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State Key Laboratory of Robotics and System，Harbin Institute of Technology，Harbin 150001，P.R.China

Abstract: An adaptive control scheme is presented，which can simultaneously realize vibration suppression and compliance control for flexible joint robot（FJR）. The proposed control scheme provides a unified formulation for both vibration suppression mode，where FJR tracks the desired position with little vibration，and compliance mode，in which FJR presents passive. Instead of designing multiple controllers and switching between them，both modes are integrated into a single controller，and the transition between two modes is smooth and stable. The stability of the closed-loop system is theoretically proven via the Lyapunov method，with the considering the dynamics uncertainties in both link side and motor side. Simulation results are presented to illustrate good performances of the proposed control scheme.

Key words：adaptive control scheme；vibration suppression；compliance；flexible joint robot（FJR）；stability

0 Introduction

Recently，interest in flexible joint robot（FJR）which possesses inherent compliance has been great?ly increased［1-4］. Unlike rigid robots，F(xiàn)JR’s joint contains an elastic element，which introduces a low?er mechanical output impedance，passive mechani?cal energy storage，and increases peak power out?put. Moreover，the inherent compliance brought by the elastic element can filter impact，protect drive?trains，and provide additional time for controller to regulate the impedance of FJR. Due to the above characteristics，F(xiàn)JRs are widely used in the field of physical interactions with the surroundings or hu?mans，such as rehabilitation robots，exoskeleton or wearable robots and assembly robots. In those appli?cations involving physical interactions，safety of FJRs and the surroundings or humans is always the most critical concern when controlling the position of FJR. Therefore，the purpose of this paper is to design a control scheme to realize the position con?trol of FJR and ensure the safety of FJR and the sur?roundings or humans.

Elastic element will bring much difficulty to po?sition control of FJR comparing with that of rigid ones. Firstly，due to the existence of elastic ele?ment，the dynamics of FJR is decoupled into two parts：Motor side and link side，which are described by a second-order nonlinear equation，thus，the or?der becomes twice that of the rigid joint. This poses a nontrivial task due to the increase of the model complexity. Secondly，the number of control inputs is strictly less than mechanical degrees of freedom，which illustrates that FJR is a nonlinear，under-actu?ated and strong coupling system. Moreover，F(xiàn)JR is prone to vibrate，which deteriorates system stability and requires extra power for position control. Sever?al approaches have been proposed to control the po?sition of FJR. Tomei［5］proved that a simple propor?tional-differential（PD）controller can globally stabi?lize about a reference position. Kim et al.［6］pro?posed a robust PD control scheme for FJR based on a disturbance observer（DOB）which was only ap?plied to the motor-side dynamics. Jin et al.［7］de?signed an adaptive tracking controller using a timedelay estimation technique. Mobayen et al.［8-9］pro?posed sliding mode control techniques for a class of fourth-order system which can describe the position control system of FJR. Besides，a variety of vibra?tion suppression control schemes have been devel?oped for FJR. As an effective method，damping dis?sipation strategy has been studied in the literature.Well-known implementations of damping dissipa?tion strategy include feedback linearization［10］，mod?el-based state-feedback controllers［11］，learning con?trol schemes［12］，and linear-quadratic regulators［13］.Other effective approaches focusing on the vibration attenuation of FJR through joint motion input in?clude input shaper［14］and singular perturbation con?trol［15］. Besides，resonance ratio control（RRC）［16］and overshoot control［17］can also effectively sup?press vibration of FJR.

The aforementioned control methods only fo?cus on the position control problem. For FJR，which is designed to closely interact with the sur?roundings，the safety between FJR and the sur?roundings should be also considered. In order to complete position control and meanwhile achieve compliance in control，hybrid position/force con?trol［18］and impedance/admittance control［19］have been proposed for robot. Nevertheless，the hard switching of hybrid position/force control may re?sult in an overall discontinuous control input，which further causes the chattering movement of robot or even compromises the safety of robot and the sur?roundings. Impedance/admittance controllers have no discontinuity during switching，but the desired impedance has to be adjusted according to the inten?tion of human. In addition，most aforementioned control schemes commonly assume the robot dynam?ics to be exactly known，and do not take the uncertain?ty of model parameters into account. Continuous con?trol techniques for the human-robot interaction have been proposed［20-21］，but these schemes have no focus on the position control with vibration suppression. As for as we know，there is no research on unified vibra?tion suppression and compliance control scheme.

In this paper，a novel continuous adaptive con?trol scheme which integrates both modes，i.e.，vi?bration suppression mode and compliance mode，in?to a single controller，is proposed for FJR. The pro?posed controller is able to automatically transit be?tween both modes，and the transition between two modes is smooth and stable. A stability analysis is performed for the closed-loop system with consider?ing the dynamics uncertainties in both link side and motor side. Simulation results are given to show the validity of the proposed control scheme. The main novelties can be summarized as follows：

（1）A unified control scheme which integrates both modes of vibration suppression and compliance into a single controller is first designed for FJR.

（2）The proposed control scheme is performed well with considering the dynamics uncertainties in both link side and motor side by means of suitable adaptive laws.

（3）The proposed control scheme is rather straightforward and easily implemented in engineer?ing.

1 Problem Description

According to the Spong’s assumptions［22］，dy?namics of FJR with high gear reduction consists of two parts coupled through the elastic element，that is，link-side dynamics

and motor-side dynamics

whereq∈Rnandθ∈Rnare the position vector of link side and motor side，respectively.M(q)∈Rn×nis the inertia matrix of link side，the Coriolis and centripetal torque vector of link side，g(q)∈Rnthe gravity vector of link side，K∈Rn×nthe stiffness matrix of FJR，andB∈Rn×nthe inertia matrix of motor side.andare the viscous friction vectors of link side and motor side，respectively.τe=JT(q)fe∈Rnis the vector of ex?ternal torque，whereJT(q)∈Rn×mis the transpose of Jacobian matrixJ(q) andfe∈Rmthe vector of continuous external force.τ∈Rnis the vector of in?put torque exerted on motor side.

Some important properties of the FJR’s dy?namics described by Eq.（1）and Eq.（2）are listed as follows：

（1）MatricesM(q)andBare positive definite；

（3）Viscous friction matricesDq=diag(dq1，…，dqn) andDθ=diag(dθ1，…，dθn) are di?agonal and positive definite，wheredqianddθiare physical parameters；

（4）Gravity vectorg(q) is linear in a set of physical parameterspq=[pq1，…，pqnq]T∈Rnqas fol?lows

whereYq(q)∈Rn×nqis a known dynamic regression matrix；

（5）First two terms of motor-side dynamics are linear in a set of physical parametersas follows

In this paper，the dynamic uncertainties in both link side and motor side are taken into consider?ation，in the sense that the dynamic modelsM(q)，C(q，?)，Dq，g(q)，B，andDθare unknown.Kis assumed to be well defined，which is obviously rea?sonable because the actual stiffness of FJR can be achieved by means of loading calibration experi?ments. Control objective is to design a unified con?trol strategy including vibration suppression mode and compliance control mode，to ensure a stable and smooth transition between both modes under Properties 1—5 and dynamic uncertainties. In addi?tion，when in vibration suppression mode，q(t) can reach the desired positionqdwith little vibration as possible，i.e.

whereeq=q-qd.

2 Controller Design

In this section，an adaptive unified controller is proposed for FJR，which smoothly integrates both modes，i.e.，vibration suppression mode and com?pliance mode，into a single controller.

2.1 Force region function and weight factor

A force region function is proposed to monitor the variation of external forcefeas［20］

whereRis a positive constant that denotes the radi?us of force region. Whenfeis inside the force re?gion，β(fe)≤0，and vice versa.

Based on force region，a weight factor is de?fined as［23］

whereκ?1 is a positive constant. An illustration of weight factorω(fe) is shown in Fig.1，wherefe=[fe1,fe2]T∈R2.

Fig.1 Illustration of ω( fe)

From Fig.1，whenfeis inside the force region such thatβ(fe)≤0，ω(fe)=1. Whenfeexceeds the force region such thatβ(fe)>0，ω(fe)=0. Dif?ferent values of weight factor will be utilized in vi?bration suppression mode and compliance mode.

2.2 Desired input for link side

Introducing the position error of motor sideeθ=θ-θd，then Eq.（1）can be rewritten as

The desired inputθdis considered as a fictitious input for link side，andeθrepresents an input pertur?bation to the link-side dynamics. Eq.（8） can be viewed as being controlled by the inputKθdwith the perturbation ofKeθ.

By using weight factorω(fe)，a desired input for link side is presented as follows

whereKp∈Rn，Kd∈Rnandγ∈Rnare positive defi?nite diagonal matrix.is the estimate ofpq，which is updated by

whereΓq∈Rnq×nqis a positive definite matrix. In Eq.（9），s=[s1，s2，…，sn]Tis the vibration suppres?sion term，with each element detailed as

whereα1i，α2i，α3iandρiare positive constants to be designed，sign(·)is a standard signum function.

Remark 1In Eq.（9），the saturated nonlinear term ofeqensures that link side has a larger control input compared with the normal proportional control term when link-side position erroreqis small；Vibra?tion suppression termsis utilized to add dissipative?ness to FJR system，which will help to suppress re?sidual vibration.

The desired fictitious inputθdintegrates vibra?tion suppression mode and compliance mode as fol?lows：

（1）Vibration suppression mode：When the ex?ternal forcefeis inside the force region whereβ(fe)≤0，ω(fe)=1，the desired input for link side becomes

where position control termKparctan(γeq) and ve?locity control termKdq? drive link side to track the desired positionqd.

（2）Compliance mode：When external forcefeis outside the force region whereβ(fe)>0，ω(fe)=0，the desired input for link side is simpli?fied as

The use of the adaptation in Eq.（10）ensures the convergence of the prediction error， thusIn addition，if the motor-side positionθtracks the desired inputθdsuch thateθ=θθd→0，we have

which is a damping system，in the sense that exter?nal forcefeis able to control link side and the link side shows compliance.

2.3 Adaptive observer and control input for motor side

In this section，control inputτfor motor side is developed such that motor-side positionθtracks the desired inputθd，i.e.，eθ→0.A sliding vector is introduced for motor side as

whereλθis a positive constant，and? is a reference vector which is defined as

Control input for motor side is now proposed as

whereKθis a positive definite matrix andthe esti?mate of motor-side uncertain dynamic parameters vectorpθ，which is updated by

whereΓθ∈R2n×2nis a positive definite matrix.

Note that the control input Eq.（18）for FJR is dependent on the derivatives of desired inputand，hence，the acceleration information?and its de?rivatives are required. We consider the uncertain dy?namics problem and propose an adaptive observer to estimate the desired inputθd，thus eliminating the requirement for acceleration and its derivatives. The adaptive observer is as follow

wheredenotes the estimate ofθd，the observation error，?an auxiliary variable，λoa positive constant，andKea positive definite matrix.Matricesanddenote the estimate ofBandDθ，respectively，which are updated through the fol?lowing two adaptation laws

Note thatis obtained by integratingwhileis obtained by differentiatingwith re?spect to time. By using the estimated signals，a new sliding vector is then defined as

where the reference vectoris defined as

Control input for motor side is now proposed as

where the uncertain parametersare updated by

3 Closed?Loop Stability Analysis

This section presents stability analysis for closed-loop system with controller Eq.（24）.

Substituting the desired inputθdinto Eq.（8）yields

Link-side scalar functionVqis proposed as

where

Taking partial derivative ofhi(eqi) with respect toeqiyields

From Eq.（29），it can be easily concluded that

Eq.（31）implies thatVqis positive definite，hence，Vqis a Lyapunov function. Differentiating Eq.（27） with respect to time and substituting Eq.（26）into it，we have

Substituting updated law Eq.（10）into Eq.（32）and using Property 2 yield

Then，motor-side dynamics can be expressed in terms ofe?sθas

Substituting Eq.（24）into Eq.（34）yields

Multiplying both sides of Eq.（20）withand using Eq.（2），we can get

Motor-side Lyapunov function is proposed as

Differentiating Eq.（37）with respect to time，and substituting Eqs.（21，25，35，36）into it yield

Next，for further stability analysis，let Lyapu?nov functionVbe constructed as

DifferentiatingVwith respect to time and sub?stituting Eqs.（11，33，38）into it，one obtains

whereWis given by

Then Eq.（41）is written as

where

Integrating Eq.（43）over[0，t]yields

Now，we have the following theorem to state the boundedness of the link-side position erroreq.

Theorem 1The proposed adaptation and con?trol schemes described by Eqs.（10，21，24，25）en?sure the stability of FJR，the boundedness of all state variables and the convergenceeθ→0，if exter?nal forcefeis bounded，and the control parameters are chosen such that

whereλmin[·] represents the minimum eigenvalues andηthe positive constant to be introduced later.

ProofLet regionΩbe defined as follows

Considering Eq.（46），in regionΩ，the follow?ing inequality holds

If condition Eq.（47）is satisfied，Qis positive.SinceQis positive，we have

Remark 2Eq.（50）demonstrates the passivi?ty of the dynamics between inputτeand outputq?.The passivity shows that at any time the energy stored in the systemV(t)-V(0) is less than or equal to the energy introduced from external force.Therefore，the proposed control scheme guarantees the safety between FJR and the surroundings.

If external forcefeis bounded，external torqueτe=JT(q)feis also bounded.Note that

From Eq.（45）and Eq.（51），we have

which also implies that

Since 2λmin[W]-1>0，we can conclude that the boundedness ofτeensures the boundedness ofq?.From Eq.（50），we have

Since bothandτeare bounded，Vis also bounded. From Eqs.（27，37，39），the boundedness ofVensures the boundedness of all state variables，thus the closed-loop system is stable. In addition，the boundedness ofand? ensures the bounded?ness ofHence，? is uniformly continuous.Then，F(xiàn)rom Eq.（44）and Eq.（45），we have

where the right side is bounded. Hence，and. Similarly，we can geteo→0. Sincetheneθ→0. That is，the convergence ofθ→θdis guaranteed.

The above stability analysis does not depend on specific values of weight factorω(fe)，and thus，it is applicable to vibration suppression mode，com?pliance control mode，and also the transition stage.If external force is further negligible，the proposed controller works in vibration suppression mode.Then，the following theorem states the convergence of link-side position error.

Theorem 2The closed-loop system gives rise to the convergence of link-side position error in vibration suppression mode，if external forcefeis negligible，and the control parameters are chosen such that Eq.（46）and the following condition are satisfied

ProofSince external forcefeis negligible，we haveτe→0. If Eq.（46）and Eq.（57）are satisfied，we have. From Eq.（41），when0，are equal to zero. Hence，andsequal zero. Inserting these results into Eq.（8）yields

The use of the adaptation in Eq.（10）ensures the convergence of the prediction error，thus solving Eq.（58）can geteq=0. Through analysis above，Eq.（58）owns the unique solutionq=qd. A direct application of the Lasalle’s invariance principle gives Theorem 2.

Remark 3It can be concluded that the de?signed controller does not need exact model parame?ters such as friction，inertia of motor and link side，Coriolis force and gravity，so it has strong practica?bility.

4 Simulation Experiments

Numerical simulation results demonstrate the effectiveness of the proposed control scheme. The simulation studies are carried out in the two-link FJR as illustrated in Fig.2.

The nominal parameters of FJR’s link side and motor side are listed in Table 1.

Table 1 Nominal parameters of FJR

Thus，the dynamic terms in Eqs.（1，2）can be presented as follows

The desired position of link side is

First of all，F(xiàn)JR is controlled to track the de?sired position without external force acting on FJR，which verifies the vibration suppression performance of the proposed scheme. Control parameters of the proposed scheme are chosen as

Comparison experiment which employs the pro?posed scheme but without vibration suppression termsand the same control parameters is carried out. In addition，RRC is also used for comparison.Simulation results are shown in Fig.3.

Fig.3 Position control for FJR

From Fig.3，it is obvious that both of two links can be driven to the desired position，and the steady state error is small through the above three schemes. However，link side experiences violent os?cillation when FJR is in charge of the proposed con?trol scheme but without vibration suppression terms. Moreover，the oscillations lead to large overshoot and long settling time. While with the proposed scheme with vibration suppression terms，residual vibration is well suppressed. Although the over?shoot will also occur，the quantities of overshoot are only 1.06% and 2.36% of two joints，and link-side position converges to the desired position at about 3 s. As a contrast，it takes approximately 4.2 s for link side to converge to the desired position，al?though no overshoot occurs with RRC.

The regulation criterion ofKpandKdis the same as PD control. The controller is sensitive to parameters must be kept close to the corresponding ones in Eq.（61）. If not，violent oscillation，exces?sive tracking time，steady-state error and even insta?bility will occur.small erroreqby increasingγ. A series of simula?tions are further carried out to test the performance of the vibration suppression termswith different set?tings of control parameters，which include the fol?lowings：

（1）Small parameters ［α11=1，α21=0.2，α31=0.9，α12=0.8，α22=0.1，α32=1，ρ1=0.3，ρ2=0.3，others remain the same as Eq.（61）］.

（2）Well-tuned parameters ［all remain the same as Eq.（61）］.

（3）Large parameters ［α11=10，α21=4，α31=9，α12=8，α22=2，α32=9.5，ρ1=5，ρ2=5，others remain the same as Eq.（61）］.

Simulation results with different settings of con?trol parameters are shown in Fig.4. When the con?trol parameters are set very small（Setting（1）），link positions experience overshoot and the over?shoot lead to large oscillation and long settling time.While with well-tuned parameters（Setting（2）），residual vibration is well suppressed. However，if the parameters are set too large（Setting（3）），it takes too long time for link to track the desired posi?tion. Besides，after debugging，the values of other

Fig.4 Performance of the proposed controller with different settings of vibration suppression term parameters

Then，the end of FJR is exerted an external force during position control，which verifies compli?ance performance of the proposed scheme. Force re?gion in Eq.（6）is specified as

wherefe1andfe2represent the forces in two coordi?nates，respectively，the radiusR=2N，and the pa?rameterκof weight factor in Eq.（7）is equal to 10.

External forcefeacting on FJR can be de?scribed as

wheret1=7 s，t2=8.5 s，δt=0.04 s，a1=1.5 anda2=-2.

The proposed scheme is employed to control FJR with control parameters as Eq.（61），and linkside positions are shown in Fig.5（a）. When the ex?ternal force exceeds the force region such thatβ(fe)>0，weight factorω(fe) decreases from 1 to 0，as shown in Fig.6（b）. Then，the proposed con?troller transits from vibration suppression mode to compliance mode，and FJR becomes passive. The external force pushes link side away from the de?sired positionpd，to guide FJR to move. When ex?ternal forcefedisappears such thatβ(fe)≤0，the controller transits to vibration suppression mode again，which drives FJR to track the desired posi?tion.

Fig.5 Unified control for FJR

Fig.6 Unified control for FJR

To eliminate the requirement of acceleration in?formation and its derivatives，the observer Eq.（20）is constructed to estimate the desired inputθd. The observation error is shown in Fig.6（a），which is less than 0.2 rad throughout the course of control.Control inputτis demonstrated in Fig.6（b）to illus?trate the transition performance of the proposed scheme.

It is seen that control input is in a proper range and the proposed scheme ensures a stable and bounded transition of control input between both modes at 7 s and 8.5 s. It’s worth noting that，con?trol input of the proposed scheme has a little noise at 8.5 s，which is mainly caused by the abrupt change of link-side desired inputθdrather than caused by the proposed control scheme.

5 Conclusions

A unified vibration suppression and compliance control scheme has been proposed for FJR. In the case of vibration suppression mode，the proposed controller allows FJR to track the desired position with better performance in terms of overshoot，set?tling time and residual vibration suppression. In the case of compliance mode，F(xiàn)JR becomes passive such that external force guides the movement of FJR when the external force exceeds the force re?gion. The smooth and automatic transition among both modes guarantees the safety between FJR and the surroundings. It has been rigorously proved that the closed-loop system is stable，with considering the dynamics uncertainties in both link side and mo?tor side. Simulation results are presented to show the effectiveness of the proposed control scheme.

The proposed control scheme does not consid?er issues such as input saturation，input constraints and unmodeled dynamics，etc. In real-world applica?tions，for practical FJRs，they always suffer from such issues. In fact，there are already effective solu?tions to these problems，as suggested in Refs.［24-25］. As a future work，unified vibration suppression and compliance control considering input saturation，input constraints and unmodeled dynamics for FJR which is more suitable for practical application will be further studied.