Design of Nominal Parameters for Robust Sensorless Force ...

IEEJ Journal of Industry ApplicationsVol.8 No.2 pp.342–351 DOI: 10.1541/ieejjia.8.342

Paper

Design of Nominal Parameters for Robust Sensorless Force ControlBased on Disturbance Observer

Hiroki Kurumatani∗ Member, Seiichiro Katsura∗ Senior Member

(Manuscript received May 2, 2018, revised Sep. 12, 2018)

The paper proposes a parameter design of a reaction force observer (RFOB) under existence of modeling er-ror/parameter fluctuation. Observer-based sensorless-force-control is a good approach to reduce phase lag in controlsystems. Hence, the performance improvement can be easily attained by such a technique. However, the RFOB doesnot always guarantee accuracy of estimated value and adds incorrect compliance on the system. Due to insufficientreport about RFOB design, its calibration is conducted based on the designer’s own experience. To calibrate the RFOBand achieve the correct force control quantitatively, the structure of the observer-based force control and physical in-terpretation of control loops should be revealed simply. The paper presents a condition to achieve the correct forcecontrol and design methodology of observers, thereby, providing a robust performance against parameter variation.

Keywords: sensorless force control, disturbance observer, nominal parameter

1. Introduction

Industrial machines are now widely used in various in-dustrial process. The machine performs with high preci-sion and high efficiency, which are difficult for human toachieve. Since the machine is good at repetitive motion, awork piece is placed into appointed area in the factory. Here-with, positioning technique benefits from above merits withthe machine while only having tolerance against external dis-turbance. For more advantage and convenience in industry,demand for force control is on the increasing (1)–(3). Decline oflabor force caused by low birth-rate, longevity, and retiringof trained engineers accelerate such requirement. Simulta-neously, adaptation of the machine into welfare/life supportfields is strongly expected. Furthermore, performance im-prove of machining is also demanded. Due to demand forhigh-variety low-volume manufacturing along diversificationof human color, flexibility for work plays a significant role (4).

Underlying problems in the force control have been ad-dressed for a long time (5) (6). Phase lag penetrated into anopen loop is critical in the force control since it determinesstability and performance. Actuator saturation, sensing la-tency, bearing friction are ones of the causes of the phase lagin a control loop. Physical dynamics such as arms, sensorsand work pieces also affect the stability and the performance.For performance improvement, effectiveness of proportional-differential (PD)/proportional-integral (PI) force controllersare analytically presented (5). However, implementation ofthe PD controller is difficult as differential value of forceinformation includes a lot of noise. The PI controller sup-presses steady-state error but performance improvement intransient-state is difficult as it is classified into 1-degree-of-freedom (DOF) controller. For such problem, 2-DOF force

∗ Department of System Design Engineering Keio University3-14-1, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

control is introduced while implementing a disturbance ob-server (DOB) (7). A DOB has a role of a type 1 servo sys-tem and decouples a tracking performance and a disturbancesuppression performance (8). Also, Kalman-filter based signalprocessing are utilized for force sensing as it provides littlelatency in filtering (9). Such technique reduces the phase lag ina control loop and contributes to design of the PD controller.

In the design of peripheral devices/control structure, de-velopment of wide-band drive systems and observer-basedforce estimation are good approaches for reduction of thephase lag (10)–(13). The reaction force observer (RFOB) removesensor dynamics from the force control system (14). In otherwords, the phase lag or noise caused by sensors can be re-duced. Reduction of components shut out extra noise. Thesensorless force control is effective for performance improve-ment as the force control requires severe condition againstthe phase lag in an open-loop (13). Furthermore, the sensorlessis also good in terms of install. The machine with narrowinstallation area benefits from above technique. Adaptationof the force control to many machines are also supported bylow installation-cost. However, estimated value of the RFOBdoes not always match to true value. This is because systemparameters used in the observer fluctuates by time-related de-terioration. Since compliance of the force controller is setfollowing to estimated value, incorrect estimation adds un-reasonable impedance on the system. Therefore, observerdesign plays an important role in the sensorless force con-trol. In industry, a traceability, which is an index of relia-bility for a certified quantitative performance, is abided tomanage a design assets (15). The index is essential index to ad-just operations of machines and analyze plant systems. How-ever, the traceability of the sensorless force control has notproven yet. In our knowledge, there are few researches forRFOB design/validation of reliability, although intentionalparameter-design of the DOB is widely researched (16)–(18). Itis true that most modern digital drives are designed under

c© 2019 The Institute of Electrical Engineers of Japan. 342

Robust Sensorless Force Control Based on Disturbance Observer（Hiroki Kurumatani et al.）

identify-then-control and able to cope with changes in plantparameters (19). However, since the RFOB is consist of sev-eral parameters and model identification is not perfect, an er-ror factor is hard to identify and its calibration is difficult inindustry where a plant system is easy to wear. Machining fre-quently causes mass reduction of drills since their bodies areshaved as long as a target object (20). When parameter vari-ation of a system occurs, it is a fatal defect in practical usebecause reliability of quality get to be not clear. With the aimof practical use, the paper reveals a structure of the sensorlessforce control and shows a physical interpretation of controlloops. In section 2, the sensorless force controller is decom-posed to simple components and the paper shows roles of thenominal parameters of observers. The paper also presentsa condition to achieve the correct force control and designmethodology of the RFOB which provides a robust perfor-mance against parameter variation is described in section 3.By simplifying the control structure and its roles, the sen-sorless force control can be physically interpreted easily anddesign method of compensators was derived. Several simu-lation and experiments are conducted for theory verification.

2. Robust Sensorless Force Control

2.1 System Setup The paper discusses about theDOB-based 2-DOF sensorless force control. The sensorlessforce control system is constructed using the DOB and theRFOB. The RFOB estimates an external force affecting ona system based on the internal model principle. When anymodeling errors do not exist, an accuracy of a disturbance-generating polynomial is limited by only a pseudo differen-tiator. The paper treats a 1-DOF linear actuator system forsimplicity of verification. In this time, the paper puts assump-tion that disturbance such as current control limit, mechani-cal friction or gravity is small enough to be ignored. A linearpower amplifier and an air slider are used for remove suchdisturbance in experimental validation.2.2 Robust Sensorless Force Control Using RFOBThe paper shows the DOB-based sensorless force con-

trol. As the DOB realizes the acceleration control and pro-vides the physical clarity for the control system, a controldesign is simplified along with the physical phenomena. Akey enabling technique for the sensorless force control is ab-straction of a disturbance-generating polynomial by the ob-server while reducing phase lag. The paper reveals how to en-sure accuracy of the disturbance-generating polynomial. Fig-ure 1 shows the control structure of the DOB-based sensor-less force control. Here, the system parameters are describedas

s Laplace operatorx Position of the motorFfric FrictionKt Torque coefficientM Motor massZe Mechanical Impedance of the environmentQ Free parameter (Q filter)©cmd Command value©ref Reference value©tab Table-valued function© Estimated value

Fig. 1. Control structure of the sensorless force control

Fig. 2. Architecture of the 2-DOF controller

Fig. 3. Architecture of the force feedback loop

©dobn Nominal parameter of DOB©rfob

n Nominal parameter of RFOB.

In this time, the friction effect is omitted in the analysis whileassuming it is enough small. The Q filter is designed to avoidmaking an algebraic loop and an improper system. Thus, itincludes pseudo differentials depending on the required num-ber of differentiation. The DOB once cancels out all externalforce and constructs the robust 2-DOF controller (21). Alter-natively, information of the reaction force is introduced intothe control system through the RFOB. Estimated value of theRFOB is feedback to the feedforward controller of the 2-DOFcontroller. Due to such a feedback, the control system attainsa back drivability and the robustness at the same time. Thusthe robust sensorless force control is achieved. Simultane-ously, the incorrect estimation declines control reliability.

Although the robustness is attained, the reliability, in de-tail whether the accurate reaction force control is realized ornot, is not guaranteed. This is because the accurate forcecontrol requires the correct disturbance-generating polyno-mial in the observer, but not the robustness. The paper con-firms which parameters determine the disturbance-generatingpolynomial. To simplify the control structure, the paper di-vides the system diagram into two parts; the 2-DOF con-troller based on the DOB and the force feedback loop by theRFOB. Figures 2 and 3 show the block diagram of the 2-DOF controller and the force feedback loop. An input-outputtransfer function and a disturbance-suppression performance

343 IEEJ Journal IA, Vol.8, No.2, 2019


of the 2-DOF controller are expressed as

xu= C f

αM

Qdob{αMs2} + (1 − Qdob){Ms2 + Ze} · · · · · (1)

xd=

1Ms2

(1−Qdob){Ms2+Ze}Qdob{αMs2}+(1−Qdob){Ms2 + Ze} · · · · (2)

α =Kt

Kdobtn

Mdobn

M, · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (3)

where u and d denote the input and the external disturbance.In the denominator of (1), the inertia αMs2 is dominantwithin the bandwidth of Qdob. The system behaves as a massof C−1

f within the bandwidth of the DOB, expressed as

xu=

1

C−1f s2. · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (4)

The numerator of (2) shows that the disturbance is suppressedwithin the bandwidth of Qdob. Thus, the 2-DOF controllerwork correctly within such the bandwidth. To achieve the ro-bust force control, the controller should be used less than thebandwidth of the DOB. On the other hand, the feedback gainof the outer loop is expressed as

−ux= Qrfob Krfob

tn

KtZe + QrfobγMs2 · · · · · · · · · · · · · · · · (5)

γ �(

Krfobtn

Kt− Mrfob

n

M

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

The main purpose of the RFOB is to find the disturbance-generating polynomial. From (5), the disturbance-generatingpolynomial is influenced by pseudo differentiation and thenominal torque coefficient. Therefore, a factor which deterio-rates the reliability is aggregated in the nominal torque coeffi-cient, even though the number of parameters which constructthe observer is two. As the other effect, the modeling errorsgenerate an acceleration feedback. The acceleration feedbackcontributes to improve a motor dynamics since it reduce theinertia of the motor. Thus, although the reliability of theRFOB tolerates against the mass fluctuation, it requires se-vere condition on setting of the nominal torque coefficient. Inother words, management of torque generation is importantfor the rigorous sensorless force control (22) (23). From the abovediscussion, the structure of the robust sensorless force con-trol is simplified as shown in Fig. 4. The control architectureis divided into two parts which denote the motor dynamicsand the environmental characteristics. Here, the force con-trol can be physically interpreted as a problem how the motorcontacts with the environment, mathematically expressed asa following governing equation;

Fcmd =

(Dm + Qrfob Krfob

tn

KtZe

)x · · · · · · · · · · · · · · · · · · · (7)

Dm �(C−1

f + γM)

s2, · · · · · · · · · · · · · · · · · · · · · · · · · · · (8)

where Dm denotes the motor dynamics. Therefore, designof the motor dynamics has a possibility to improve the sys-tem performance. The force gain controls the motor inertiaand determines response speed. As other method, impedancecontrol is one of the technique to design the motor dynam-ics. It is good approach when a contact object is known in

Fig. 4. Simplified structure of the force control

advance. In other words, arbitrary pole assignment can beachieved while degrading a back drivability. It should benoted that the robustness of the total system is guaranteedwithin the bandwidth of the DOB and control design shouldbe conducted within the lower bandwidth than that.2.2.1 Design of Motor Dynamics As mentioned be-

fore, there are many ways to design the motor dynamics.Several researches have been reported that the performanceimprovement can be achieved by parameter design of theDOB (17) (18). As the intentional modeling error of the DOBadds a phase compensator in the feedforward loop, it workslike a PD controller. Therefore, a stability margin and re-sponse speed increase. However, the PD controller can bedesigned for the force controller and its design is very sim-ple. Furthermore, the design simplicity plays a significantrole since a bandwidth to be compensated is determined bythe environmental characteristics. For vibration control ofa multi-mass resonant system, the phase-lead effect by theDOB always increases a stability margin as phase does notcross over π due to pairs of resonance and anti-resonance (16).However, the force control requires regional phase compen-sation and a reasonable phase compensator. For this problem,the phase-compensator obtained by the intentional modelingerror is difficult to adjust a bandwidth to be compensated.The detailed discussion is conducted in Appendix. From theabove discussion, the paper design the DOB with using thenominal parameters.

Parameter design of the RFOB also changes the motor dy-namics. It generates the acceleration feedback loop and fluc-tuates the motor inertia. The negative feedback heavier theinertia and the positive feedback lighten that. In other words,design of the nominal mass works like the force gain. Asthe high force gain increases response speed, there is dangerof system destabilization (discussed in section 4). Therefore,the detailed design method is explained in the latter section.2.2.2 Abstraction of Environmental CharacteristicsIn the RFOB, the disturbance-generating polynomial is

affected by the nominal torque coefficient and the Q filter.Therefore, the nominal torque coefficient should match to thetrue value. This problem can not be compensated/addressedfrom the controller. It denotes that brush DC motors or inte-rior permanent magnet synchronous motors which are diffi-cult to control torque in principle take a lot of costs to achievethe correct force control. On the other hand, surface per-manent magnet synchronous motors which easily meet suchcondition is suitable for the control. The bandwidth of the Qfilter should be set wider as possible to reduce the phase lagin the control loop. Its design should take the noise fromthe sensors or pseudo differentiation into account. Above



Table 1. Simulation parameters

Parameter Description Value

Kt Torque coefficient 15.0 N/AM Motor mass 1.0 kgZe Mechanical impedance of the environment 400 + 10.8sC f Force gain 1.5

Kdobtn Nominal torque coefficient of the DOB 15.0 N/A

Krfobtn Nominal torque coefficient of the RFOB 12.0/15.0/18.0 N/A

Mdobn Nominal motor mass of the DOB 1.0 kg

Mrfobn Nominal motor mass of the RFOB 0.8/1.0/1.2 kggdis Cut-off frequency of the DOB 1500 rad/sgreac Cut-off frequency of the RFOB 1000 rad/s

bandwidth is set to lower than that of the DOB to perform therobust force control. The wideband Q filter can be achievedby using a high-speed processor as it provides low noise sig-nal.2.3 Simulation For verification of the theory, the pa-

per conducted simulation. The paper supposes that a mo-tor contact with an environment which has a stiffness of400.0 N/m and a viscous of 10.8 N·s/m in the initial state. Asensor used in the simulation is only a position sensor. In im-plementation, sampling time of a controller and the sensor isset at 0.1 ms and there were no approximations as shown in(4). An effect of a friction is not considered in the simulation.A force command is set at 0.1 N and the paper shows resultsof true values which denote only a reaction force from anenvironment. Assuming to use a displacement sensor, the Qfilters of DOB and RFOB are designed as 2nd-order low-passfilter;

Qdob =g2

dis

s2 + 2 gdiss + g2dis

· · · · · · · · · · · · · · · · · · · · · · · (9)

Qrfob =g2

reac

s2 + 2 greacs + g2reac, · · · · · · · · · · · · · · · · · · · · (10)

where gdis and greac express the cut-off frequencies of theDOB and the RFOB. The parameters used in the simulationare shown in Table 1. The nominal values of the torque coef-ficient and the motor mass are chosen so that the values take80% and 120% of each true value.

The simulation results are shown in Figs. 5–7. To checkthe isolation of the environmental characteristics loop and theacceleration feedback loop, the results are arranged based onthe ratios of the nominal to the true torque coefficient, 0.8,1.0 and 1.2, respectively. There are stationary errors betweenthe force command and the true reaction force in Figs. 5and 7, while the reaction force converges to the force com-mand in Fig. 6. The values of nominal motor mass changethe response speed. In other words, it is revealed that thedisturbance-generating polynomial is determined by the ratioof the nominal to the true torque coefficient and is not dependon the nominal motor mass. From these figures, the responsespeed becomes fast as the ratio of the nominal to the truemass gets larger. It is equivalent to increase of the positiveacceleration feedback and reduction of the motor inertia. Itis also confirmed in these figures that the convergence timegets faster as the ratios of the nominal to the true torque co-efficient is small. This is because the acceleration feedbackgain is determined by the difference between the ratio of themotor mass and the ratio of the torque coefficient. From theabove results, the validity of the theory is verified.

Fig. 5. Simulation results of reaction force (Krfobtn /Kt = 0.8)



3. Analysis of Mass Fluctuation

The paper presents the analyses of the system for controllerdesign. As mentioned in the previous section, the nominalparameters of the RFOB affect on the disturbance-generatingpolynomial and the gain of the acceleration feedback. It en-ables to identify the error factor of the observer-based forcecontrol at a certain level under the identify-then-control strat-egy. However, although the variation of the torque coefficientcan be neglected as it is generally small, the mass fluctuationcan not be ignored in practical situation. From these per-spectives, the paper shows the effect of the mass fluctuationmathematically. The rigorous input-output transfer functionof the total system is represented as

x

Fcmd=

1Dx f (s)

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (11)

Dx f (s) = Qrfob

{(C−1

f + γM)

s2 +Krfob

tn

KtZe

}

+(Qdob − Qrfob

)C−1

f s2

+(1 − Qdob

) (αC f

)−1{s2 +

Ze

M

}. · · · · · · (12)

The ratio of the nominal torque coefficient to the true oneshould be 1.0 to realize the accurate acceleration control.Here, one of the conditions derived from the Routh-Hurwitzstability criterion is expressed as

C−1f +

Krfobtn

KtM > Mrfob

n . · · · · · · · · · · · · · · · · · · · · · · · · · (13)



Fig. 8. Bode diagrams of the open/closed loop

Other conditions are omitted since these are mainly deter-mined by the bandwidth of force sensing and the nominalparameters have little impact on them. In the case of α > 1.0,the influence of the environmental characteristics to the ac-celeration controller is attenuated. However, it is noted thatthere are an upper limit of Mdob

n and a lower limit of Kdobtn

due to effect of the friction or the sensor noise. In ideal con-dition, it is desirable that the acceleration feedback derivedfrom the RFOB is not be exist and an acceleration feedbackloop is additionally designed. However, the mass fluctuationcan not be neglected and should be considered in the controldesign. Thus, in order to design the acceleration feedbackwhich does not make system unstable, the nominal parame-ter design of the RFOB can be conducted quantitatively thandesign of the additional feedback loop. As the one of theproblems caused by the mass fluctuation, a variation of themotor dynamics, which directly affects the control perfor-mance, is remarkable. Figure 8 shows the Bode diagramsof the open loop and the closed loop (Fres/Fcmd) with themass fluctuation of 0%, ±20% and ±40%. This analysis usedthe control parameters listed in Table 1, and the Q filter wasdesigned as shown in (9) and (10). As the environment, thespring-damper system which has a stiffness of 2500.0 N/mand a viscous of 10.8 N·s/m was used. The figure shows thevariation of the resonance frequency. This is because the gaincross frequency of the open-loop transfer function moves dueto effect of the mass fluctuation. The phase increase of thegreen and purple lines in the open loop is caused by theirzeros. The stability margin decreases as M became small.Since the resonant frequency shifts with wide scale, the gainstabilization is difficult to attain. In contrast, it is conceivablethat the phase stabilization is effective for suppression of theresonances.

Fig. 9. Nyquist plot of the system L1(s)

4. Parameter Design in the RFOB

To implement the phase stabilization, the paper checks theopen loop transfer function. As the design parameters areQrfob, Krfob

tn and Mrfobn , the paper introduces three way to de-

sign. Although the modeling error of the torque coefficientdegrades the reliability of the force control, the paper showsdesigns including even that model. From another point ofview, this technique adds impedance feedback on the con-trol system. In other words, this design contributes to per-formance improvement in the form of the impedance control.Therefore, it is not irrelevant consideration in this research.4.1 Design of the Q Filter First, the paper consider

about the case of Qrfob design. In cases where the feedbackgain is represented as (5), an open loop transfer function is

L1(s) = QrfobγMs2 +

KrfobtnKt

Ze

D1(s)· · · · · · · · · · · · · · · · · · · (14)

D1(s)=QdobC−1f s2+

(1−Qdob

) (αC f

)−1{s2+

Ze

M

}.

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

When Krfobtn is well calibrated, the Nyquist plot of the open

loop transfer function is shown in Fig. 9. A configurationof this verifivation is same as the previous section. Themodeling error generates zeros, in detail stable zeros underM > Mrfob

n and unstable zeros under M < Mrfobn . When

M > Mrfobn , the problem is how to ensure the phase margin

since the gain margin is infinite. In another case M < Mrfobn , a

theme is also ensuring the phase margin while the gain mar-gin has finite value. This is because the open-loop transferfunction get to be the non-minimum phase system due to theunstable zero. In the design, the main issue is to deal with thecase M < Mrfob

n and to enlarge a tolerance against the down-ward mass fluctuation. When the mass fluctuation-range issmall, the design is simplified by setting the Mrfob

n small sothat the design requires only ensuring of the phase margin.Here, it is noted that the small Mrfob

n leads to a slow responsefor the upward mass fluctuation as the acceleration feedbackgain is decrease. It should be noted that L1(s) can be designedby tuning C f . The difference between them are whether it is aseries compensator or a feedback compensator, i.e. existenceof zeros in the closed loop.4.2 Design of the Torque Coefficient Model Sec-

ond, the paper shows the case of Krfobtn design. As mentioned

before, the design of this parameter change the disturbance-generating polynomial and reliability of the force control.However, the force control with this design is equivalent tothe control with additional impedance feedback. In other



words, it provides velocity feedback, the acceleration feed-back and so on. This design should ensure the consistency ofthe parameter on DC components. To design Krfob

tn , the pa-per decomposes the outer feedback loop. The outer feedbackloop can be described as

−ux= Qrfob Krfob

tn

Kt(Ms2 + Ze) − QrfobMrfob

n s2. · · · · · (16)

By adding the second term of (16) into the inner loop, theopen loop transfer function is represented as

L2(s) =Krfob

tn

Kt

Qrfob(Ms2 + Ze

)D2(s)

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

D2(s) = Qrfob{(

C−1f − Mrfob

n

)s2

}+

(Qdob − Qrfob

)C−1

f s2

+(1 − Qdob

) (αC f

)−1{s2 +

Ze

M

}. · · · · · · · (18)

Since C f is generally designed as C−1f < M, (20) can have un-

stable poles and the stability and the stability margin shouldbe checked by the Nyquist diagram. Figure 10 shows theNyquist diagram of the open loop transfer function L2(s).When the number of the unstable poles is two, the stabilitycondition is described as follows; the gain should be largerthan 1.0 when the phase is −180◦. The pseudo gain marginis expressed by a distance between the point when the phaseis −180◦ and (−1, j0). While the phase margin is difficult tojudge. An adjustment of the filter characteristics may be alsohard because the change of the Nyquist plot and the poles ofthe system L2(s) that follow the filter design is unpredictable.4.3 Design of the Inertia Model Third, the paper de-

scribes the the case of Mrfobn design. By adding the first term

of (16) into the inner loop, the open loop transfer function isexpressed as

L3(s) = −Mrfobn

Qrfobs2

D3(s)· · · · · · · · · · · · · · · · · · · · · · · · · (19)

D3(s) = Qrfob

{(C−1

f +Krfob

tn

KtM

)s2 +

Krfobtn

KtZe

}

+(Qdob − Qrfob

) {C−1

f s2}

+(1 − Qdob

) (αC f

)−1{s2 +

Ze

M

}. · · · · · · · (20)

When Krfobtn is well calibrated, the Nyquist plot of the open

loop transfer function L3(s) is shown in Fig. 11. The stabi-lization problem is simply explained as how to obtain boththe gain and the phase margin. As it does not affect on thedisturbance-generating polynomial unlike the Qrfob or Krfob

tndesign and deals the mass fluctuation directly. If the con-troller needs to consider the environmental characteristicsstrongly, Mrfob

n design is effective as it provides the clearphysical interpretation. This method is stabilization of ac-celeration feedback and ensures the phase margin around theresonant frequency.

5. Experiments

The paper conducts the force control which applied thephase stabilization. The purpose of the experiments is to at-tain the robust stability against the mass fluctuation ±15% ofthe static mass. Now the paper adopts the Qrfob design and the



Fig. 12. Experimental setup

proportional force-controller for the design simplicity. Anyserial compensators are not installed. The nominal torquecoefficients are well calibrated in advance. Mass fluctuationis virtually reproduced by setting the nominal motor mass de-pending on the situation; a virtual motor mass Mv is set as itsfluctuated value get to be the static motor mass. Since theDOB enables to manage the motor dynamics within its band-width, above operation does not have a significant impact onthe verification.5.1 Experimental Setup The experimental set up is

shown in Fig. 12. A rod-type linear motor S080Q (GMCHILLSTONE, CO., LTD.), is used as an actuator. A lin-ear power-amplifier TA310 (Trust Automation, Inc.), whosecontrol bandwidth is 5.0 kHz, is utilized as a motor driverin order to provide low-noise output and a quick response.The system uses an air slider to realize a frictionless mo-tion. A position sensor of the motor is consist of a linearencoder LIP281, a linear scale LIP201 and an interpolationdivider EIB392 (all of them are products of HEIDENHAINCo.), which has resolution of 31.25 pm. Control operationis processed on Power PMAC (Delta tau Co.) and the con-trol program is written by C language on it. Sampling time isset at 55.6 μs. Due to high-resolution sensing and fast signal-processing, the phase-lead compensator can be designed. Acontact object is an elastic object which has a resonance fre-quency of 26.38 rad/s when Mrfob

n is equal to a static motormass. The experimental parameters are shown in Table 2 andQdob was designed as (9). For the simple control design, the



Table 2. Experimental parameters

Parameter Description Value in

C f Force gain 2.5Kdob

tn Nominal torque constant of the DOB 4.3 N/AKrfob

tn Nominal torque constant of the RFOB 4.3 N/AMs Static mass 0.8 kgMv Virtual mass 0.94/0.8/0.70 kgMdob

n Nominal mass constant of the DOB Mv kgMrfob

n Nominal mass constant of the RFOB 0.85Mv kggdis Cut-off frequency of the DOB 1500 rad/sgreac Cut-off frequency of the RFOB 1000 rad/s

Fig. 13. Experimental results (mass fluctuation −15%)

Fig. 14. Experimental results (mass fluctuation 0%)

mass model of the RFOB is set at lower-limit value of themass fluctuation. Since the design requires only acquisitionof the phase margin, the paper designed Qrfob with phase leadcompensator as

Qrfob =1.25s + 30.14

s + 30.14

g2reac

s2 + 2 greacs + g2reac. · · · · · · (21)

The designed phase compensator cover the bandwidth of res-onance. As the phase compensator increases a gain in highfrequency domain and noise effect, the filter is experimen-tally designed with some adjustment. The paper conductedthe contact motion with force commands of 0.3/0.2/0.4 Nby 2.0 seconds and confirms force/position responses. Theforces are measured by the RFOB, which is separated fromthe controller and has bandwidth of 1000 rad/s.5.2 Experimental Results Figures 13–15 show the

experimental results of the force/position response. Each fig-ure shows the responses with the mass fluctuation of −15%,

Fig. 15. Experimental results (mass fluctuation +15%)

0% and +15%, respectively. The red lines denotes phase-stabilized responses, while the green lines represents originalresponses with setting Mrfob

n = Mv and no compensator. Thefigures show that the environment affects the total system andmakes responses oscillatory. Since increase of the mass con-tributes to the negative feedback of the acceleration, the res-onant frequency get to decrease as mass fluctuation increase.On the whole, the green lines vibrate strongly for a long timecompared with the red lines. This is because the phase marginof the total system is insufficient. In the Fig. 13, the red lineshows the performance of only the phase compensation with-out the modeling error, while the green lines present the per-formance with the positive acceleration feedback due to themodeling error. Due to the above reason, the red lines showthe smaller oscillation and faster convergence than the greenline. Since the phase margin get to be low when the massfluctuates to downward direction, it should be compensatedmainly. Likewise, the red lines in the Figs. 14 and 15 showsthe well convergence speed. Although the intentional settingMrfob

n = 0.85Mv generates the negative feedback of the accel-eration, the phase compensator compensates such the effectand provides the quick convergence. From these results, itis found that the phase stabilization contributes to the perfor-mance improvement under the existence of the mass fluctua-tion. As discussed in section 4, the responses are rapidly con-verged by ensuring the phase margin while the gain margin isnot taken into account. From the view point of the design, itrequires only the phase lead effects and easily attains the ro-bust performance. The robust stability is also enhanced alongwith the increase of the phase margin. Therefore, the robustsensorless force control is achieved.

6. Conclusions

The paper clarified the structure of the sensorless forcecontrol. The reliability of the reaction force control is ag-gregated in the nominal torque coefficient, while the nomi-nal motor mass generates the acceleration feedback. Thus,the error factor which deteriorates the performance can bedetectable. In general, the variation of the torque co-efficient can be neglect but the mass fluctuation can notbe ignored. It causes the fluctuation of a resonant fre-quency. The paper explains the phase stabilization meth-ods with considering the design of the three parame-ters in the RFOB. The features of the methods are also



described and one is applied to the system. The experi-mental results show the improvement of the robust stabil-ity and performance associated with increase of the phasemargin. The paper presented the reliability and clearphysical interpretation of the robust sensorless force control.

AcknowledgmentThis work was partially supported by JSPS KAKENHI

Grant Number 18H03784.

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Appendix

The phase-compensation effect obtained by the intentionaldesign of the nominal parameters of the DOB is not good forperformance improvement of the force control. There are twonoteworthy reasons; (i) The design freedom of the DOB is in-adequate to individually design the robustness and the phasecompensator. (2) The Q-filter should be designed while ded-icating to attain a good disturbance-suppression performancesince the bandwidth of the DOB defines the bandwidth of therobust force control. By intentional design of the nominalparameters, robustness, sensitivity function and bandwidthto be compensated by an appeared phase compensator arechanged. The robustness and the sensitivity have comple-mentary relationship but the compensated bandwidth has norelation. However, only one parameter α exists in intentionaldesign of that. In addition, the Q-filter defines the poles of afeedback controller of the 2-DOF controller and it should bedesigned according to a target system. It is better to add anadditional phase compensator in a loop. In this appendix, thepaper describes why the intentional design of that is not goodfor the force control.1. Phase Compensator by DOB DesignThe section 2.2.1 discussed about the phase-compensation

effect by parameter design of the DOB. When the sys-tem does not contact with an environment, the phase-compensation effect can be obtained as

xu=

α

αQdob + (1 − Qdob)1s2. · · · · · · · · · · · · · · · · · · · (A1)

when Q is a 1st-order low pass filter, (A1) gets to be

xu= α

s + gdis

s + αgdis

1s2· · · · · · · · · · · · · · · · · · · · · · · · · · · · · (A2)

Qdob �gdis

s + gdis. · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (A3)

This result matches to the previous researches (16)–(18). By set-ting α > 1.0, phase-lead effect can be obtained. This phe-nomena appear regardless of order of the Q filter. For sim-plicity, the paper discusses about a case with using the 1st-order filter. From above equation, a bandwidth to be com-pensated are determined by gdis and α. Since gdis set athigh value as possible to attain the robustness, a substan-tial design-parameter is α. However, its value is limited asit increases the equivalent cutoff frequency of the DOB anda proportional gain of the feedforward controller in high fre-quency domain. In other word, its design degrades a toler-ance against noise. Thus, design freedom is not high. Fur-thermore, reaction-force feedback in contact motion is alsoregarded as the modeling error of the inertia from perspective



app. Fig. 1. Generalized 2-DOF controller

of the DOB. When the motor contacts with the environment,the phase-lag effect is changed as

xu=

αMs2(s + gdis)(Ms2 + Ze)s + αMgdis s2

1s2. · · · · · · · · · · · · · · · (A4)

The above equation provides inarticulacy in design of thephase compensator. From discussion in section 4, the perfor-mance improvement of the force control requires the regionalphase compensation. Therefore, design simplicity is impor-tant in practical use. To obtain such phase compensation, thePD force controller or the series phase compensator is goodin design simplicity.2. Robust 2-DOF Control by using the DOBThe paper showed that the design freedom of the DOB

is not enough to individually design the robustness and thephase compensator. It should be noted that the DOB shouldbe designed not to acquire the phase compensation effect butto obtain the robustness. Since the bandwidth of the DOBdefines the bandwidth of the robust force control, the Q-filter should be designed while dedicating to attain a gooddisturbance-suppression performance. The Q-filter definesthe poles of a feedback controller of the 2-DOF controllerand it should be designed according to a target system.

Let us considering a 2-DOF controller shown in app. Fig. 1,where P, C, N, D and K are a plant, a feedback controller,an irreducible numerator and denominator of P satisfying

P = N D−1 = D−1N, · · · · · · · · · · · · · · · · · · · · · · · · · · · (A5)

and a tracking controller which makes N and D proper, re-spectively. The controllers which make the system stable canbe expressed by using free parameters X, Y and Q as

C = (Y − QN)−1(X + QD), · · · · · · · · · · · · · · · · · · · · (A6)

which fulfill a following Bezout’s lemma

Y D + XN = I. · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (A7)

r, d and y denote a command, a disturbance and an outputsof this system, respectively. This controller outputs y = NKrin ideal condition. Now, the paper puts an input to the systemand an output of the feedback controller as u and u. Theseparameters is expressed as

u = DKr + u · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (A8)

u = (Y − QN)−1(X + QD)(NKr − y). · · · · · · · · · · (A9)

Here, a regulation with respect to u can be found as

u = Y−1(QNu + (X + QD)(NKr − y)) · · · · · · · · ·(A10)

= Y−1(QN(u + DKr) + X(NKr − y)) − QDy).

· · · · · · · · · · · · · · · · · · · · ·(A11)

Then, following equations are derived;

app. Fig. 2. 2-DOF controller in the form of the DOB

app. Fig. 3. Equivalent 2-DOF controller

u = DKr + Y−1X(NKr − y) + Y−1Q(Nu − Dy)

· · · · · · · · · · · · · · · · · (A12)

u = Y−1X(NKr − y) + Y−1Q(Nu − Dy). · · · · · · (A13)

By using the Bezout’s lemma (A7), we obtain

u = Y−1(Kr − Xy + Q(Nu − Dy)). · · · · · · · · · · · (A14)

Then, the generalized 2-DOF controller can be rewritten asapp. Fig. 2. This architecture is similar to that of the DOB.Here, the paper puts an output of Q as β, which fulfill

β = Q(NY−1β − DP(Y−1β − d)) = QNd. · · · · · (A15)

To realize command tracking and disturbance rejection, thesystem should satisfy following conditions;

y = NKr = PY−1 Kr = r · · · · · · · · · · · · · · · · · · · · · (A16)

Y−1β = Y−1QNd ≈ d. · · · · · · · · · · · · · · · · · · · · · · · (A17)

Now, let us get to the point, how to achieve the accelerationcontrol by using the DOB. For the acceleration control, rel-ative degree of N and D is 0 and hence K can get constantvalue. To ensure certain tracking, K should be equal to N−1

and Y should be D−1. Here, comparing the architecture ofthe DOB and app. Fig. 2, it is proven that the DOB is a 2-DOF controller when the free parameter X = O. On theother hand, remaining free parameter Q should be designedsuch that Y−1QN ≈ 1 to achieve the disturbance suppression.Considering a sensitivity function and a complementary sen-sitivity function, Q is designed with a low-pass filter Qdob as

Q = YQdobN−1 = D−1QdobN−1. · · · · · · · · · · · · · · · (A18)

Therefore, the DOB is a 2-DOF controller which has follow-ing controllers;

K = N−1 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (A19)

C= (D−1−QN)−1QD= (I−Qdob)−1Qdob P−1n .

· · · · · · · · · · · · · · · · · (A20)

An equivalent block diagram of the DOB is shown inapp. Fig. 3, where Pn denotes a nominal model of the plant.Here, the output of this system is expressed as



app. Fig. 4. Acceleration controller by the DOB

y ={(I−Q)+QPP−1

n

}−1 (PP−1

n r−(I−Q)Pd)

· · · · · · · · · · · · · (A21)

≈ r − (I − Q) Pd (when Pn = P). · · · · · · · · · (A22)

It is just the same output with the DOB. When the nominalmodel well describes the plant, the acceleration controller isconstructed as app. Fig. 4. This result shows that there is lit-tle design freedom in the DOB design since it should be de-signed as it has a disturbance-generating-polynomial to at-tain the robustness. It should be noted that a 1st-order low-pass filter for Qdob is a parameter of the Luenberger observerwhile assuming a derivative of a disturbance is 0, namely itis for a system influenced by steady disturbance (8). Althoughthe DOB with the 1st-order low-pass filter provide the phase-compensation effect, Qdob should not be restricted in the formof the first-order low-pass filter. It is better to design an addi-tional compensator and shape a closed loop characteristic.

Hiroki Kurumatani (Member) received the B.E. degree in system de-sign engineering and the M.E. degree in integrateddesign engineering from Keio University, Yokohama,Japan, in 2015 and 2017, respectively. Since 2015, hehas been a Ph.D. course student at Keio University,Yokohama, Japan. His research interests include sys-tem design of electromechanical integrated systemsand industrial electronics. He is a Member of IEEJ,as well as IEEE.

Seiichiro Katsura (Senior Member) received the B.E. degree in sys-tem design engineering and the M.E. and Ph.D. de-grees in integrated design engineering from Keio Uni-versity, Yokohama, Japan, in 2001, 2002 and 2004,respectively. From 2003 to 2005, he was a ResearchFellow of the Japan Society for the Promotion of Sci-ence (JSPS). From 2005 to 2008, he worked at Na-gaoka University of Technology, Nagaoka, Niigata,Japan. Since 2008, he has been at Keio University,Yokohama, Japan. In 2017, he was a Visiting Re-

searcher with the Laboratory for Machine Tools and Production Engineering(WZL) of RWTH Aachen University, Aachen, Germany. His research inter-ests include applied abstraction, human support, data robotics, wave system,systems energy conversion, and electromechanical integration systems. Prof.Katsura serves as an Associate Editor of the IEEE Transactions on IndustrialElectronics. He was the recipient of the IEEJ Distinguished Paper Awards in2003 and 2017, IEEE Industrial Electronics Society Best Conference PaperAward in 2012, and JSPS Prize in 2016.


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