A Robust Method Based on Dual Encoders to Eliminate Velocity

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A Robust Method Based on Dual Encoders to Eliminate Velocity

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actuators
Article
A Robust Method Based on Dual Encoders to Eliminate Velocity Ripple for Modular Drive Joints
Qiang Xin 1,2 , Chin-Yin Chen 2,*, Chongchong Wang 2, Guilin Yang 2, Chi Zhang 2, Zaojun Fang 2 and Chun Lung Philip Chen 3,4
1 College of Materials Sciences and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing 100049, China; [email protected]
2 Zhejiang Key Laboratory of Robotics and Intelligent Manufacturing Equipment Technology, Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo 315201, China; [email protected] (C.W.); [email protected] (G.Y.); [email protected] (C.Z.); [email protected] (Z.F.)
3 School of Computer Science & Engineering, South China University of Technology, Guangzhou 510006, China; [email protected]
4 Faculty of Science and Technology, University of Macau, Macau 999078, China * Correspondence: [email protected]; Tel.: +86-0574-8760-2663
Received: 16 November 2020; Accepted: 8 December 2020; Published: 10 December 2020
Abstract: Velocity ripple is one of the common problems of modular drive joints, which easily induces vibration and noise and affects motion accuracy. In order to improve the motion control accuracy, a robust method based on dual encoders to eliminate velocity ripple is proposed in this paper. The method contains a velocity ripple elimination controller (VREC), a rigid-body velocity solver (RBVS), and a proportional–integral (PI) controller. Feeding back the VREC output to the PI controller based on the rigid-body velocity obtained from the weighted sum of dual encoders in the RBVS, an equivalent system damping term was added into the system. Therefore, the velocity ripple can be suppressed effectively with the adjustable damping term composed of control parameters. Above all, the proposed method has only one more parameter to further eliminate velocity ripple compared to the pure PI method and, meanwhile, has apparent advantages over the conventional method, such as fewer parameters and full frequency ripple elimination, as well as robustness to input disturbance and modular drive joint load inertia changes. This proposed method’s effectiveness is verified by simulations in MATLAB and experiments in the modular drive joint platform.
Keywords: modular drive joint; velocity ripple; dual encoders; system damping; robustness

1. Introduction
Modular drive joints are now widely used in industrial robots and collaborative robots, integrating high efficiency and high power density permanent magnet synonym motors (PMSMs), harmonic drive, torque sensors, shafts, bearings, dual encoders, and other components [1–3]. The PMSM is widely employed in modular drive joints due to its advantages: fast dynamics, high efficiency and reliability, and a favorable torque to inertia ratio [4–6]. However, the torque ripple in the PMSM and the stiffness of the modular drive joint influenced by the flexpline of harmonic drive easily induce velocity ripple [7,8]. Furthermore, the velocity ripple will result in vibration, noise, and other similar problems, which are major factors affecting the accuracy of the motion control system [9]. In the last few years, many kinds of research have been done on the accuracy of motion control, where the velocity ripple has been found to be a major issue [10].
The factors that cause the velocity ripple of modular drive joints can mainly be divided into two categories. The first category is caused by the position measuring errors because the feedback

Actuators 2020, 9, 135; doi:10.3390/act9040135

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velocity signal is usually calculated from the measured position signal [11,12]. Many algorithms were introduced to correct the position measuring error under certain conditions but with no systematic description for measuring the position measuring error on the velocity ripple in the servo system [13,14]. The other category is caused by the inevitable parasitic torque ripple in the PMSM, such as cogging and flux harmonics, leading to velocity ripples, vibrations, acoustic noise, and poor response performance in motion control systems [5,15].
Aiming to minimize torque ripple and realize velocity ripple elimination, the methods can also mainly be classified in two ways: one way is to optimize the design or improve the body structure of the PMSM [16–19], and the other way is to depend on control algorithms. The former mainly involves skewing the slot or magnet, ensuring a fractional number of slots per pole, and improving the winding distribution. These methods are proven to be effective for eliminating the velocity ripple but require complex production processes. Besides, once the designed and optimized body structure is produced, the modular drive joint cannot be modified, which results in a higher production cost [20,21]. The control algorithms of the latter way mainly contain sensorless methods and sensor-based methods. The sensorless methods control the current phase indirectly by controlling the voltage–current phase deference based on the V/f control [22–24]. However, if the load inertia varies depending on the joint position, such as robot posture changing, the other velocity ripple will occur easily [25].
On the contrary, the sensor-based methods are the main approaches to eliminating velocity ripple, which mainly include traditional proportion-integral-differential (PID) methods and improved PID methods, intelligent methods, and model-based methods. The traditional methods, such as PI velocity control and cascade control structure (CCS) based on PI controllers, are simple and easily achieve velocity ripple but with limited efficiency [5]. The intelligent methods include: adaptive fuzzy control methods [2,26,27] and neural network algorithms [28] with robust performance, iterative learning control (ILC) methods as a model-free control strategy to suppress velocity ripple and that is robust to noise [29,30], and a linear parameter varying H∞ velocity control method with robustness against disturbance characteristics [31]. These methods achieve robustness against velocity ripple and disturbances, but with a large amount of data calculation and poor real-time performance. In recent years, model-based methods have been developed, including model predictive control (MPC) methods, observer-based control methods, sliding mode control methods, and so on. These methods have advantages such as robustness, simple modeling, and the ability to handle control variable constraints to ensure the system’s satisfactory performance [7].
Among the model-based methods, the methods based on the equivalent rigid-body velocity method [32] and self-resonance cancellation (SRC) methods [33,34] are more popular. The equivalent rigid-body velocity method aims to add system damping to eliminate velocity ripple. The SRC method aims to achieve a rigid-body system by the weighted sum of sensor signals: dual encoders (motor-side encoder and link-side encoder). Base on the dual encoders of the modular drive joint, the rigid-body velocity can be obtained. The conventional method based on equivalent rigid-body velocity added an equivalent damping term after the closed loop to suppress velocity ripple. However, it only can suppress a limited frequency range ripple of velocity. To improve the conventional equivalent rigid-body velocity method, a method based on a state observer to suppress velocity ripple in the full frequency range ripple was proposed in [35]. As for the SRC methods, there are fewer control parameters but there are high requirements for system model identification accuracy, which is challenging to apply in practical engineering applications with the low accuracy of system identification.
A method combined with the ideas of the equivalent rigid-body velocity method and SRC method is proposed to add system damping with fewer control parameters in this paper. The proposed method contains a rigid-body velocity solver (RBVS), a velocity ripple elimination controller (VREC), and a proportional–integral (PI) controller. As the feedback velocity can be motor velocity or link velocity, the proposed method can be further classed as a motor-side controller and link-side controller. In the proposed method, the RBVS obtains a rigid-body velocity based on the signals of dual encoders (motor velocity measured by the motor-side encoder and link velocity measured by the link-side

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==

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mm mm

 

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 

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. . .. ..
The physical meanings of Jm, Bm, Jl, Bl, K, D, τm, τj, τdis, θm, θl, θm, θl, θm, θl are motor

inertia, motor viscous damping, load inertia, load viscous damping, joint stiffness, joint viscous

damping, input torque, joint torque, input disturbance, motor position, link position, motor velocity,

link velocity, motor acceleration, and link acceleration, respectively. The transfer function from input

.

.

torque τm to motor velocity θm and link velocity θl in the Laplace domain can be formulated as follows.

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.

 PTM(s) = θm =

1 · 1 · s2+ωa2 · ωr2



τm

Jm+Jl s

ωa2 s2+ωr2

Figure 2. The two-inertia. model block diagram of the modular drive joint.

(2)

 PTL(s) = τθml = Jm1+Jl · 1s · s2ω+rω2r2

The physical meanings of Jm , Bm , Jl , Bl , K, D, τ m , τ j , τ dis , θm , θl , θm , θl , θm , θl are motor

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inertia, motor viscous damping, load inertia, load viscous damping, joint stiffness, joint viscous damping,

input torque, joint torque, input disturbance, motor position, link position, motor velocity, link velocity,

motor acceleration, and link acceleration, respectively. The transfer function from input torque τ m to motor velocity θm and link velocity θl in the Laplace domain can be formulated as follows.

 P

(s) = θm

 =

1



1

 ⋅

s2

+ ωa2



ωr 2

 

TM


τ m  Jm + Jl s   ωa2 s2 + ωr2 

 

θ  1 1   ω 2 

(2)

 PTL 

(s)

=

τ

l m

=

 

Jm

+ Jl



s

 



 

s

2

r
+ ωr2

 

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TM


τ m  Jm + Jl s   ωa2 s2 + ωr2 

 

θ  1 1   ω 2 

(2)

 PTL 

(s)

=

τ

l m

=

 

Jm

+ Jl



s

 



 

s

2

r
+ ωr2

 

where ωa = K Jl , ωr = ωa 1+ Jl Jm .
In order to verify that this two-inertia model can fit the actual modular drive joint well, a swept sine current signal is utilized to estimate the joint frequency response, with a load inertia of Jl = 2.26
kg·m2. In experiments, the sampling frequency is 1 kHz. As for the modular drive joint, the current torque constant is tested to be 0.17 N·m/A. The input saturation current is set to 10 A. The maximum velocity of the motor side is 2500 rpm. Moreover, the gear ratio of the harmonic drive is 160. After the identification experiment, the joint frequency response from input torque τ m to motor velocity θm and link velocity θl can be obtained as shown in Figure 3a,b, respectively.

(a)

(b)

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nonlinear and time-varying factors, such as the nonlinear torques, frictions, and damping effects in

the motor side and transmission, and the varying efficiency of harmonic drive (60–75% depending

on ratio, velocity, and lubricant). In particular, the stiffness and damping of harmonic drive are

related to the motion velocity. Therefore, the two-inertia model cannot fit the phase of actual system

characteristics well [37]. (Ha)owever, the amplitude–frequency fitting accu(br)acy can accurately fit the

system’s

anti-resonance and resonance characteristics, proving
Figure 3. Modular drive joint frequency responses: (a) from

tτhattothθe

two-inertia
; (b) from τ

mtoodθe.l

can

reflect

m

m

m

l

In Figure 3, the black lines are acquired by actual measurement data of the experiment. The red lines are fitted frequency response curves of the two-inertia model. The motor-side velocity response has a clear anti-resonance and resonance behavior from the identified results, whereas the link-side velocity only has a resonance behavior. In fact, the actual joint dynamic model is subjected to several

nonlinear and time-varying factors, such as the nonlinear torques, frictions, and damping effects in the motor side and transmission, and the varying efficiency of harmonic drive (60–75% depending on ratio, velocity, and lubricant). In particular, the stiffness and damping of harmonic drive are related to the motion velocity. Therefore, the two-inertia model cannot fit the phase of actual system Acchtuaartaorcste20r2is0t, i9c,s13w5 ell [37]. However, the amplitude–frequency fitting accuracy can accurately f5itofth19e system’s anti-resonance and resonance characteristics, proving that the two-inertia model can reflect the joint’s dynamic characteristics well. The anti-resonance frequency ωa and resonance frequency tωher joairnetl’oscdaytendamatic19chHazraacntder2is1tiHcszw, reelslp. eTchtievaenlyti.-Nreesxotn, athneceidferenqtiufieendcypaωraamanedterressoofntahnicsemfroedquuleanrcdyriωvre are located at 19 Hz and 21 Hz, respectively. Next, the identified parameters of this modular drive joint are summarized in Table 1. joint are summarized in Table 1.

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Symbol
Symbol
Jm BJmm BJml
Jl
BBll
K
D D

Name
Name
motor inertia motormvoitsocroiunesrdtiaamping
motor viscous damping
llooaadd iinneertritaia llooaadd vviissccoouussddaammpipnigng
joinntt ssttiifffnfnesesss jojoiinntt vviissccoouussddaammpipnigng

Value
Value
7.34 kg·m2 337.2.384Nkg·m·m·s2/rad
33.28 N·m·s/rad
22.2.266kkgg·m·m2 2 55 NN·m·m·s·/sr/ardad 344,,000000NN·m·m/r/ardad 1100 NN·m·m·s·/sr/ardad

33.. Controller Design

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velocity and rigid-body velocity. As well motor velocity and rigid-body velocity.

as the pure As well as

rtihpeplpeu, nreisritphpeleer,ronr

between link velocity is the error between

and link

rigid-body velocity. Therefore, according to the joint velocity, the proposed method can be classed as

velocity and rigid-body velocity. Therefore, according to the joint velocity, the proposed method can

the motor-side controller and link-side controller, as shown in Figure 4a,b, respectively.

be classed as the motor-side controller and link-side controller, as shown in Figure 4a,b, respectively.

(a)

(b)

FFiigguurree44.. BBlloocckk ddiiaaggrraamm ooff tthhee pprrooppoosseedd mmeetthhoodd:: ((aa)) tthhee mmoottoorr--ssiiddee ccoonnttrroolllleerr bbaasseedd oonn mmoottoorr vveelloocciittyy ffeeeeddbbaacckk;;((bb))tthheelliinnkk--ssiiddeeccoonnttrroolllleerrbbaasseeddoonnlliinnkkvveellooccitityyffeeeeddbbaacckk..

IInn

FFiigguurree

44,,tthheemmooddeellPMPMdedfienfeinsetsratnrasnfesrfefrunfucnticotniofnrofmrominpinuptutot rtqourqeuτem

τtom

.
mtotmorovtoelrovcietlyocθitmy.

Mθea. nMwehanilwe,hthile,mthoedmeloPdMeLl

dPefindeseftirnaenssfterar nfusfnecrtifoun cftrioomn

.
fmroomtormvoetloorcivteyloθcmityto

θlinktovellionckitvyeθl.ol,ciatsy

m

ML

m

sθhlo,wasnsihnoEwqnuaintioEnqu(3a)t.ion (3).



.

 PM = θm =

1 · 1 · s2+ωa2 · ωr2



τm

Jm+Jl s s2+ωr2 ωa2



.

(3)

 PML = θθ. ml = s2ω+aω2a2

Besides, the PI controller and coefficients α and β of the RBVS are defined as follows, which will be discussed in the next section:

 

PI = Kp +

Ki s

  α =

Jm s+Bm

(J +J )s+(B +B )

(4)

  β =

ml

m

Jl s+Bl

l



( Jm + Jl )s+(Bm +Bl )

by the motor-side encoder. Link velocity is measured by the link-side encoder, as shown in Figure 5a.

( ) Jmθm + Bmθm = τ m − K (θm −θl ) + D θm −θl



( ) Jlθl + Blθl = K (θm −θl ) + D θm −θl

Actuators 2020, 9, 135

6 of 19

τ m = ( Jms + Bm )θm + ( Jl s + Bl )θl

(5)



3.1. Rigid-Boθdry=Velocity Soτlvmer Design =

Jm s + Bm

θm +

Jl s + Bl

θ l

 ( Jm + Jl ) s + ( B.m + Bl ) ( Jm + Jl ) s + ( Bm + Bl ) ( Jm + Jl ) s + ( Bm + Bl )

The rigid-body velocity θr is obtained based on the weighted sum of motor velocity and link

veTlohceitytrianntshferrifguind-cbtioodnyfvroelmocitnypsuotlvtoerr.qCuoemτbminetdowriigthidt-hbeotdwyov-ienleorctiatymθordeclathnabt ehaosbbtaeiennegdivbeyn: in

.

Equation

(1θ),

rigid-body

velocity 1

θr

can

be

deriJvesd+frBom

Equatθion

(5).

The Jinsp+utBdisturbanθce

τdis

is

not

discussed atr t=his time and is set to zero= for simpmlicity. mAmong thme+m, motor vl elocilty
motor-sideτ menco( Jdmer+. LJiln)ks +ve(lBomci+tyBils) me(aJsmur+eJdl b) sy+th(eBmlin+kB-sl i)dτemenc(oJdme+r, Jals)ssh+o(wBnm

is i+n

measlured by
BFlig) τumre 5a.

th(e6)

From

Equations 

(J2m)θ.a. mnd+(B6m),θ.thme=Boτdme−dKia(θgrma−mθslh) o+wDs

tθ.hme

−trθa. nl sfer

function

from

input

torque

τ m

to motor velocity

θ m

,Jlθi..nk+vBelθ.oc=ityK

(θθ l

a−nθd )ri+giDd-bθ.od−yθ.velocity

θ r

in Figure 5b, as the blue line, red

 l l l l

ml .

ml .

(5)

line,

and

thick

black 

liτnme,=re(sJpmesc+tivBeml)yθ. mA+s t(hJels

m+oBdl)uθllar

drive

joint

identified

results

listed

in

Table

1,

the motor viscous damθ. rp=ing is hiτgmher than= link vJmiss+cBomus daθ.mmp+ing, wJhlsi+chBl leadsθ. lto the magnitude of

motor

velocity

 frequency

re(Jsmp+oJnl)ss+e(Bbme+inBgl)

low( Jemr+tJhl )as+n(Bthm +e Blli)nk

velo(Jcmi+tyJl)fsr+e(qBmu+eBnlc) y

response.

θl

τm

θr

τm θm

τm

(a)

(b)

FigFuirgeu5r.eB5l.oBcklodckiadgiraagmraomf tohfethriegridig-bido-dbyodvyelvoecliotycistyolsvoelrv:e(ra:)(tah)ethweewigehitgehdtesdumsumof omfomtoortovrelvoecliotcyitaynadnldink

velloinciktyvferloomcitdyufraolmendcoudaleresn;c(obd) ethrse;

B(bo)dteheplBoot dfreopmlotτfmrotmo

τθmm

.
t,ofrθomm,

frτommtτomθtlo,

.
aθnl,danfrdomfromτ m

τmtotoθθr. r..

.
The transfer function from input torque τm to rigid-body velocity θr can be obtained by:

.

.

.

θr =

1

=

Jms + Bm

θm +

Jls + Bl

θl

(6)

τm (Jm + Jl)s + (Bm + Bl) (Jm + Jl)s + (Bm + Bl) τm (Jm + Jl)s + (Bm + Bl) τm

From Equations (2) and (6), the Bode diagram shows the transfer function from input torque

.

.

.

τm to motor velocity θm, link velocity θl and rigid-body velocity θr in Figure 5b, as the blue line,

red line, and thick black line, respectively. As the modular drive joint identified results listed in Table 1,

the motor viscous damping is higher than link viscous damping, which leads to the magnitude of

motor velocity frequency response being lower than the link velocity frequency response.
.
As stated by the Bode diagram from input torque τm to rigid-body velocity θr, there are no

anti-resonance and resonance behaviors. Consequently, the system from input torque τm to rigid-body
.
velocity θr is an ideal rigid-body system without any velocity ripple. The velocity obtained from the

weighted sum of motor velocity and link velocity is an ideal rigid-body velocity.

3.2. Velocity Ripple Elimination Controller Design

3.2.1. Motor-Side Controller Design
Based on the rigid-body velocity discussed in Section 3.1, the motor-side controller can be
.
simplified as a PI controller and a VREC. The VREC feeds back the velocity u to the desired velocity θd input to the PI controller. The pure ripple m is obtained from the error between motor velocity and rigid-body velocity after the proportional gain K tuning, as shown in Figure 6.

3.2.1. Motor-Side Controller Design
Based on the rigid-body velocity discussed in Section 3.1, the motor-side controller can be simplified as a PI controller and a VREC. The VREC feeds back the velocity u to the desired velocity
Actuθadtorisn2p0u20t,t9o, 1t3h5e PI controller. The pure ripple m is obtained from the error between motor veloc7itoyf 19
and rigid-body velocity after the proportional gain K tuning, as shown in Figure 6.

FiFgiugurere66. .BBlloocckkddiiaaggrraamm ooff tthhee mmoottoorr--ssiiddeeccoonntrtorolllelre.r.

In IFnigFuigreu6re, t6h,ethsiegsniaglnmal ismpuirseptuhreertihpeplreipopflme ooftomr ovteolrocvietyloacfittyerabfteeirngbecionngtcroolnletrdolbleydKb. yThKe s. yTmhebol u issytmheboslumu oisf mthoetosur mveloofcmityotaonrdvepluocreityripanpdlepmu.reTrhipepinlepumt .dTishtue ribnapnuctedτidsitsuirsbannoctedτisdcisusissendotfor simdpislcicuistys.edThfoernstihmepolipceitny-.lToohpentrtahnesofeprefnu-lnocotpiotnraPnospfeenrmf(usn)cftrioomn tPhoependme(ssi)refdrovmelothceitdyeθ.sdirteodmveoltoocritvyeloθcd ity θ. mtocamnobteorgviveeloncitnyEθquactaionnb(e7g).iven in Equation (7).
m

( ) Pm m (s) =θ. θmm =

PPI ωIωr 2r 2

s2 +s2ω+a2 ωa2

( ) Popen op(esn) = θ. θdd = ((JJmm++JlJl))ss⋅· s(s2 2++ωrω2 r2) ωa2ωa2

(7) (7)

The closed-loop transfer function The closed-loop transfer function P

Pclomsem((ss))

from the desired velocity . θd to motor velocity. from the desired velocity θ to motor velocity θ

θ m can

can be derived from the following equactloisoens.

d

m

be derived from the following equations.

( ) . θd − u

Pm open

= θm.

 

θdu −=.θu

Popenm +m

=

θm

 
( ) 

u=m θ=mKm+. θm−θ.

 

m. =

K

θmm− .

θrr .

 

θrθ=r =ααθθmm++ ββθθll

 

θ. l θ=l =θ.θmmPPMMLL

(8) (8)

PclosemP(csl)os=emθm(s=)

=

.
θm
.

m

=

Pm open

Popen m

θd 1 + Pθopden 1 + 1K+(P1o−peαnm−[1β+PKM(L1−)α−βPML)]

( ) PI s2PI+(sω2+2ωaω2)ω2 r2

(9)

( ) =

( )(Jm+Jl)(as2+ωrr2)ωa2s
PI(s2+ωa2)Jωr2+ J s2 + ω(J2msω+B2ms)(s2+ωa2)−(J s+B )ωa2

(9)

=

1+ (Jm+J

)(s2

+ωr

m
2 )ωa

2

s

l

1+K

1− r[(Jm+a J

)s+(Bm +B

ll
)](s2 +ωa 2 )

( ) ( ) PI s2 + ωa2 ωlr2   ( Jms + Bm ) l s2 + ωa2 l − ( Jls + Bl )ωa2 

( ) ( ) In

order

to

analyze

th1e+c(loJmse+dJ-ll)oosp2 +trωarn2 sωfea2rsfu1n+cKtio1n−pe(rJfomr+mJal )nsc+e(oBfmt+heBlm)ost2o+r-ωsiad2 e

controller, 

the

viscous damping terms Bm, Bl, D are set to zero for simplicity. Then the closed-loop transfer function

Pclosem(s) can be simplified, as in Equation (10).



.

 P

m(s)

=

θm
.

=

PIωr 2

(s2 +ωa 2 )

 close

θd (Jm+Jl)s·(s2+2ξmωrs+ωr2)+PIωr2 ωa2

 ξm = P2I ωKr ωωar22 − 1 + (Jm+ωJrl)ωa2

(10)

Based on the modular drive joint identified results, the resonance frequency ωr is always larger than the anti-resonance frequency ωa, which leads to ωr2/ωa2 being more than 1 and ξm being positive permanently. Compared with the open-loop transfer function given in Equation (7), the closed-loop transfer function has an added equivalent damping term 2ξmωrs and an equivalent natural resonance term PIωr2, which increase system damping and system response performance, respectively. With the control parameter K being positive, the velocity ripple can be eliminated by the
increased equivalent damping.

than the anti-resonance frequency ωa , which leads to ωr2 ωa2 being more than 1 and ξm being
positive permanently. Compared with the open-loop transfer function given in Equation (7), the closed-loop transfer function has an added equivalent damping term 2ξmωr s and an equivalent natural resonance term PIωr2 , which increase system damping and system response performance, Actrueastopresc2t0i2v0e,l9y, .1W35 ith the control parameter K being positive, the velocity ripple can be eliminated 8byof 19 the increased equivalent damping.

3.2.2. Link-Side Controller Design 3.2.2. Link-Side Controller Design

Similarly, when the pure ripple n Similarly, when the pure ripple

ins

obtained from the error between link velocity and rigid-body is obtained from the error between link velocity and rigid-

velboocdityy,vtehleorceityco, uthlderbeecoaulilndkb-esida elicnokn-stirdoellecrontotreollilmerintoateelitmheinvaetelotchietyvreilpopcilteybryipfpeleedbinygfebeadcikntghbisacekrror

antdhilsinekrrvoerlaoncditylintok tvheeloPcIitcyotnotrthoellePrI, caosnsthroolwlenr, ians Fshigouwrne 7in. Figure 7.

FFigiguurree77.. BBlloocckk ddiiaaggrraammoofftthheelilninkk-s-isdideecocnotnrtorlolellre.r.

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u issytmhebsoul mu oifslitnhke svuemlocoiftylinakndveploucrietyriapnpdlepnu.reThrieppinlepunt .dTishteurinbpanuct ediτsdtius ribsannoctedτisdcisusissendotfodrissciumspselidcity.

Thfeoropseimn-ploliocpityt.raTnhseferopfuennc-ltoioonp

Ptorpaennsmf(esr )

ffuronmctiothne

dPeopsenimre(sd)

.
vefrloomcitythθed

dtoesliirnekdvveeloloccitiytyθ. lθcdantobelignikven

byvEeqloucaittyionθl (1c1a)n. be given by Equation (11). .

Popenl(s) = θθ. l =

PIωr2 PIω 2

(11)

Pl open

(s)

=

θθ dl

=

(Jm + Jl)sr· (s2 + ωr2)
(J + J )s⋅ s2 +ω 2

(11)

d

m

l

r

.

.

The closed-loop transfer function The closed-loop transfer function

Pclose l (s) P l (s)

from from

the the

desired velocity desired velocity

θθ d

to to

link velocity link velocity

θθ l

can can

be

derived from the following.

close

d

l

be derived from the following.

 θ. d −. u Popenm = θ. l

 

u = θl + n



..

 n = K θl − θr

(12)

 .

.

.

 

θr

= αθm + βθl

 θ. m = PθM. lL

Pclose l (s)

.
θl

Popen l

= = .

θd 1+Popenl 1+K 1− PMα L −β

PIωr 2

( Jm + Jl )(s2 +ωr 2 )s

= 1+

PIωr 2

1+K 1− (Jms+Bm)(s2+ωa2)−(Jls+Bl)ωa2

( Jm + Jl )(s2 +ωr 2 )s

[( Jm + Jl )s+(Bm +Bl )]ωa 2

(13)

Correspondingly, the viscous damping terms Bm, Bl, D are ignored for simplicity to analyze the closed-loop transfer function performance of the link-side controller. Therefore, the closed-loop transfer function Pclosel(s) can be simplified, as shown in Equation (14).



.

 P

l(s) =

θl
.

=

PIωr 2

 close

θd (Jm+Jl)s·(s2+2ξlωrs+ωr2)+PIωr2

 ξl = −2Kω·Pr I

(14)

Compared to the open-loop transfer function given in Equation (11), the closed-loop transfer function adds an equivalent damping term 2ξlωrs and an equivalent natural resonance term PIωr2. When the K is negative, the added equivalent damping term becomes positive, and then the system damping can be increased. With the control parameter K and PI tuning, the velocity ripple can be
eliminated by the increased equivalent damping.

ξl = 

2ωr

Compared to the open-loop transfer function given in Equation (11), the closed-loop transfer

function adds an equivalent damping term 2ξlωr s and an equivalent natural resonance term PIωr2 .

When the K is negative, the added equivalent damping term becomes positive, and then the system

Actuators 2020, 9, 135
damping can be increased. With the control parameter

K

and

PI

9 of 19
tuning, the velocity ripple can

be eliminated by the increased equivalent damping.

3.3. Controller Parameter Analysis 3.3. Controller Parameter Analysis
In the proposed method, as discussed above, the motor-side controller and link-side controller have onInlythoenpermopoorseedcomntertohlopd,aarasmdiescteurssKedthaabnovaeP, tIhme metohtoodr-.siTdheecoPnItmroelltehroadndcalinnka-lssiodeinccolnutdroelltewr ohakviends of ofenelydboancekm(moroetocornvterololcpitayrafmeeedtebracKk atnhdanlinakPvI emloecthitoydf.eTehdebaPcIkm),eatshoshdocwann ainlsoFiignuclrued8eat,bw,oreksipnedcstiovfely.
feedback (motor velocity feedback and link velocity feedback), as shown in Figure 8a,b, respectively.

(a)

(b)

Figure 8F.igBuloreck8.dBialogcrkamdiaogfrPamI moef tPhIomde: t(hao)db:a(sae)dbaosnedmoontomrovteolrovcietlyocfieteydfbeeadckb;ac(bk;) (bba)sbeadseodnolinnlkinvkevloecloitcyitfyeedback. feedback.
Firstly, the PI parameters need to be designed before the proposed method application. Based on the Finirtsetglyr,atlhoef PsqI upaarraemd eetrerrosrn(eISedE)tocrbiteerdieosnig[n3e8d],bthefeorperothpeoprtrioopnoasleadnmd einthtoegdraapl pglaicinatioofnt.hBeamseodtor velooncitthyefienetdebgaraclkoPfIscqounatrreodlleerrraorre(dISeEs)igcnrietderaiosnK[p38=], 4th80e apnrodpKori t=ion2a4l0a0n. dThinetpegroraplogratiionnoafl tahnedminotteogrral
gain of the link velocity feedback PI controller are designed as Kp = 168 and Ki = 1200.

3.3.1. Parameters of Motor-Side Controller

For a modular drive joint, the elimination of velocity ripple is mainly to suppress velocity ripple occurring on the link side. This is based on the closed-loop transfer function from the desired velocity to motor velocity with the motor-side controller, as expressed in Equation (10) and Figure 6. Naturally, the transfer function from the desired velocity to link velocity can be given by Equation (15).

 

Pclosem(s) =

.
θl
.

=

.
θm
.

· PML =

PIωr 2



θd

θd

( Jm + Jl )s·(s2 +2ξm ωr s+ωr 2 )+PIωr 2

 ξm = P2I ωKr ωωar22 − 1 + (Jm+ωJrl)ωa2

(15)

The closed-loop transfer function from the desired velocity to link velocity has an added equivalent damping term 2ξmωrs and an equivalent natural resonance term PIωr2 compared to the open-loop transfer function. The parameter K can be expressed by Equation (16).

/ K = 2ωr ξm − ωr2

ωr2 − 1

(16)

PI

(Jm + Jl)ωa2 ωa2

When the damping ratio ξm of the closed-loop transfer function is under damping, critical damping, and over damping, such as ξm = 0.2, ξm = 1, ξm = 2, as well as the special damping ratio ξm = 0.707, the parameter K can be obtained from Equation (16) based on the identified parameters listed in Table 1 and the above parameters Kp and Ki, as shown in Table 2.

Table 2. The parameter K derived from different damping ratios ξm.

Under Damping Under Damping Critical Damping Over Damping

ξm

0.2

0.707

1

2

K

0.08

0.87

1.41

3.28

The closed-loop Bode diagrams of motor-side controller from the desired velocity to motor velocity with different parameters K are shown in Figure 9.

Under Damping Under Damping Critical Damping Over Damping

ξ m

0.2

0.707

1

2

K

0.08

0.87

1.41

3.28

ActuatoTrhs 2e02c0l,o9s,e1d35-loop Bode diagrams of motor-side controller from the desired velocity to 1m0 oofto19r velocity with different parameters K are shown in Figure 9.

Open-loop

(a)

(b)

FFiigguurree 99. .BBooddeeplpoltootf omfomtoor-tsoird-esicdoentcroonllterrolflreormfrθ.odmto θθ.dm wtoithθdm iffweriethntdpiaffrearmenetteprsaKra:m(ae)tearms plKitu:d(ea–) farmeqpuleitnucdyec–hfraerqaucteenrcisyticch; a(bra)cptheraisseti–cf;r(ebq)upehnacysec–hfraerqacuteenricsyticch. aracteristic.

AAss the parameter KK inincrceraesaesses(t(htheeddaammppininggraratitoioininccrreeaasseess),),tthheessyysstteemm ddaammppiinngg ggrraadduuaallly
iinnccrreeaassees, wwhhich lleeaads ttoo tthhee vveelloocciitty rriipppple bbeeiinng ddaammped, hhowever,, tthhiiss ddeeccrreeaasseess tthhee cclloosseedd--lloooopp ccoonntrol bandwwiiddtthh..TThheepapraarmametetrerKK as wweelllaassththeeprporpooprotriotinoanlaalnadnidntiengtreaglrgaal ignainK pKpanandd KKii aarree
ddeessiiggnneeddtotobebethtohsoesieniTnabTlaeb3l.eU3n. dUenrdtheer ctuhrerecnutrpreanrat mpaetrearms,ettheersa,mthpelitaumdeplmituardgeinmaanrdgpinhaasnedmpahrgaisne omf atrhgeincloofstehde-lcoloopsefdr-elqouopenfcreyqrueesnpcoynrseesaproenssueffiarceiesunftf.icTiheneta. Tmhpeliatmudpelitmuadregminaargnidn panhdaspehmasaergminaragrien 2a2r.e92d2B.9adnBda5n8d.75d8e.7g,dreegsp, reecstpiveecltyi,vwelhyi,cwhhpircohvpersotvheesstyhsetesmysitsemstaibslsetaabftleeracfltoesrecdlo-lsoeodp-lcooonptrcooln. trol.

Table 3. Parameters of motor-side controller.

Symbol
Kp Ki K

Name
proportional gain integral gain gain of VREC

Value
480 2400 1.3

3.3.2. Parameters of Link-Side Controller

The link-side controller, shown in Figure 7, shows that the closed-loop transfer function from the

desired velocity to link velocity is expressed in Equation (14). Based on the foregone proportional and

integral gain of Kp = 168 and Ki = 1200, the parameter K of the link-side controller can be obtained by

Equation (17).

K = − 2ωr ξl

(17)

PI

Similarly, when the damping ratio ξl of the closed-loop transfer function represents under damping, critical damping, and over damping, such as ξl = 0.2, ξl = 1, ξl = 2, as well as the special damping ratio ξl = 0.707, the different parameters K can be summarized in Table 4, based on the identified parameters listed in Table 1 and the above parameters Kp and Ki.

Table 4. The parameter K derived from different damping ratios ξl.

Under Damping Under Damping Critical Damping Over Damping

ξl

0.2

K

−0.22

0.707 −0.98

1 −1.52

2 −3.21

The closed-loop Bode diagram of the link-side controller from the desired velocity to link velocity with different parameters K are shown in Figure 10.
In a similar way, with parameter K increasing (damping ratio increasing), the system damping gradually increases, which lets the velocity ripple eliminate but decreases the closed-loop control bandwidth. Since there is no anti-resonance point in the transfer function from the desired velocity to link velocity, the phase will drop quickly, and the system is easy to diverge [39]. Therefore, the control parameters of the link-side controller should be very conservative. The parameter K as well as the proportional–integral gain Kp and Ki are set in Table 5.
Velocity RippleVelocityDriveEquivalentMethod