⎪⎨

⎪⎪⎩

(23)


euroθ1 f1 θ1 _θ1 μ1(t) t1113872 1113873 + k1μ1

euroθ2 f2 θ2 _θ2 μ2(t) t1113872 1113873 + k2μ2

euroθ3 f1 θ3 _θ3 μ3(t) t1113872 1113873 + k3μ3

(24)





x1 t

x2 800 sin 03t +1360

1113874 1113875lowast π1113874 1113875


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)












TDControl

lawb0

G (s)v

z1

z2

3z

y

minus

minusminus e

err

LESO

minus

1s

1b

1s

1s

μ0 μ

β1

β2

β3



TD

Delt

a hig

h-sp

eed

para

llel r

obot

minus

minusminus minus

LESO 2

TD Control law 1

minus

minusminus e1

e1

minus

LESO 1

v1

z11 z21z31

z32

z33

z22

z23z13

z12

v2

v3 TD Control law 2

Control law 2

minus

minusminus minus

LESO 3

μ2

μ1

3micro

μ01

μ02

μ03

1b01

1b02

1b03

b01

b02

r2r2

r3

e2

e3

e3

r3

e2

b03





1000500 1000

500Position y (mm)

0


05

101520253035

Posit

ion z (

mm

)

(a)


ndash1000

ndash800

ndash600

ndash400

ndash200

0

200

400

600

800

1000

Posit

ion y (

mm

)

(b)

Figure 7 Continued


ndash5

0

5

10

15

20

25

30

35Po

sitio

n z (

mm

)


(c)

ndash5

0

5

10

15

20

25

30

35

Posit

ion z (

mm

)


(d)


1000

500

0

ndash500

Posit

ion y

(mm

)

ndash1000

0

5

10

15

20

25

30

35

Posit

ion z (

mm

)

1000

Position x (mm)




ndash60

ndash40

ndash20

0

20

40

60

Am

plitu

de (m

m)

5 10 15 20 25 300Time (s)



Joint 1

010203040

Posit

ion

(mm

)5 10 15 20 25 300

Time (s)


(a)

Joint 2

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(b)

Joint 3

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(c)


1000500 1000

500Position y (mm)

0


05

101520253035

Posit

ion z (

mm

)




Joint 1

010203040

Posit

ion

(mm

)5 10 15 20 25 300


(a)

Joint 2

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(b)

Joint 3

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(c)


1000500 1000

500Position y (mm)

0



05

101520253035

Posit

ion z (

mm

)



5 Conclusion



Data Availability




Acknowledgments


References









































euroθ minusMminus1


G(θ) minus Mminus1



(3)












_x1 x2

_x2 x3 + b0μ

_x3 h + _f

(5)


e z1 minus θ



_z3 minusβ3f3(e)

(6)

μ minusz3 + μ0

b0 (7)



B1

B2

B3

C3 C1

C2

A3

A2

A1

LBC

O R

r

Mobile platform

Driven arm

Active arm

Fixed platform

P


+

+

+ +

+

+

+

++

+ +

X1 (s)

X2 (s)

X3 (s)

Y1 (s)

Y2 (s)

Y3 (s)

G11 (s)

G12 (s)

G13 (s)

G21 (s)

G22 (s)

G33 (s)

G31 (s)

G32 (s)

G23 (s)







z1 β1s

2+ β2s + β3

s3

+ β1s2

+ β2s + β3θ +

b0s

s3

+ β1s2

+ β2s + β3μ

z2 β2s + β3( 1113857s

s3

+ β1s2

+ β2s + β3θ +

b0 s + β1( 1113857s

s3

+ β1s2

+ β2s + β3μ

z3 β3s

2

s3

+ β1s2


b0β3s3

+ β1s2

+ β2s + β3μ

(9)


μ0 1b0



μ 1b0

s3

+ β1s2

+ β2s + β3s3

+ β1 + k d( 1113857s2

+ β2 + kp + k dβ11113872 1113873skp + k ds1113872 1113873v

kpβ1 + k dβ2 + β31113872 1113873s2

+ kpβ2 + k dβ31113872 1113873s + kpβ3s3 + β1s

2+ β2s + β3

⎡⎣ ⎤⎦ (11)


One has

G1(s) kp + k ds (12)

G2(s) s3

+ β1s2

+ β2s + β3s3

+ β1 + k d( 1113857s2

+ β2 + kp + k dβ11113872 1113873s (13)

H(s) kpβ1 + k dβ2 + β31113872 1113873s

2+ kpβ2 + k dβ31113872 1113873s + kpβ3

s3

+ β1s2

+ β2s + β3

(14)


Gb(s) G1(s)G2(s)G(s)( 1113857b0( 1113857

1 + G2(s)G(s)( 1113857b0)H(s)((15)




G(s) k0

s2

+ a1s + a2 (16)



LADRC

LADRC

Deltaparallel robot

Xd Xd

θd θd

θ θ

θd2 θd2

θ2 θ2

θd3 θd3

θ3 θ3

θd1 θd1

θ1 θ1


G1 (s) G2 (s) G (s)1b0

H (s)

f0 ()yv

ndash

ndash



Gb(s) kp + k ds1113872 1113873 s

3+ β1s

2+ β2s + β31113960 1113961k0

b0 s3

+ β1 + k d( 1113857s2

+ β2 + kp + k dβ11113872 1113873s1113960 1113961 s2

+ a1s + a21113960 11139611113966 1113967 + k0 kpβ1 + k dβ2 + β31113872 1113873s2

+ kpβ2 + k dβ31113872 1113873s + kpβ31113960 1113961

(17)


D(s) b0s5

+ b0 β1 + k d + a1( 1113857s4

+ b0 β2 + kp + k dβ1 + a1 β1 + k d( 1113857 + a21113872 1113873s3


+ b0a2 β2 + kp + k dβ11113872 11138731113960

+ k0 kpβ2 + k dβ31113872 11138731113961s + k0kpβ3

(18)


D0 b0

D1 b0 β1 + k d + a1( 1113857

D2 b0 β2 + kp + k dβ1 + a1β1 + a1k d + a21113872 1113873

D3 b0a1 β2 + kp + k dβ11113872 1113873 + b0a2 β1 + k d( 11138571113960

+ k0 kpβ1 + k dβ2 + β31113872 11138731113961


D5 k0kpβ3

(19)


s5

D0 D2 D4

s4

D1 D3 D5

s3

B31 B32

s2

B41 B42

s1

B51

s0

B61

B31 D1D2 minus D0D3

D1

B32 D1D4 minus D0D5

D1


B31

B42 D5


B41

B61 D5

(20)




1 + G2(s)1b0

Gn(s)(1 + δG)H(s) 0 (21)




G2(s)Gn(s)H(s)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(22)








Y1(s) G11(s)X1(s) + G12(s)X2(s) + G13(s)X3(s)

Y2(s) G21(s)X1(s) + G22(s)X2(s) + G23(s)X3(s)

Y3(s) G31(s)X1(s) + G32(s)X2(s) + G33(s)X3(s)

⎧⎪⎪⎨

⎪⎪⎩

(23)


euroθ1 f1 θ1 _θ1 μ1(t) t1113872 1113873 + k1μ1

euroθ2 f2 θ2 _θ2 μ2(t) t1113872 1113873 + k2μ2

euroθ3 f1 θ3 _θ3 μ3(t) t1113872 1113873 + k3μ3

(24)





x1 t

x2 800 sin 03t +1360

1113874 1113875lowast π1113874 1113875


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)












TDControl

lawb0

G (s)v

z1

z2

3z

y

minus

minusminus e

err

LESO

minus

1s

1b

1s

1s

μ0 μ

β1

β2

β3



TD

Delt

a hig

h-sp

eed

para

llel r

obot

minus

minusminus minus

LESO 2

TD Control law 1

minus

minusminus e1

e1

minus

LESO 1

v1

z11 z21z31

z32

z33

z22

z23z13

z12

v2

v3 TD Control law 2

Control law 2

minus

minusminus minus

LESO 3

μ2

μ1

3micro

μ01

μ02

μ03

1b01

1b02

1b03

b01

b02

r2r2

r3

e2

e3

e3

r3

e2

b03





1000500 1000

500Position y (mm)

0


05

101520253035

Posit

ion z (

mm

)

(a)


ndash1000

ndash800

ndash600

ndash400

ndash200

0

200

400

600

800

1000

Posit

ion y (

mm

)

(b)

Figure 7 Continued


ndash5

0

5

10

15

20

25

30

35Po

sitio

n z (

mm

)


(c)

ndash5

0

5

10

15

20

25

30

35

Posit

ion z (

mm

)


(d)


1000

500

0

ndash500

Posit

ion y

(mm

)

ndash1000

0

5

10

15

20

25

30

35

Posit

ion z (

mm

)

1000

Position x (mm)




ndash60

ndash40

ndash20

0

20

40

60

Am

plitu

de (m

m)

5 10 15 20 25 300Time (s)



Joint 1

010203040

Posit

ion

(mm

)5 10 15 20 25 300

Time (s)


(a)

Joint 2

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(b)

Joint 3

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(c)


1000500 1000

500Position y (mm)

0


05

101520253035

Posit

ion z (

mm

)




Joint 1

010203040

Posit

ion

(mm

)5 10 15 20 25 300


(a)

Joint 2

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(b)

Joint 3

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(c)


1000500 1000

500Position y (mm)

0



05

101520253035

Posit

ion z (

mm

)



5 Conclusion



Data Availability




Acknowledgments


References













































z1 β1s

2+ β2s + β3

s3

+ β1s2

+ β2s + β3θ +

b0s

s3

+ β1s2

+ β2s + β3μ

z2 β2s + β3( 1113857s

s3

+ β1s2

+ β2s + β3θ +

b0 s + β1( 1113857s

s3

+ β1s2

+ β2s + β3μ

z3 β3s

2

s3

+ β1s2


b0β3s3

+ β1s2

+ β2s + β3μ

(9)


μ0 1b0



μ 1b0

s3

+ β1s2

+ β2s + β3s3

+ β1 + k d( 1113857s2

+ β2 + kp + k dβ11113872 1113873skp + k ds1113872 1113873v

kpβ1 + k dβ2 + β31113872 1113873s2

+ kpβ2 + k dβ31113872 1113873s + kpβ3s3 + β1s

2+ β2s + β3

⎡⎣ ⎤⎦ (11)


One has

G1(s) kp + k ds (12)

G2(s) s3

+ β1s2

+ β2s + β3s3

+ β1 + k d( 1113857s2

+ β2 + kp + k dβ11113872 1113873s (13)

H(s) kpβ1 + k dβ2 + β31113872 1113873s

2+ kpβ2 + k dβ31113872 1113873s + kpβ3

s3

+ β1s2

+ β2s + β3

(14)


Gb(s) G1(s)G2(s)G(s)( 1113857b0( 1113857

1 + G2(s)G(s)( 1113857b0)H(s)((15)




G(s) k0

s2

+ a1s + a2 (16)



LADRC

LADRC

Deltaparallel robot

Xd Xd

θd θd

θ θ

θd2 θd2

θ2 θ2

θd3 θd3

θ3 θ3

θd1 θd1

θ1 θ1


G1 (s) G2 (s) G (s)1b0

H (s)

f0 ()yv

ndash

ndash



Gb(s) kp + k ds1113872 1113873 s

3+ β1s

2+ β2s + β31113960 1113961k0

b0 s3

+ β1 + k d( 1113857s2

+ β2 + kp + k dβ11113872 1113873s1113960 1113961 s2

+ a1s + a21113960 11139611113966 1113967 + k0 kpβ1 + k dβ2 + β31113872 1113873s2

+ kpβ2 + k dβ31113872 1113873s + kpβ31113960 1113961

(17)


D(s) b0s5

+ b0 β1 + k d + a1( 1113857s4

+ b0 β2 + kp + k dβ1 + a1 β1 + k d( 1113857 + a21113872 1113873s3


+ b0a2 β2 + kp + k dβ11113872 11138731113960

+ k0 kpβ2 + k dβ31113872 11138731113961s + k0kpβ3

(18)


D0 b0

D1 b0 β1 + k d + a1( 1113857

D2 b0 β2 + kp + k dβ1 + a1β1 + a1k d + a21113872 1113873

D3 b0a1 β2 + kp + k dβ11113872 1113873 + b0a2 β1 + k d( 11138571113960

+ k0 kpβ1 + k dβ2 + β31113872 11138731113961


D5 k0kpβ3

(19)


s5

D0 D2 D4

s4

D1 D3 D5

s3

B31 B32

s2

B41 B42

s1

B51

s0

B61

B31 D1D2 minus D0D3

D1

B32 D1D4 minus D0D5

D1


B31

B42 D5


B41

B61 D5

(20)




1 + G2(s)1b0

Gn(s)(1 + δG)H(s) 0 (21)




G2(s)Gn(s)H(s)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(22)








Y1(s) G11(s)X1(s) + G12(s)X2(s) + G13(s)X3(s)

Y2(s) G21(s)X1(s) + G22(s)X2(s) + G23(s)X3(s)

Y3(s) G31(s)X1(s) + G32(s)X2(s) + G33(s)X3(s)

⎧⎪⎪⎨

⎪⎪⎩

(23)


euroθ1 f1 θ1 _θ1 μ1(t) t1113872 1113873 + k1μ1

euroθ2 f2 θ2 _θ2 μ2(t) t1113872 1113873 + k2μ2

euroθ3 f1 θ3 _θ3 μ3(t) t1113872 1113873 + k3μ3

(24)





x1 t

x2 800 sin 03t +1360

1113874 1113875lowast π1113874 1113875


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)












TDControl

lawb0

G (s)v

z1

z2

3z

y

minus

minusminus e

err

LESO

minus

1s

1b

1s

1s

μ0 μ

β1

β2

β3



TD

Delt

a hig

h-sp

eed

para

llel r

obot

minus

minusminus minus

LESO 2

TD Control law 1

minus

minusminus e1

e1

minus

LESO 1

v1

z11 z21z31

z32

z33

z22

z23z13

z12

v2

v3 TD Control law 2

Control law 2

minus

minusminus minus

LESO 3

μ2

μ1

3micro

μ01

μ02

μ03

1b01

1b02

1b03

b01

b02

r2r2

r3

e2

e3

e3

r3

e2

b03





1000500 1000

500Position y (mm)

0


05

101520253035

Posit

ion z (

mm

)

(a)


ndash1000

ndash800

ndash600

ndash400

ndash200

0

200

400

600

800

1000

Posit

ion y (

mm

)

(b)

Figure 7 Continued


ndash5

0

5

10

15

20

25

30

35Po

sitio

n z (

mm

)


(c)

ndash5

0

5

10

15

20

25

30

35

Posit

ion z (

mm

)


(d)


1000

500

0

ndash500

Posit

ion y

(mm

)

ndash1000

0

5

10

15

20

25

30

35

Posit

ion z (

mm

)

1000

Position x (mm)




ndash60

ndash40

ndash20

0

20

40

60

Am

plitu

de (m

m)

5 10 15 20 25 300Time (s)



Joint 1

010203040

Posit

ion

(mm

)5 10 15 20 25 300

Time (s)


(a)

Joint 2

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(b)

Joint 3

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(c)


1000500 1000

500Position y (mm)

0


05

101520253035

Posit

ion z (

mm

)




Joint 1

010203040

Posit

ion

(mm

)5 10 15 20 25 300


(a)

Joint 2

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(b)

Joint 3

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(c)


1000500 1000

500Position y (mm)

0



05

101520253035

Posit

ion z (

mm

)



5 Conclusion



Data Availability




Acknowledgments


References









































Gb(s) kp + k ds1113872 1113873 s

3+ β1s

2+ β2s + β31113960 1113961k0

b0 s3

+ β1 + k d( 1113857s2

+ β2 + kp + k dβ11113872 1113873s1113960 1113961 s2

+ a1s + a21113960 11139611113966 1113967 + k0 kpβ1 + k dβ2 + β31113872 1113873s2

+ kpβ2 + k dβ31113872 1113873s + kpβ31113960 1113961

(17)


D(s) b0s5

+ b0 β1 + k d + a1( 1113857s4

+ b0 β2 + kp + k dβ1 + a1 β1 + k d( 1113857 + a21113872 1113873s3


+ b0a2 β2 + kp + k dβ11113872 11138731113960

+ k0 kpβ2 + k dβ31113872 11138731113961s + k0kpβ3

(18)


D0 b0

D1 b0 β1 + k d + a1( 1113857

D2 b0 β2 + kp + k dβ1 + a1β1 + a1k d + a21113872 1113873

D3 b0a1 β2 + kp + k dβ11113872 1113873 + b0a2 β1 + k d( 11138571113960

+ k0 kpβ1 + k dβ2 + β31113872 11138731113961


D5 k0kpβ3

(19)


s5

D0 D2 D4

s4

D1 D3 D5

s3

B31 B32

s2

B41 B42

s1

B51

s0

B61

B31 D1D2 minus D0D3

D1

B32 D1D4 minus D0D5

D1


B31

B42 D5


B41

B61 D5

(20)




1 + G2(s)1b0

Gn(s)(1 + δG)H(s) 0 (21)




G2(s)Gn(s)H(s)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(22)








Y1(s) G11(s)X1(s) + G12(s)X2(s) + G13(s)X3(s)

Y2(s) G21(s)X1(s) + G22(s)X2(s) + G23(s)X3(s)

Y3(s) G31(s)X1(s) + G32(s)X2(s) + G33(s)X3(s)

⎧⎪⎪⎨

⎪⎪⎩

(23)


euroθ1 f1 θ1 _θ1 μ1(t) t1113872 1113873 + k1μ1

euroθ2 f2 θ2 _θ2 μ2(t) t1113872 1113873 + k2μ2

euroθ3 f1 θ3 _θ3 μ3(t) t1113872 1113873 + k3μ3

(24)





x1 t

x2 800 sin 03t +1360

1113874 1113875lowast π1113874 1113875


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)












TDControl

lawb0

G (s)v

z1

z2

3z

y

minus

minusminus e

err

LESO

minus

1s

1b

1s

1s

μ0 μ

β1

β2

β3



TD

Delt

a hig

h-sp

eed

para

llel r

obot

minus

minusminus minus

LESO 2

TD Control law 1

minus

minusminus e1

e1

minus

LESO 1

v1

z11 z21z31

z32

z33

z22

z23z13

z12

v2

v3 TD Control law 2

Control law 2

minus

minusminus minus

LESO 3

μ2

μ1

3micro

μ01

μ02

μ03

1b01

1b02

1b03

b01

b02

r2r2

r3

e2

e3

e3

r3

e2

b03





1000500 1000

500Position y (mm)

0


05

101520253035

Posit

ion z (

mm

)

(a)


ndash1000

ndash800

ndash600

ndash400

ndash200

0

200

400

600

800

1000

Posit

ion y (

mm

)

(b)

Figure 7 Continued


ndash5

0

5

10

15

20

25

30

35Po

sitio

n z (

mm

)


(c)

ndash5

0

5

10

15

20

25

30

35

Posit

ion z (

mm

)


(d)


1000

500

0

ndash500

Posit

ion y

(mm

)

ndash1000

0

5

10

15

20

25

30

35

Posit

ion z (

mm

)

1000

Position x (mm)




ndash60

ndash40

ndash20

0

20

40

60

Am

plitu

de (m

m)

5 10 15 20 25 300Time (s)



Joint 1

010203040

Posit

ion

(mm

)5 10 15 20 25 300

Time (s)


(a)

Joint 2

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(b)

Joint 3

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(c)


1000500 1000

500Position y (mm)

0


05

101520253035

Posit

ion z (

mm

)




Joint 1

010203040

Posit

ion

(mm

)5 10 15 20 25 300


(a)

Joint 2

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(b)

Joint 3

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(c)


1000500 1000

500Position y (mm)

0



05

101520253035

Posit

ion z (

mm

)



5 Conclusion



Data Availability




Acknowledgments


References












































Y1(s) G11(s)X1(s) + G12(s)X2(s) + G13(s)X3(s)

Y2(s) G21(s)X1(s) + G22(s)X2(s) + G23(s)X3(s)

Y3(s) G31(s)X1(s) + G32(s)X2(s) + G33(s)X3(s)

⎧⎪⎪⎨

⎪⎪⎩

(23)


euroθ1 f1 θ1 _θ1 μ1(t) t1113872 1113873 + k1μ1

euroθ2 f2 θ2 _θ2 μ2(t) t1113872 1113873 + k2μ2

euroθ3 f1 θ3 _θ3 μ3(t) t1113872 1113873 + k3μ3

(24)





x1 t

x2 800 sin 03t +1360

1113874 1113875lowast π1113874 1113875


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)












TDControl

lawb0

G (s)v

z1

z2

3z

y

minus

minusminus e

err

LESO

minus

1s

1b

1s

1s

μ0 μ

β1

β2

β3



TD

Delt

a hig

h-sp

eed

para

llel r

obot

minus

minusminus minus

LESO 2

TD Control law 1

minus

minusminus e1

e1

minus

LESO 1

v1

z11 z21z31

z32

z33

z22

z23z13

z12

v2

v3 TD Control law 2

Control law 2

minus

minusminus minus

LESO 3

μ2

μ1

3micro

μ01

μ02

μ03

1b01

1b02

1b03

b01

b02

r2r2

r3

e2

e3

e3

r3

e2

b03





1000500 1000

500Position y (mm)

0


05

101520253035

Posit

ion z (

mm

)

(a)


ndash1000

ndash800

ndash600

ndash400

ndash200

0

200

400

600

800

1000

Posit

ion y (

mm

)

(b)

Figure 7 Continued


ndash5

0

5

10

15

20

25

30

35Po

sitio

n z (

mm

)


(c)

ndash5

0

5

10

15

20

25

30

35

Posit

ion z (

mm

)


(d)


1000

500

0

ndash500

Posit

ion y

(mm

)

ndash1000

0

5

10

15

20

25

30

35

Posit

ion z (

mm

)

1000

Position x (mm)




ndash60

ndash40

ndash20

0

20

40

60

Am

plitu

de (m

m)

5 10 15 20 25 300Time (s)



Joint 1

010203040

Posit

ion

(mm

)5 10 15 20 25 300

Time (s)


(a)

Joint 2

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(b)

Joint 3

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(c)


1000500 1000

500Position y (mm)

0


05

101520253035

Posit

ion z (

mm

)




Joint 1

010203040

Posit

ion

(mm

)5 10 15 20 25 300


(a)

Joint 2

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(b)

Joint 3

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(c)


1000500 1000

500Position y (mm)

0



05

101520253035

Posit

ion z (

mm

)



5 Conclusion



Data Availability




Acknowledgments


References









































TD

Delt

a hig

h-sp

eed

para

llel r

obot

minus

minusminus minus

LESO 2

TD Control law 1

minus

minusminus e1

e1

minus

LESO 1

v1

z11 z21z31

z32

z33

z22

z23z13

z12

v2

v3 TD Control law 2

Control law 2

minus

minusminus minus

LESO 3

μ2

μ1

3micro

μ01

μ02

μ03

1b01

1b02

1b03

b01

b02

r2r2

r3

e2

e3

e3

r3

e2

b03





1000500 1000

500Position y (mm)

0


05

101520253035

Posit

ion z (

mm

)

(a)


ndash1000

ndash800

ndash600

ndash400

ndash200

0

200

400

600

800

1000

Posit

ion y (

mm

)

(b)

Figure 7 Continued


ndash5

0

5

10

15

20

25

30

35Po

sitio

n z (

mm

)


(c)

ndash5

0

5

10

15

20

25

30

35

Posit

ion z (

mm

)


(d)


1000

500

0

ndash500

Posit

ion y

(mm

)

ndash1000

0

5

10

15

20

25

30

35

Posit

ion z (

mm

)

1000

Position x (mm)




ndash60

ndash40

ndash20

0

20

40

60

Am

plitu

de (m

m)

5 10 15 20 25 300Time (s)



Joint 1

010203040

Posit

ion

(mm

)5 10 15 20 25 300

Time (s)


(a)

Joint 2

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(b)

Joint 3

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(c)


1000500 1000

500Position y (mm)

0


05

101520253035

Posit

ion z (

mm

)




Joint 1

010203040

Posit

ion

(mm

)5 10 15 20 25 300


(a)

Joint 2

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(b)

Joint 3

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(c)


1000500 1000

500Position y (mm)

0



05

101520253035

Posit

ion z (

mm

)



5 Conclusion



Data Availability




Acknowledgments


References









































ndash5

0

5

10

15

20

25

30

35Po

sitio

n z (

mm

)


(c)

ndash5

0

5

10

15

20

25

30

35

Posit

ion z (

mm

)


(d)


1000

500

0

ndash500

Posit

ion y

(mm

)

ndash1000

0

5

10

15

20

25

30

35

Posit

ion z (

mm

)

1000

Position x (mm)




ndash60

ndash40

ndash20

0

20

40

60

Am

plitu

de (m

m)

5 10 15 20 25 300Time (s)



Joint 1

010203040

Posit

ion

(mm

)5 10 15 20 25 300

Time (s)


(a)

Joint 2

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(b)

Joint 3

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(c)


1000500 1000

500Position y (mm)

0


05

101520253035

Posit

ion z (

mm

)




Joint 1

010203040

Posit

ion

(mm

)5 10 15 20 25 300


(a)

Joint 2

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(b)

Joint 3

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(c)


1000500 1000

500Position y (mm)

0



05

101520253035

Posit

ion z (

mm

)



5 Conclusion



Data Availability




Acknowledgments


References









































Joint 1

010203040

Posit

ion

(mm

)5 10 15 20 25 300

Time (s)


(a)

Joint 2

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(b)

Joint 3

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(c)


1000500 1000

500Position y (mm)

0


05

101520253035

Posit

ion z (

mm

)




Joint 1

010203040

Posit

ion

(mm

)5 10 15 20 25 300


(a)

Joint 2

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(b)

Joint 3

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(c)


1000500 1000

500Position y (mm)

0



05

101520253035

Posit

ion z (

mm

)



5 Conclusion



Data Availability




Acknowledgments


References









































Joint 1

010203040

Posit

ion

(mm

)5 10 15 20 25 300


(a)

Joint 2

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(b)

Joint 3

ndash1000ndash500

0500

1000

Posit

ion

(mm

)

5 10 15 20 25 300Time (s)


(c)


1000500 1000

500Position y (mm)

0



05

101520253035

Posit

ion z (

mm

)



5 Conclusion



Data Availability




Acknowledgments


References









































5 Conclusion



Data Availability




Acknowledgments


References

























































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ResearchonaLADRCStrategyforTrajectoryTrackingControlof ...downloads.hindawi.com/journals/mpe/2020/9681530.pdf · ResearchArticle ResearchonaLADRCStrategyforTrajectoryTrackingControlof