back-drivability improvement of geared system based on

IEEJ Journal of Industry ApplicationsVol.9 No.5 pp.475–485 DOI: 10.1541/ieejjia.9.475

Paper

Back-drivability Improvement of Geared System Based onDisturbance Observer and Load-side Disturbance Observer

Shuang Xu∗ Member, Minoru Yokoyama∗ Student Member

Tomoyuki Shimono∗∗ Senior Member

(Manuscript received July 2, 2018, revised Feb. 18, 2020)

This paper proposes a method of improving the back-drivability of torque control according to the dynamic character-istic of forward-drivability based on the concept of proposed equivalent expression. The control strategy is composed ofdisturbance observer (DOB), a load-side DOB (LDOB) and the designed controller for torque control. The DOB real-izes robust motion control by compensating for the friction and modeling error of the system, and the LDOB estimatesthe external torque input from the load-side. According to the designed controller, the proposed method constructs theback-drivability as forward-drivability via each transfer function of the equivalent expression. In addition, a scalingfactor is introduced to quantitatively adjust the back-drivability to adapt to different application demands. Moreover,the decision basis of parameters in the proposed control is discussed and the effect on the control system is clarified.The experimental results are compared with the conventional torque control method, and show the consistency in thequantitative improvement of back-drivability effectively with the scaling factor.

Keywords: geared two-inertia system, back-drivability, disturbance observer, load-side disturbance observer

1. Introduction

Geared two-inertia system is widely used in a lot of mecha-tronic systems because of its benefit of increasing torque out-put in the conditions of restricted space, mass, and motor ca-pacity. Recently, the demand of force control applicationswhich need to realize flexible interaction between system andenvironment are increasing. It is also expected that gearedmotors are utilized in force or torque control applications tosave cost or overcome output limitation.

In the industrial robot applications and cooperative oper-ations with humans require the structural flexibility to adaptsoft or stiff interaction between system and environment (1)–(3).In human support system and rehabilitation robotics, human-robot interaction must be able to transfer high torques at lowvelocity (feature of gearbox) and need high back-drivabilityto achieve safety (4)–(6). However, gears introduce friction,backlash, and noise, which make the back-drivability ofmotor low, especially the friction. Therefore, the back-drivability of geared two-inertia system is clarified as the tar-get of improvement in this paper.

To solve the problem of back-drivability deterioration,mainly two kinds of approaches such as hardware design andfriction compensation, have been proposed. In hardware de-sign, a novel mechanical designed robot arm, which is ap-plied with the series elastic actuator (SEA) retaining high

∗ Graduate School of Engineering, Yokohama National Univer-sity79-5, Tokiwadai, Hodogaya-ku, Yokohama, Kanagawa 240-8501, Japan

∗∗ Faculty of Engineering, Yokohama National University79-5, Tokiwadai, Hodogaya-ku, Yokohama, Kanagawa 240-8501, Japan

back-drivability, allows people to manipulate the load termi-nal freely and easily (7). A direct-dive high thrust spiral motorwith high back-drivability has been developed (8). However,back drivable actuator with complicated structure and diffi-cult control demands nominally lead disadvantages of highcost, long design cycle, restricted applying and so on. In theother approach, there is a common method of compensatingfor gear friction by using feed-forward control with a nec-essary force sensor to detect the interaction force betweenhuman and actuator (9). Furthermore, a new methodology tocompensate for both kinetic and breakaway friction withoutforce sensor is presented (10). However, the solutions based onfriction compensation always rely on the detection of frictionmodel, which are sometimes difficult to estimate from com-plicated mechanical systems. Besides, there is also a pro-posal of high back-drivable control based on joint torque andbacklash that is avoided using friction model (11). Nonetheless,this strategy only works on the condition of back driving.

According to the above, these authors offered a kind ofthought of patch to compensate the disadvantage of gearedtwo-inertia system so that to improve the back-drivability.Some other authors choose another way that is based on re-construction of dynamics characteristics. This strategy is re-quired to give a mathematical definition of back-drivabilityand evaluate it via implementing of designed controller.For example, a conceptual model of back-drivability suchas the ability for interactive transmission of force betweeninput and output is proposed (12), and a concept of idealback drivable motions evaluated from the load-side exter-nal torque to the load-side velocity and acceleration are alsoclaimed (13) (14). The conventional force control with P.I.D. con-troller is widely used in the industrial machines and roboticssuch as pseudo I-PD torque control using load-side torque

c© 2020 The Institute of Electrical Engineers of Japan. 475

Back-drivability Improvement for Geared Two-inertia System（Shuang Xu et al.）

observer and torsion torque sensor (13), and torque feedbackcontrol with PI controller using new torque sensor with lin-ear encoder (15). In this paper, authors also focused on the viewof constructing the definition of back-drivability to evaluatethis index, and give the comparison of proposed control andconventional method (16).

This paper defines a concept of back-drivability evaluatedfrom the load-side external torque to the motor-side accelera-tion, which is based on the conception of equivalent forward-drivability. A widely used disturbance observer (DOB) (17)

is employed to attain the robust control and friction com-pensation in the condition of lacking exactly correct frictionmodel (18). And then, the external disturbance of load side isestimated using load-side disturbance observer (LDOB) (19).The proposed control constructs the characteristic of back-drivability as the transfer function of forward-drivability by adesigned controller. Moreover, in some rehabilitation robotsof upper limb, the patients are required to hold the terminal ofrobot to follow rehabilitative exercises. In this case, the back-drivability as a sensitive index of force transmission from theload terminal to the motor output, is expected to be adjustedto make different rehabilitation course. Therefore, a scalingfactor α is introduced to achieve an adjustable sensitivity ofthe performance of the back-drivability to adapt to this kindof applications.

2. Geared Two-inertia System and Equivalent In-ertia

In this paper, a geared two-inertia system is discussed.Based on the equations description of this system, the con-cept of equivalent inertia from angular acceleration of motorto input torque is explained. Then in detail, the equivalentback-drive and forward-drive inertia of the system are math-ematically discussed to show the deterioration of back-driveinertia. The concept of back-drivability is also clarified inmathematics.2.1 Structure of Geared Two-inertia System A sys-

tem including geared connection between motor and load isoften modeled as a geared two-inertia system. The schematicview is shown in Fig. 1. The block diagram of geared two-inertia system is shown in Fig. 2. To simplify the calcula-tions, the motor side and load side viscosity coefficients areassumed as zero, which are like Dm = Dl = 0. The followingequations show the motion description for geared two-inertiasystem.

Jmθ̈m = τdri − 1Grτa · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (1)

Jlθ̈l = τa − τdisl · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (2)

τa = Ks

(1

Grθm − θl

)· · · · · · · · · · · · · · · · · · · · · · · · · · · · (3)

Where:Jm: exact motor inertiaJl: exact load inertiaτdri: driving torque of motorτm: motor torqueKs: the connector stiffnessθm, θl: motor angle and load angleτa: torsional torque for load arm

Fig. 1. The model of geared two-inertia system

Fig. 2. Block diagram of geared two-inertia system

Fig. 3. Equivalent inertia of forward-drive and back-drive of two-inertia system

τdisl : disturbance torque from load side2.2 Concepts of Forward-drive and Back-drive In

dynamics of rotational motor, torque is required to generateangular acceleration which is inversely proportional to its in-ertia. The concepts of forward-drive inertia and back-driveinertia are shown in Fig. 3, and the construction of each in-ertia is shown as Fig. 4. The equivalent forward-drive inertiaJF(s) and back-drive inertia JB(s) are brought into consid-eration to discuss these motions based on geared two-inertiasystem, which are described as Eq. (4) and Eq. (5).

JF(s) =τdri

θ̈m=

JmG2r (Jls2 + Ks) + JlKs

G2r (Jls2 + Ks)

· · · · · · · · · · (4)

JB(s) =τdis

l

θ̈m= − JmG2

r (Jls2 + Ks) + JlKs

KsGr· · · · · · · · (5)

Where:JF(s): equivalent inertia of forward-driveJB(s): equivalent inertia of back-driveDrivability is defined as an index of transmission perfor-

mance from effected torque to motor acceleration. Basedon this definition, the drivability is expressed as the recip-rocal expression of equivalent inertia. In this paper, the back-drivability is discussed as performance of torque transmissionfrom the load terminal input to the motor angular acceleration

476 IEEJ Journal IA, Vol.9, No.5, 2020


Fig. 4. Equivalent block diagram of Fig. 2

Table 1. Parameters for simulation of step responses

Nominal motor inertia Jmn 5.8 × 10−8 kgm2

Nominal load inertia Jln 1.39 × 10−5 kgm2

Nominal torque constant of drive motor Ktn 0.0139 N/ANominal connection constant Ksn 1250

Reducer ratio Gr 84Control period Ts 0.1 ms

output, which is shown as Eq. (6). It is easy to know, thesmaller of equivalent inertia, the better drivability in bothforward-drive and back-drive.

GB(s) =1

JB(s)=θ̈m

τdisl

· · · · · · · · · · · · · · · · · · · · · · · · · · · (6)

2.3 Comparison of Forward and Back Motion Char-acteristics In this section, motion characteristics are dis-cussed according to the expression of forward-drivability andback-drivability. The simulated step responses are drawn inthe condition of the experimental parameters shown as Ta-ble 1. The forward-drivability and back-drivability are calcu-lated via the definition of drivability and parameters, whichshown as Eq. (7) and Eq. (8).

GF(s) =θ̈mτdri=

JlG2r s2 + KsG2

r

JmJlG2r s2 + JmG2

r Ks + JlKs· · · · · (7)

GB(s) =θ̈m

τdisl

= − KsGr

JmJlG2r s2 + JmG2

r Ks + JlKs· · · · (8)

According to the calculation of amplification, only constantitems are need to consider as following:

AGF = 20 log10

(KsG2

r

JmG2r Ks + JlKs

)

= 20 log10(1.6675 × 107) = 144.4 · · · · · · · · · · · (9)

AGB = 20 log10

(∣∣∣∣∣∣−KsGr

JmG2r Ks + JlKs

∣∣∣∣∣∣)

= 20 log10(1.985 × 105) = 105.9 · · · · · · · · · · · (10)

The acceleration of motor in Fig. 5 and Fig. 6 are the forward-drivability and back-drivability of geared two-inertia systemin the condition of input torque as 1 Nm, which are consistentwith equation expressions.

In next section, the proposed method of improving back-drivability is based on expression of forward-drivability. Adesigned controller is considered with the goal of construct-ing a same motion characteristic with forward-drive.

3. Strategy of Torque Control for ImprovingBack-drivability

In this section, the motion characteristic of forward-drive

Fig. 5. Step response of forward-drive in mechanical control

Fig. 6. Step response of back-drive in mechanical control

with DOB is seen as the goal of construction firstly. Andthen the proposed back-drivability is designed to consistentwith referenced forward-drivability. At last, the discussionof scaling factor α, which is introduced to adjust the back-drivability, is given in both situations of continuous systemand discrete system.3.1 DOB The block diagram of torque feedback con-

trol with DOB is shown as Fig. 7. In general, disturbanceobserver introduces an acceleration controller to estimate andcancel the disturbance torque as quickly as possible. The esti-mated disturbance torque is obtained from the motor velocityθ̇m and current reference Iref shown in the deeper grey block



Fig. 7. Block diagram of forward-drive feedback con-trol with DOB

of Fig. 7. The disturbance torque τdism is denoted as Eq. (11).

τdism = (Jm−Jmn)θ̈m+(Ktn−Kt)I

ref +τ f ric+τreac · · · · (11)

τ̂dism =

gmdis

s + gmdisτdis

m · · · · · · · · · · · · · · · · · · · · · · · · · · · · (12)

Where,τ f ric: coulomb frictionτreac: torsional reaction torque to motor

Kt = Ktn, Jm = Jmn, Jl = Jln · · · · · · · · · · · · · · · · · · · · (13)

From above construction of DOB, the difference betweennominal value and exact value in first and second term arecompensated by τ̂dis

m which is given through a low-pass fil-ter with cut-off frequency gmdis in Eq. (12). Reaction forceobserver (RFOB) (20) is also a useful tool to estimate reactionforce by eliminating friction force τ f ric without force sen-sor. In this research, a RFOB just simply estimates all dis-turbance torques without consideration of friction model. Inother words, not only friction but also reaction disturbanceare as feedback from RFOB to input. Therefore, as Eq. (13),the nominal value of inertia of motor, load and torque con-stant (Jmn, Jln, Ktn) are used in this paper to replace identifi-cation of the exact value of them. In the calculation of the restof this paper, the difference between exact value and nominalvalue would be ignored to simplify the representation.

The third term of friction makes the back-drivability ofgeared two-inertia system deteriorate. The friction of systemcan be compensated in the condition lacking exactly correctfriction model for the sake of DOB (18). Since the DOB is ableto reject disturbances, it increases the motor impedance sig-nificantly (21). It brings to good effect to forward-drive becausethe compensated torque is contributed to drive side of mo-tor to generate more torque. However, in back-drive, the fedback torsional reaction torque τreac is performed as the motorimpedance to reduce the velocity of motor output instead. Inother words, it is a tradeoff between dynamics properties and

friction compensation while keeping off the complicated cal-culation. At the same time, a way of remedy is employed byselecting a lower cut-off frequency of DOB in this paper toreduce the high frequency components of estimated torsionalreaction torque.

Considering the robustness of system, DOB is one ofthe widely recognized robust motion control compared withcomplicated sliding control and fuzzy control. However,DOB brings an estimation delay to system but a high-performance DOB is not intended for geared two-inertia sys-tem (22). Even though the motion control with DOB could notachieve high robustness against a step torque input from load-side, it still worth to use DOB to reduce the calculation ofconstructing the characteristic of back-drivability in proposedmethod.3.2 Referenced Forward-drive with DOB From

this system of Fig. 7, which neglects the effect of friction,the equation of forward-drivability is calculated as shownEq. (14):

GFD(s) =θ̈mτdri=

G2r (Jls2 + Ks)

JmG2r (Jls2 + Ks) + JlKs

= GF(s) · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (14)

It is shown that, if we consider as the effected torque andonly describe the expression of plant of motor, the expres-sion of back-drivability is same with open loop of mechan-ical forward-drive. However, the effect of parameters suchas force gain and cut-off frequency could not clarified in thisequation. Therefore, taking τcmd into the expression shownas Eq. (15).

τdri=τcmdKf Jm+(1−Kf Jm)JlKsgmdis

G2r (Jls2+Ks)(s+gmdis)

θ̈m

· · · · · · · · · · · · · · · · · · · (15)

The new forward-drivability is given for showing the trans-mission performance from command torque to output motoracceleration. G

′FD(s) is shown as Eq. (16).

G′FD(s) =

θ̈mτcmd

=

Kf JmG2r (Jls2 + Ks)(s + gmdis)

JmG2r (Jls2 + Ks)(s + gmdis) + JlKss + Kf JmJlKsgmdis

=bFD3s3 + bFD2s2 + bFD1s + bFD0

aFD3s3 + aFD2s2 + aFD1s + aFD0· · · · · · · · · · · · · (16)

bFD3 = Kf JmG2r Jl, bFD2 = Kf JmG2

r Jlgmdis,

bFD1 = Kf JmG2r Ks, bFD0 = Kf JmG2

r Ksgmdis,

aFD3 = JmG2r Jl, aFD2 = JmG2

r Jlgmdis,

aFD1 = Ks(JmG2r + Jl), aFD0 = JmKsgmdis(G

2r + Kf Jl)

From the new expression of forward-drivability, we can findthat it is a high-order dynamics representation.3.3 LDOB In order to design a proposed back-

drivability, a load-side disturbance observe is needed to im-plement for estimating the external torque to achieve torquecontrol. Some different structures of load-side disturbanceobserve are proposed to estimate load-side input. A newmulti encoder based disturbance observer (MEDOB) is pro-posed with the advantage of no necessary to identify the nom-inal connection coefficient Ksn

(23), which is not suitable to



Fig. 8. Block diagram of designed back-drive controlbased on DOB and LDOB

proposed method due to the complex structure. In this paper,a similar structure of DOB is used for load-side disturbanceobserver shown as Eq. (17).

τ̂disl =

gldis

s + gldis

((Ks − Ksn)

(θl − 1

Grθm

)+ (Jl − Jln)sθ̇l

+ τdisl

)· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (17)

The LDOB is calculated from difference between the nom-inal torsional torque and nominal load-inertia torque. It isestimated though a low-pass filter with cut-off frequency ofgldis, so that goes through the designed controller to feed-back to reference torque input of motor. In other words, theproposed control is achieved via a semi-closed loop controlfrom estimated external torque to driving motor using thedesigned controller H(s) shown as Fig. 8. The conventionaltorque control with torque gain Ct that compared in this pa-per is shown as Fig. 9 (16). A third-order low-pass filter withsame cut-off frequency gdl is implemented using accelerationcontrol with DOB. In following, the progressing of consider-ation to design the proposed back-drivability is discussed tocompare to conventional control method.3.4 Control Design for Back-drivability First, con-

structing the proposed back-drivability to consistent with ref-erenced forward-drivability. A scaling factor is introduced torealize adjusted drivability such as Eq. (18).

GBD(s) = −α G′FD(s) (α > 0) · · · · · · · · · · · · · · · · · · · (18)

The minus sign is the direction of input signal. Calculatingthe expression of back-drivability of Fig. 8. It is shown asEq. (19).

GBD(s)=−H(s)JmG2

r (Jls2+Ks)(s+gmdis)gldis

s+gldis+GrKss

JmG2r (Jls2+Ks)(s+gmdis)+JlKss

= − bBD3s3+bBD2s2+bBD1s+bBD0

aBD4s4+aBD3s3+aBD2s2+aBD1s+aBD0· · · · · · (19)

Bringing Eq. (16) and Eq. (19) into Eq. (18) as Eq. (20).

bBD3s3 + bBD2s2 + bBD1s + bBD0

aBD4s4 + aBD3s3 + aBD2s2 + aBD1s + aBD0

Fig. 9. Block diagram of conventional torque controlbased on load-side torque observer

= −αbFD3s3+bFD2s2+bFD1s+bFD0

aFD3s3+aFD2s2+aFD1s+aFD0(α>0) · · · · · (20)

Before solving this equation to get the expression of H(s), itis easy to find that the order of numerator in back-drivabilityis not consistent with the order of forward-drivability. So, alow-pass filter is implemented to the control of forward-drivethat is defined as GFDL(s) to achieve the same dynamics char-acteristic as Eq. (21).

GFDL(s) =gLPF

s + gLPFG′FD(s) · · · · · · · · · · · · · · · · · · · · (21)

And then the design strategy is rewritten as Eq. (22).

GBD(s) = −αGFDL(s) (α > 0) · · · · · · · · · · · · · · · · · · (22)

Expanding Eq. (22), Eq. (23) is the complete expression ofconstructing proposed back-drivability.

bBD3s3 + bBD2s2 + bBD1s + bBD0

aBD4s4 + aBD3s3 + aBD2s2 + aBD1s + aBD0

= −α gLPF

s + gLPF

bFD3s3 + bFD2s2 + bFD1s + bFD0

aFD3s3 + aFD2s2 + aFD1s + aFD0

= −αbFDL3s3+bFDL2s2+ · · · +bFDL0

aFDL4s4+aFDL3s3+ · · · +aFDL0(α>0) · · · · (23)

bFDL3 = gLPFbFD3, bFDL2 = gLPFbFD2,

bFDL1 = gLPFbFD1, bFDL0 = gLPFbFD0,

aFDL4 = aFD3, aFDL3 = aFD2 + gLPFaFD3,

aFDL2 = aFD1 + gLPFaFD2, aFDL1 = aFD0 + gLPFaFD1,

aFDL0 = gLPFaFD0

Therefore, the solution of expression H(s) is shown asEq. (24). By the way, for simplifying the calculation, we letthe cut-off frequency of LDOB (gldis) to be consistent to low-pass filter gLPF in this paper.

H(s) =

αKfJmG2

r (Jls2+Ks)(s+gmdis)+JlKss

JmG2r (Jls2+Ks)(s+gmdis)+JlKss+Kf JmJlKsgmdis



− Kss(s + gldis)JmGrgldis(Jl s2 + Ks)(s + gmdis)

=bH6s6+bH5s5+bH4s4+bH3s3+bH2s2+bH1s+bH0

aH6s6+aH5s5+aH4s4+aH3s3+aH2s2+aH1s+aH0

· · · · · · · · · · · · · · · · · · · (24)

bH6 = αKf J2mG4

r J2l gLPF ,

bH5 = 2αKf J2mG4

r J2l gmdisgLPF − JmG3

r JlKs,

bH4 = αKf JmG2r JlgLPF(JmG2

r Jlg2mdis+2JmG2

r Ks+JlKs)

− JmG3r JlKs(gmdis + gldis),

bH3 = JmG2r JlKsgmdisgLPF(4αKf JmG2

r + αKf Jl −Gr)

−GrK2s (Jl + JmG2

r ),

bH2=αKf J2mG4

r Ks(2Jlg2mdis gLPF+KsgLPF)−Gr JlK

2s gldis

+ Kf JmGr JlK2s (αGrgLPF − gmdis)

− JmG3r K2

s (gmdis + gldis),

bH1 = αKf JmG2r K2

s gmdisgLPF(Jl + 2JmG2r )

− JmGrK2s gmdisgldis(Kf Jl +G2

r ),

bH0 = αKf J2mG4

r K2s g

2mdis gLPF ,

aH6 = J2mG4

r J2l gldis,

aH5 = 2J2mG4

r J2l gmdisgldis,

aH4 = JmG2r Jlgldis(JmG2

r Jlg2mdis + 2JmG2

r Ks + JlKs),

aH3 = JmG2r JlKsgmdisgldis(Kf JmJl + 4JmG2

r + Jl),

aH2 = J2mG2

r JlKsg2mdis gldis(Kf Jl + 2G2

r )

+ JmG2r K2

s gldis(JmG2r + Jl),

aH1 = JmG2r K2

s gmdisgldis(Kf JmJl + 2JmG2r + Jl),

aH0 = J2mG2

r K2s g

2mdis gldis(G

2r + Kf Jl)

Finally, the expanded expression of back-drivability GBDH(s)is shown as Eq. (25) by introducing Eq. (24),

GBDH(s)=−α (bBDH9s9+ · · · +bBDH4s4)+ · · · +bBDH0

(aBDH10s10+ · · · +aBDH5s5)+ · · · +aBDH0

· · · · · · · · · · · · · · · · · · · (25)

bBDH0 = αKf JmG2r KsgmdisgLPF = bFDL0,

aBDH0 = JmKsgmdisgLPF(G2r + Kf Jl) = aFDL0

The 10-orders polynomial coefficients of transfer functionGBDH(s) is the expanded form as Eq. (19). The higher com-ponents in brackets are suppose to be zero. According tothe constant items, the amplification of referenced forward-drivability is as follow:

AGFDL = 20 log10

(bFDL0

aFDL0

)= 20 log10

(αKf

1 + 1.97−9Kf

)

� 20 log10(αKf )

= AGBDH · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (26)

According to the small value of Kf , the item 1.97−9Kf �1,so that the amplification of referenced forward-drivability isonly in proportion to torque gain Kf and scaling factor α.

Here, let’s review the constant items of continuous expres-sion of controller H(s):

bH0 = αKf J2mG4

r K2s g

2mdis gLPF ,

aH0 = J2mG2

r K2s g

2mdis gldis(G

2r + Kf Jl)

AGH = 20 log10

(bH0

aH0

)= 20 log10

(αKf

1 + 1.97−9Kf

)

� 20 log10(αKf ) · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (27)

It is shown that Eq. (27) is same with the Eq. (26), whichmeans the designed back-drivability contributes from thecontroller H(s) entirely.3.5 Design for Proposed Back-drivability Acco-

rding to above discussion theoretically, back-drivability asEq. (25) is suppose to realize not only same amplitude butalso dynamics characteristic with the forward-drivabilitywith low-pass filter as Eq. (21) in the condition of α = 1.In order to focus the discussion on scaling factor α, anotherimpact factor Kf is suppose to be decided. Since the inputtorque can not be selected too big in consideration of gear ra-tio, the torque gain of forward-drive is decided as the 0.0001multiple of 1/Jm (Kf = 1724.1) to simplified the calculation.So that Eq. (26) is rewritten as Eq. (28):

AGFDL =AGBDH �20 log10(αKf )=64.7 (α=1) · · · · · (28)

Nevertheless, in experiments, the controller H(s) is describedin C programming. Since the designed controller is too com-plicated to realize factorisation, the signal discretization isapplied to motion control. In this research, Tustin transformis utilised. The transform principle is shown as Eq. (29),

s =2Ts

1 − z−1

1 + z−1· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (29)

where:s: operator of continuous signalz: operator of discrete signalTs: sampling period of discretizationThe discretized controller is suppose to be expressed as

Eq. (30) with discrete operator,

H(z)=bdH0+bdH1z−1+bdH2z−2+ · · · bdH6z−6

adH0+adH1z−1+adH2z−2+ · · · adH6z−6· · · · · (30)

where:adH0, adH1, ...: parameters of each order of discrete integral

operator, the numerical subscript denotes the orderbdH0, bdH1, ...: parameters of each order of discrete differ-

ential operator, the numerical subscript denotes the orderz−1, z−2, ...: periods of signal delay, the numerical supscript

denotes how many periods delayedTherefore, the full orders of representation in Eq. (25)

would be kept (Higher components in brackets are not zero.),and take part in the discretization. As a result, the expectedeffect on the system considered from constant items in dis-cretized controller H(z) are as follow.

AGH (z) = 20 log10

(bdH0

adH0

)

= 20 log10

(3.16×1012αKf −4.44×106Kf −2.49×1018

3.16×1012

)

� 20 log10(αKf − 7.87 × 105) (Kf = 1724.1) · · · · (31)

The minus item of Eq. (31) is the effect of higher compo-nents that not shown in continuous system. For realizing ex-pected quantitatively increasing of back-drivability, αKf issuppose to be satisfied to be multiple of 7.87 × 105. As aresult, α0 = 456.6 is selected as a standard value of scalingfactor. In next section, the value of multiple of α0 are used toimplement to experiments.



Fig. 10. Equivalent block diagram of full dynamics

4. Discussion

The design strategy and method of proposed back-drivecontrol are discussed in previous two sections. In this section,the dynamics of back-drive motion and the design method ofeach parameter in controller are summarized.4.1 Dynamics of Proposed Control Figure 10 is the

equivalent block diagram of in the condition of both refer-enced forward control and proposed back-drive control. Inthis figure, the control blocks in the red frame is the refer-enced forward dynamics with DOB and feedback control,while the control blocks in the blue frame is the proposedback-drive dynamics in this paper. Obviously, the proposedback-drive dynamics includes an inner loop of DOB for com-pensation and a feed-forward loop to motor with designedcontroller H(s).

In the view of expressions, Eq. (32) shows the full dynam-ics descriptions of proposed control method. In this research,the authors only discussed the situation of load side distur-bance torque input while τcmd = 0.

θ̈m =

[GFDL(s) 0

0 GBD(s)

] [τcmd

τdisl

]· · · · · · · · · · · (32)

4.2 Parameters Design In this paper, there are threedifferent cut-off frequencies implemented in controller forDOB, LDOB and low-pass filter, respectively. However, evenif there is no physical meaning to discuss the stability in thispaper, the three parameters of cut-off frequency matter thestability and robustness of the proposed system. The designmethods of cut-off frequency based on the root locus of back-drivability are explained one by one in this section.

The compensation of DOB is discussed at first. It is easyto observe the cut-off frequency of DOB affects the locationof poles of Eq. (19). In Fig. 11, the circle and cross symbolsshow the zeros and poles affected by the increasing of gmdis.Obviously, the complex conjugate poles are getting far awayto the negative half plane and close to the real axis, while thereal poles are also getting far away from origin.

Secondly, the item of (s + gldis) is cancelled in the com-bination of Eq. (19) and (24). Therefore, the parameter gldis

supposes no matter to stability of the system that is verifiedas Fig. 12. The circle and cross symbols show the zeros andpoles affected by the increasing of gldis. The poles have nochange of location.

The last one, in Fig. 13, the circle and cross symbols showthe zeros and poles affected by the increasing of gLPF . Thereal poles have no change of location, while the complex con-jugate poles are getting close to the imaginary axis.

Therefore, the Fig. 12 and Fig. 13 show that if gldis andgLPF are assumed as same value as 1000 rad/s, the system re-alizes better effect of vibration suppression. In addition, the

Fig. 11. Root Locus of feed-forward system using pro-posed controller. Parameter: gmdis = 100–1000 rad/s

Fig. 12. Root Locus of feed-forward system using pro-posed controller. Parameter: gldis = 1000–3000 rad/s

Fig. 13. Root Locus of feed-forward system using pro-posed controller. Parameter: gLPF = 1000–3000 rad/s

parameter of gmdis is playing a pair of contradictory roles thatdifficult to decide. The farther the complex conjugate polesare away from imaginary axis, the better for vibration sup-pression. At the same time, the real pole closer to origin isrequired for slowly decaying of components. Finally, by trialand error, the smaller value 100 rad/s is decided for gmdis.4.3 Scaling Factor In this paper, the performance

of back-drivability that is evaluated via speed variation in-tuitively is discussed based on parameter of scaling factorα. Obviously, the item of α affects only numerator of the



Fig. 14. Root Locus of feed-forward system using pro-posed controller. Parameter: α = 2α0–8α0

expression in the combination of Eq. (19) and (24). There-fore, the scaling factor α supposes no matter to stability ofsystem that is verified as Fig. 14.

Actually, theoretically the scaling factor α looks like anunrestricted parameter, it does have limitation of selectionbased on experimental results. Even if the scaled up inputtorque from external to feed-forward to motor, the driver ofmotor could not generate enough current to follow it. There-fore, once the scaling factor is selected larger than the driv-ing ability of motor, the back-driving system would keep thelimited maximum velocity of motor and be out of control.In experimental setup, the drive motor has the restricted ve-locity at about 750 rad/s because of velocity protection ofmotor driver. Based on this situation, the range of scal-ing factor is selected under the limitation of max velocity asα < 9α0 = 4109.4 in next section.

5. Experiments

In this section, both conventional method and proposedcontrol method are actually implemented in different condi-tions by two groups of cases. The effectiveness of proposedmethod is clarified by comparing to conventional experimentresults.5.1 Experimental Setup In experiments, the over-

view of proposed control method, which described based onthe equipment of geared two-inertia system such as Fig. 15, is

shown as Fig. 16. The utilized parameters refer to Table 2. Inthis experiment, a Maxon DC motor (2.0 W type) with 1 : 84gear ratio is selected as the drive motor. The inertial loadwhich consists of another Maxon DC motor (150 W type), isconnected to the drive motor by a coupling. Two angle en-coders are implemented to both motor and load side with theresolution proportion following the gear reduction. The angleencoder of drive motor is 512 plus/r, while the angle encoderof load motor is 32768 plus/r. Via this design, the externaltorque input could be emulated with torque output of loadmotor.5.2 Simulation and Experiment Results of Forward

and Back Drivabilities In Section 3, the simulated re-sults of forward-drivability and back-drivability in mechani-cal motion had been shown. The results of two drivabilities inreferenced forward drive and proposed back drive are givenin the case as Table 3. The nominal torque constant of loadmotor Ktnl is implemented to simulations.

Case 1-1 and case 1-2 are shown the simulation results asFig. 17 and Fig. 18, the same acceleration of motor are real-ized in referenced forward-drive and proposed control. Bothof forward and back drive are achieved 86.2 rad/s2 of accel-eration, which means the same drivability. In experiment re-sults of case 1-1 and case 1-3 shown as Fig. 19, the scalingfactor α is approximately selected as two times of α0 to sat-isfy with the same scaling as referenced forward-drive. Itis shown that the back-drivability realize almost 80 rad/s2,while the forward-drivability is lower than simulated results(only around 60) rad/s2.5.3 Experiments for Comparison of Back-drivability

from Conventional and Proposed Method In this sec-tion, the effect of scaling factor α is discussed in proposedcontrol and compared with conventional method by changing

Fig. 15. The equipment of geared two-inertia systemused in experiments

Fig. 16. Block diagram of proposed back-drive with designed controller



Table 2. Control parameters

Nominal motor inertia Jmn 5.8 × 10−8 kgm2

Nominal load inertia Jln 1.39 × 10−5 kgm2

Nominal torque constant of drive motor Ktn 0.0139 N/ANominal torque constant of load motor Ktnl 0.0302 N/A

Nominal connection constant Ksn 1250Reducer ratio Gr 84Torque gain K f 1724.1

Cut-off frequency of DOB gmdis 100 rad/sCut-off frequency of LDOB gldis 1000 rad/sCut-off frequency of LPF gLPF 1000 rad/s

Control period Ts 0.1 msStandard scaling factor α0 456.6

Table 3. Comparison of forward-drivability and back-drivability in simulation and experiments

case 1-1 τcmd = 0.05 Nm(t ≤ 1)case 1-2 τdis

l = 0.05 Nm(t ≤ 1)(α = 1)case 1-3 τdis

l = −0.05 Nm(t ≤ 1)(α = 2α0 = 913.2)

Fig. 17. Case 1-1: step response of forward-drive in ref-erenced forward control in simulation

Table 4. Comparison of back-drivability in conven-tional and proposed method

case 2-1 τdisl = 0.05 Nm Ct = 0.05/Jm , 0.1/Jm, 0.2/Jm, 0.5/Jm, 1/Jm

case 2-2 τdisl = −0.05 Nm α = 2α0, 4α0, 6α0, 8α0 (α0 = 456.6)

torque gain Ct. Case 2 shown in Table 4 is given in a constanttorque input for different performance of back-drivability.The input torque is given during t=0-1s too.

Figure 20 is the group of velocities in different torque gainwhile Fig. 21 is the groups of velocities in different scalingfactor. We are going to discuss the back-drivability in termof conventional method which is defined from the load-sidetorque to velocity of load. When the torque gain is selectedas lower values from 1

20Jmto 1

2Jm(Author selected torque gain

as 120Jm

in conventional method.), the velocity of load is in-

creased little. However, keeping on largen the gain to 1Jm

, theload velocity is improved a lot with un-clarified proportionaland accompanied by oscillation. In contrast, the load velocity

Fig. 18. Case 1-2: step response of back-drive in pro-posed control in simulation

Fig. 19. Case 1-1,1-3: experimental comparison offorward-drivability and back-drivability in condition ofα = 913.2

Fig. 20. Comparison of velocity by increasing torque gain

of proposed method is proportionally increased along withthe scaling factor.

It is easy to see the back-drivability performance plot in3D figure in Fig. 22 and Fig. 23. From Fig. 22, the per-formance of back-drivability in conventional method is dis-played as a broken line shown as the black one with ar-row since the un-clarified proportion of torque gain Ct. In



Fig. 21. Comparison of velocity by increasing scaling factor

Fig. 22. Comparison of back-drivability by torque gain

Fig. 23. Comparison of back-drivability by scaling factor

Fig. 23, the approximate exponential line of improving ofback-drivability is shown as red one with arrow.

This is the discussion of comparison of back-drivabilityimprovement between conventional and proposed method.As we discussed about the effect of DOB in section 3.1, theDOB not only compensates the modeling error and frictionbut also the torsional reaction torque from load side, whichis performed as the motor impedance to reduce the veloc-ity of motor output in back-drive. The torque gain of torquefeedback control is not proportional to back-drivability dueto two terms of effect torques on the motor in opposite re-action. Only if the torque gain is selected big enough toovercome the other torque component, the back-drivabilitycan not be shown as proportional. Furthermore, it is still notshown quantified improved in big gain due to big gain alsolead to oscillation of output velocity. In proposed method,the scaling factor α is designed as the proportion of forward-drivability at the beginning. The two terms of effect torques

on the motor in opposite reaction has been compensated bycontroller H(s). In spite of the order difference in experi-ment according to the constructal strategy, the improvementof back-drivability of proposed method could realize quanti-fied improvement by adjusting scaling factor.

6. Conclusion

In this paper, the back-drivability is discussed from theview of construction of control which is described based onforward-drive and back-drive equivalent inertias. The trans-fer function and the mathematical description of forward-drivability and back-drivability are given. In geared two-inertia system, the drivability of back-drive is lower thanthe forward-drive due to the increasing gear ratio and fric-tion. The control system achieves torque feedback controlwith designed controller H(s) in the conditions of the fric-tion is compensated by DOB while the estimated torque ofload side is realized by LDOB. It is constructed for proposedback-drivability with the target of agreeing with referencedforward-drivability. A scaling factor is introduced to realizeadjusted performance of drivability. The decision basis ofparameters appeared in proposed control are discussed andalso clarified the effect on the control system. The construc-tal back-drivability is compared with referenced forward-drivability and conventional torque control method in exper-iments. In the proposed method, back-drive can achieve thesame level drivability as forward-drive, and also can achievequantified improvement of back-drivability by adjusting scal-ing factor.

AcknowledgmentThis research was supported in part by the New Energy and

Industrial Technology Development Organization (NEDO)of Japan.

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Shuang Xu (Member) received the B.E. degree in electronic engineer-ing from Chengxian College, Southeast University,Nanjing, China, in 2008, and the M.E. degree in elec-trical and computer engineering from Yokohama Na-tional University, Yokohama, Japan, in 2015, whereshe is currently working toward the Ph.D. degree. Herresearch interests include geared two-inertia motorand scaled bilateral linear motion control.

Minoru Yokoyama (Student Member) received the B.E. and M.E.degrees in electrical and computer engineering fromYokohama National University, Yokohama, Japan, in2015 and 2017, respectively. He is currently workingtoward the Ph.D. degree at Yokohama National Uni-versity. His research interests include mechatronics,motion control and haptics.

Tomoyuki Shimono (Senior Member) received the B.E. degree in me-chanical engineering from Waseda University, Tokyo,Japan, in 2004 and the M.E. and Ph.D. degrees inintegrated design engineering from Keio University,Yokohama, Japan, in 2006 and 2007, respectively.Since 2009, he has been with Faculty of Engineering,Yokohama National University, Yokohama, where heis currently an Associate Professor. He has also beena project leader of Kanazawa Institute of IndustrialScience and Technology since 2016. His research in-

terests include haptics, motion control, medical and rehabilitation robots,and actuators.


back-drivability improvement of geared system based on

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