global task coordinate frame-based contouring control of linear …byao/papers/tie_11_gtcf.pdf ·...

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 58, NO. 11, NOVEMBER 2011 5195

Global Task Coordinate Frame-Based ContouringControl of Linear-Motor-Driven Biaxial Systems

With Accurate Parameter EstimationsChuxiong Hu, Member, IEEE, Bin Yao, Senior Member, IEEE, and Qingfeng Wang

Abstract—This paper presents a global task coordinate frame(TCF) (GTCF)-based integrated direct/indirect adaptive robustcontouring controller for linear-motor-driven biaxial systems thatachieves both stringent contouring performance and accurate pa-rameter estimations. In contrast to the past research works thoseuse various locally defined TCFs “attached” to the desired contourto approximately calculate the contouring error for feedback con-troller designs, this paper first formulates the contouring controlproblem using a recently developed GTCF, in which calculationof the contouring error is rather accurate and not affected by thecurvature of the desired contour. A physical-model-based indirect-type parameter-estimation algorithm is then synthesized to obtainaccurate online estimates of unknown physical model parame-ters. An integrated direct/indirect adaptive robust controller withdynamic-compensation-type fast adaptation is also constructedto preserve the excellent transient and steady-state performanceof the direct adaptive robust control (ARC) designs. Compara-tive experimental results show that the proposed GTCF-basedintegrated direct/indirect ARC algorithm not only achieves thebest contouring performance but also possesses rather accurateestimations of physical parameters.

Index Terms—Adaptive control, contouring control, linearmotor, parameter estimation, robust control, task coordinateframe (TCF).

I. INTRODUCTION

L INEAR MOTOR drive systems are ideally suited forindustrial applications such as material handling and

machining that demand high-speed/high-accuracy positioning

Manuscript received September 9, 2010; revised December 26, 2010;accepted February 14, 2011. Date of publication March 7, 2011; date of currentversion September 7, 2011. This work was supported in part by the U.S.National Science Foundation under Grant CMMI-1052872, by the NationalBasic Research and Development Program of China under 973 Program Grant2007CB714000, and by the Ministry of Education of China through a ChangJiang Chair Professorship. This paper was presented in part at the 2011American Control Conference, San Francisco, CA, June 29–July 1.

C. Hu is with the State Key Laboratory of Fluid Power Transmission andControl, Zhejiang University, Hangzhou 310027, China, and also with theState Key Laboratory of Tribology, Department of Precision Instruments andMechanology, Tsinghua University, Beijing 100084, China (e-mail: [email protected]).

B. Yao is with the School of Mechanical Engineering, Purdue University,West Lafayette, IN 47907 USA, and also with the State Key Laboratory of FluidPower Transmission and Control, Zhejiang University, Hangzhou 310027,China (e-mail: [email protected]).

Q. Wang is with the State Key Laboratory of Fluid Power Transmissionand Control, Zhejiang University, Hangzhou 310027, China (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2011.2123859

or motion tracking [1]–[6]. The ultimate performance of amulti-axis machine is often measured not by the motion track-ing accuracy of each individual axis but rather by its ability tofollow certain patterns and orientations (i.e., desired contour)in space, which involves the coordinated motion of all axesof the machine [7]–[10]. To meet the increasingly demandingrequirements of higher productivity, high-speed linear-motor-driven machines have been the choice of modern industry.Furthermore, to have better quality of manufactured products,those machines have to be equipped with controllers that caneffectively coordinate the motion of all their axes for precisecontour following in space.

Recognizing the lack of axes coordination in the traditionalmachine tool controls, Koren [11] first proposed the cross-coupled control strategy by feeding back the real-time com-puted contouring error to generate additional corrective actionin the individual axis controllers to strengthen the coordinationof axes. Later on, the contouring control problem is formu-lated in a task coordinate frame (TCF) by using either theconcept of generalized curvilinear coordinates introduced in[12] and [13] or the locally defined coordinates “attached” tothe desired contour proposed in [7]. Under TCF formulations,a control law could be designed to assign different dynamicsto the normal and tangential directions relative to the desiredcontour. This formulation has been used in a large number ofrecent publications [8], [14]. However, the presented controltechniques cannot explicitly deal with parametric uncertaintiesand uncertain nonlinearities.

During the past decade, an adaptive robust control (ARC)framework has been developed by Yao and Tomizuka in [15]and [16] to provide a rigorous theoretic framework for theprecision motion control of systems with both parametric un-certainties and uncertain nonlinearities. In [1] and [17], theproposed ARC algorithm is experimentally tested on a linearmotor. Global stability is also guaranteed even in the pres-ence of actuator saturation and short duration of very largedisturbances [18], [19]. In all those papers, excellent position-tracking performances have been obtained in experiments forthe individual axes of linear motor drive systems. In [20]–[22],the ARC strategy and the local TCF (LTCF) formulation in [7]have been integrated to develop various contouring controllersfor high-speed biaxial linear-motor-driven gantries under bothparametric uncertainties and uncertain nonlinearities. Althoughsufficient for some applications, they are not well suited forapplications which demand not only good output tracking

0278-0046/$26.00 © 2011 IEEE

5196 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 58, NO. 11, NOVEMBER 2011

performance but also accurate online parameter estimationsfor secondary purposes such as machine component healthmonitoring and prognosis. Accurate parameter estimations arealso achieved in our previous work using the LTCF formulation[23], which is valid only for applications with very small actualtracking errors and small curvatures.

This paper focuses on the synthesis of Global TCF (GTCF)-based ARC contouring controllers that explicitly take intoaccount large extents of parametric uncertainties and achievenot only stringent contouring performance but also accurateestimates of physical parameters for other second goals suchas machine health monitoring and prognosis. Specifically, toaddress the problems associated with LTCF-based contouringcontrollers [20], [23] for large curvatures, our recently de-veloped orthogonal GTCF [24] will be used. The presentedGTCF is globally defined based on the geometry of the de-sired contour only and has nothing to do with the specificdesired trajectory to be tracked on the contour. Furthermore,the calculation of the contouring error is rather accurate andnot affected by the curvature of the desired contour. The in-tegrated direct/indirect ARC (DIARC) strategy [25], [26] isthen employed to construct a coordinated motion controllerfor a biaxial gantry which can explicitly handle a large de-gree of parametric uncertainties and uncertain nonlinearitiestheoretically. The resulting DIARC controller achieves notonly excellent contouring performance but also accurate onlineparameter estimations. Comparative experimental results havebeen obtained on a linear-motor-driven high-speed industrialbiaxial gantry. The proposed GTCF-based DIARC contouringcontroller, the GTCF-based direct ARC (DARC) contouringcontroller, and the LTCF-based DIARC scheme in [23] are allimplemented. The obtained experimental results also confirmthat the proposed GTCF-based DIARC contouring controllerachieves not only the best contouring performance but alsoaccurate parameter estimations in practical applications.

II. PROBLEM FORMULATION

A. Orthogonal GTCF

Traditionally, the TCF for a biaxial system is locally definedat the desired position Pd(xd(t), yd(t)) based on the desiredtrajectory on the desired contour given by

qd(t) = [xd(t), yd(t)]T (1)

where qd(t) represents the Cartesian coordinates of the desiredposition at time t. Such a definition depends not only on thegeometry of the desired contour but also on the desired motionto be tracked on the contour (i.e., the parameterization of thedesired trajectory by the time t). As such, when the actualposition at time t (i.e., q = [x(t), y(t)]T, where x(t) and y(t)are the Cartesian coordinate values of the actual position) dif-fers from its desired value of qd(t) significantly, the calculatedcontouring error could be far different from the actual one. Forexample, for the actual position at Pa and the desired positionat Pd shown in Fig. 1, the traditional LTCF would be thecoordinate frame represented by the dashed lines at Pd and the

Fig. 1. Globally defined orthogonal TCF.

calculated contour error would be the projection of the positiontracking error vector eq = q − qd along the normal directionof the desired contour at Pd, i.e., nd of LTCF. Such a calculatedcontour error is totally different from the actual contour errorεc, the distance between the actual position Pa(x(t), y(t))and Pc(xc(t), yc(t)), in which Pc represents the projectionof Pa(x(t), y(t)) onto the desired contour along its normaldirection, as shown in Fig. 1. Intuitively, the actual contouringerror εc depends on the geometry of the desired contour andthe actual path of the system only. It has nothing to do withthe desired trajectory to be tracked on the contour, which isaffected by other factors such as the speed of the desired motionon the contour as well. To overcome this problem, in [24], anorthogonal GTCF is introduced solely based on the geometry ofthe desired contour. Specifically, for the desired contour shownin Fig. 1 and defined by

f(x, y) = 0 (2)

where f is a known smooth shape function, a curvilinearcoordinate along the normal direction of the desired contouris defined as

rc(x, y) =f(x, y)√f2

x + f2y

(3)

where fx = ∂f/∂x and fy = ∂f/∂y. With this curvilinearcoordinate, the desired contour can be described by simplyconstraining the value of such a curvilinear coordinate to zeroin GTCF, i.e., rc(x, y) = 0. The normal direction of the desiredcontour is also the same as the axis direction of this curvilinearcoordinate in GTCF. It is further shown in [24] that the coordi-nate value of the actual position in GTCF along this curvilinearcoordinate (i.e., rc(x(t), y(t))) is essentially the same as theactual contouring error εc.

With the aforementioned definition of the curvilinear coor-dinate representing the contouring error, the other curvilinearcoordinate rm(x, y) is defined based on the curve length on the

HU et al.: GTCF-BASED CONTOURING CONTROL OF LINEAR-MOTOR-DRIVEN BIAXIAL SYSTEMS 5197

desired contour between the reference point and the projectionof the actual position at the desired contour along its normaldirection. For example, the curve lengths s0 and st shownin Fig. 1 represent the other curvilinear coordinate valuesof the initial position Pa(x(0), y(0)) and the actual positionPa(x(t), y(t)), respectively. Further details are given in [24].Overall, the proposed GTCF vector r = [rc, rm]T is defined bythe following curvilinear coordinate transformation:

r = h(q) =[

rc(x, y)rm(x, y)

]. (4)

It is shown in [24] that, on the desired contour, the direc-tional vector along the first curvilinear coordinate rc, [∂rc/∂x,∂rc/∂y]T, is the same as the unit vector along the normal direc-

tion of the desired contour, [fx/√

f2x + f2

y , fy/√

f2x + f2

y ]T,

and that the directional vector along the second curvilinearcoordinate rm, [∂rm/∂x, ∂rm/∂y]T, is the same as the unitvector along the tangential direction of the desired contour,

[−(fy/√

f2x + f2

y ), fx/√

f2x + f2

y ]T. It is thus obvious that, in

the proposed GTCF, the original contour tracking problem isdecoupled into a regulation problem along the curvilinear coor-dinate rc and a trajectory tracking problem along the curvilinearcoordinate rm. Furthermore, around the desired contour, theJacobian matrix of the curvilinear coordinate transformation (4)can be approximated by

J =[ ∂rc

∂x∂rc

∂y∂rm

∂x∂rm

∂y

]≈

⎡⎣ fx√

f2x+f2

y

fy√f2

x+f2y

− fy√f2

x+f2y

fx√f2

x+f2y

⎤⎦ (5)

which is unitary for all values of x and y, i.e., J−1 = JT.

B. Coupled System Dynamics in GTCF

The dynamics of a biaxial linear-motor-driven gantry can bedescribed by [21]

Mq + Bq + F(q) = u + d (6)

where M = diag[M1,M2] and B = diag[B1, B2] are the 2 ×2 diagonal inertia and damping matrices, respectively; u is the2 × 1 vector of control input and d is the 2 × 1 vector ofunknown nonlinear functions due to external disturbances ormodeling errors such as the force ripple; F(q) is the 2 × 1vector of nonlinear friction, and the model used in this paperis given by F(q) = ASf (q), where A = diag[A1, A2] is the2 × 2 diagonal friction coefficient matrix and Sf (·) is avector-valued smooth function, i.e., Sf (q) = [Sf (x), Sf (y)]T.Define the approximation error as F = F − F. Then, (6) can berewritten as

Mq + Bq + ASf (q) = u + dn + d (7)

where dn = [dn1, dn2]T is the constant offset value ofd1 = d + F and d = d1 − dn is the variation over time.Noting (4)

r = Jq(t), r = Jq(t) + Jq(t). (8)

Thus, the system dynamics in the proposed GTCF can beobtained from (7) as

Mtr + Btr + Ctr + AtSf (q) = ut + dt + Δ (9)

where

Mt =JMJ−1 Bt = JBJ−1 Ct = −JMJ−1JJ−1

At =JA dt = Jdn ut = Ju Δ = Jd. (10)

It is well known that (9) has several properties [23]: (P1)Mt is a symmetric positive definite (s.p.d.) matrix withμ1I ≤ Mt ≤ μ2I, where μ1 and μ2 are two positive scalars;(P2) The matrix Nt = Mt − 2Ct is a skew-symmetric ma-trix. In other words, sTNts = 0 ∀s; and (P3) Mt, Bt, Ct,At, dt, and ut in (9) can be linearly parameterized by aset of unknown parameters defined as θ = [θ1, . . . , θ8]T =[M1,M2, B1, B2, A1, A2, dn1, dn2]T.

In general, the parameter vector θ may not be known exactly(i.e., the payload of the gantry depends on tasks). However,the extent of parametric uncertainties can be predicted, and thefollowing practical assumption can be made.1

Assumption 1: Extent of parametric uncertainties and uncer-tain nonlinearities is known. More precisely

θ ∈ ΩθΔ= {θ : θmin ≤ θ ≤ θmax}

Δ ∈ ΩΔΔ=

{Δ : ‖Δ‖ ≤ δΔ

}(11)

where θmin = [θ1min, . . . , θ8min]T and θmax =[θ1max, . . . , θ8max]T are known constant vectors and δΔ

is a known function.The control objective is to synthesize a control input ut such

that q = [x, y]T tracks a desired contour qd(t) = [xd, yd]T

which is assumed to be at least second-order differentiable.In the proposed task coordinate system, such an objective isachieved by simply regulating rc to zero and letting rm followthe desired trajectory of rmd(t) = rm(xd(t), yd(t)).

III. DARC CONTOURING CONTROL LAW

Although the system dynamics in Cartesian space given by(6) is decoupled, the transformed system dynamics in the taskspace by (9) are strongly coupled and highly nonlinear as theJacobian matrix J in (10) is of full matrix and nonlinearfunction of the position. In this section, a DARC contouringcontrol law is firstly presented to directly deal with suchstrongly coupled nonlinear system dynamics. The control lawalso explicitly takes into account the large extent of parametricuncertainties and uncertain nonlinearities assumed in (11).

1For simplicity, the following notations are used: •i for the ith component ofthe vector •, •min for the minimum value of •, and •max for the maximumvalue of •. The operation ≤ for two vectors is performed in terms of thecorresponding elements of the vectors.


A. Projection-Type Adaptation Law

Let θ denote the estimate of θ and θ denote the estimationerror (i.e., θ = θ − θ). In view of (11), the following adap-tation law with discontinuous projection modification can beused [15]

˙θ = Proj

θ(Γτ) (12)

where Γ > 0 is a diagonal matrix, τ is an adapta-tion function to be synthesized later, and Proj

θ(•) =

[Projθ1

(•1), . . . , P rojθ8

(•8)]T represents the standard projec-tion mapping given by

Projθi

(•i) =

⎧⎨⎩

0, if θi = θimax and •i > 00, if θi = θimin and •i < 0•i, otherwise.

(13)

It can be shown [27] that, for any adaptation function τ ,

the projection mapping used in (13) guarantees (P4) θ ∈ ΩθΔ=

{θ : θimin ≤ θ ≤ θimax} and (P5) θT(Γ−1Projθ(Γτ ) − τ ) ≤

0 ∀τ .

B. DARC Contouring Control Law Synthesis

Define a switching-function-like quantity as

s = e + Λe = r − req, reqΔ= rd − Λe (14)

where e = r(t) − rd(t) is the output tracking error andΛ > 0 is a diagonal matrix. Define a positive semidefinitefunction

V (t) =12sTMt(r)s. (15)

Differentiating V yields

V (t)=sT[ut−Mtreq−Btr−Ctreq−AtSf (q)+dt+Δ

](16)

where reqΔ= rd − Λe and (P2) is used to eliminate the term

(1/2)sTMt(r)s. From (P3)

Mtreq+Btr+Ctreq+AtSf (q)−dt =−Ψ(r, r, req, req)θ(17)

where Ψ(r, r, req, req) is a 2 × 8 matrix of known functions,commonly referred to as the regressor. Thus

V (t) = sT[ut + Ψ(r, r, req, req)θ + Δ

]. (18)

Noting (18), the following ARC law is proposed:

ut = ua + us, ua = −Ψ(r, r, req, req)θ (19)

where ua is the adjustable model compensation needed forperfect tracking and us is a robust control law to be synthe-sized later. Substituting (19) into (18) and then simplifying theresulting expression leads to

V (t) = sT[us − Ψ(r, r, req, req)θ + Δ

]. (20)

The robust control function us consists of two terms

us = us1 + us2, us1 = −Ks (21)

where us1 is used to stabilize the nominal system and us2 is arobust feedback used to attenuate the effect of model uncer-tainties for a guaranteed robust control performance. NotingAssumption 1 and the uniformly boundedness of the onlineparameter estimation θ due to the use of the projection-typeadaptation law (12), there exists a us2 such that the followingtwo conditions are satisfied:

i sT{us2 − Ψ(r, r, req, req)θ + Δ

}≤ η

ii sTus2 ≤ 0 (22)

where η is a design parameter that can be arbitrarily small.One smooth example of us2 satisfying (22) is given by us2 =−(1/4η)H2s, where H is a smooth function satisfying H ≥‖θM‖‖Ψ(r, r, req, req)‖ + δΔ and θM = θmax − θmin. Thefollowing theorem summarizes the control performance of theproposed DARC contouring control law.

Theorem 1: Suppose that the adaptation function in (13) ischosen as

τ = ΨT(r, r, req, req)s. (23)

Then, the ARC control law in (19)–(22) guarantees thefollowing:

1) In general, all signals are bounded. Furthermore, thepositive semidefinite function V (t) in (15) is bounded by

V (t) ≤ exp(−λt)V (0) +η

λ[1 − exp(−λt)] (24)

where λ = 2σmin(K)/μ2, in which σmin(·) denotes theminimum eigenvalue of a matrix.

2) After a finite time t0, if there exist parametric uncertain-ties only, i.e., Δ = 0, ∀t ≥ t0, then, in addition to resultA, a zero steady-state tracking error is also achieved, i.e.,e → 0 and s → 0 as t → ∞.

The proof is presented in Appendix I.

IV. DIARC CONTOURING CONTROL LAW

In this section, the DIARC control strategy in [25] is appliedto synthesize a contouring control law for the system (9) thatachieves not only excellent output tracking performance as inthe previous section but also accurate physical parameter esti-mation in implementation. As in the DARC contouring controllaw design, the first step is to use a projection-type adaptationlaw structure (12) to achieve a controlled learning or adaptationprocess. However, unlike DARC designs, the least-squares-typeadaptation law will be used to achieve better convergence ofparameter estimations and the adaptation rate matrix will betime-varying and nondiagonal. As such, the simple discontin-uous projection mapping (13) in DARC designs cannot be usedtheoretically since such a discontinuous projection mapping isvalid only for diagonal adaptation rate matrix Γ. Instead, thestandard projection mapping in adaptive control [27] should be


used to keep the parameter estimates within the known boundedset Ωθ, the closure of the set Ωθ. It can be verified that (P4) and(P5) are still preserved with such a projection. The details aregiven in [23] and [25] and omitted here.

A. DIARC Contouring Control Law Synthesis

Through the use of projection-type adaptation law, the pa-rameter estimates are bounded with known bounds, regardlessof the estimation function τ to be used. This property will beused in this section to synthesize a DIARC contouring controllaw that achieves a guaranteed robust transient performance andsteady-state contouring accuracy regardless of how the physicalparameters will be estimated.

The proposed DIARC control law has the following form:

ut =ua + us ua = ua1 + ua2 us = us1 + us2

ua1 = − Ψ(r, r, req, req)θ us1 = −Ks (25)

where ua1 is the adjustable model compensation needed forperfect tracking with θ being the online estimates of phys-ical parameters to be detailed later, ua2 is a fast dynamiccompensation term to be synthesized in the following, andus1 is used to stabilize the nominal system, which is chosento be a simple proportional feedback with K being an s.p.d.matrix for simplicity. Moreover, us2 is a feedback used toattenuate the effect of model uncertainties for a guaranteedrobust performance.

For V given by (15), substituting (25) into (18) and simpli-fying the resulting expression lead to

V = sT[ua2 + us − Ψ(r, r, req, req)θ + Δ

]. (26)

Define a constant dc and time-varying function d∗(t)such that

Jdc + d∗(t) = −Ψ(r, r, req, req)θ + Δ. (27)

Conceptually, (27) lumps the disturbance and the modeluncertainties due to parameter-estimation error together anddivides it into the low-frequency component dc and the higherfrequency components d∗(t), so that the low-frequency com-ponent dc can be compensated through the fast adaptation ofDARC design [16]. Substituting (27) into (26)

V = sT[ua2 + us + Jdc + d∗(t)

]. (28)

Choose the fast compensation term ua2 as

ua2 = −Jdc (29)

where dc represents the estimate of dc updated by

˙dc = Projdc

(γdJTs),∣∣∣dc(0)

∣∣∣ ≤ dcmax (30)

in which dcmax is a preset bound for dc(t) and γd is a 2 ×2 constant diagonal matrix. As in DARC in Section III, the

projection mapping in (30) guarantees that |dc(t)| ≤ dcmax ∀t.Substituting (29) into (28)

V = sT[−Ks + us2 − Jdc + d∗(t)

]. (31)

Similar to (22), the robust term us2 is now chosen to satisfythe following two conditions:

i sT[us2 − Jdc + d∗(t)

]≤ η

ii sTus2 ≤ 0 (32)

where η is a design parameter that can be arbitrarily small.One smooth example of us2 satisfying (32) is given byus2 = −(1/4η)h2s, where h is a smooth function satisfy-ing h ≥ ‖dcmax‖ + ‖θM‖‖Ψ(r, r, req, req)‖ + δΔ and θM =θmax − θmin.

Theorem 2: When the DIARC control law (25) withthe projection-type adaptation law (12) is applied, regardless ofthe estimation function τ to be used, in general, all signals in theresulting closed-loop system are bounded and the contouringerror and output position tracking error are guaranteed to havea prescribed transient performance and steady-state accuracy inthe sense that the positive semidefinite function V (t) definedby (15) is bounded by (24).

The proof is similar to the proof of result A of Theorem 1.

B. Estimation of Physical Parameters

The DIARC control law achieves a guaranteed transient andsteady-state performance regardless of the estimation functionτ to be used. Thus, this section focuses on the constructionof suitable estimation functions τ so that an improved steady-state performance—asymptotic output tracking or zero steady-state contouring and position tracking error—can be obtainedeven when all physical parameters are unknown. In addition,it is desirable to have online parameter estimates convergeor stay close to their true values so that they can be usedfor other purposes such as machine component health mon-itoring. To this end, in this section, it is assumed that thesystem has parametric uncertainties only, i.e., assuming d = 0in (7).

Let Hf (s) be the transfer function of any filter with a relativedegree larger than or equal to 1 (e.g., Hf (s) = 1/(τfs + 1)).Then, applying the filter to both sides of (7), we obtain

uf = ΥTf θ (33)

where

uf =[

u1f

u2f

]ΥT

f =[

xf , 0, xf , 0, Sff (x), 0, 1f , 00, yf , 0, yf , 0, Sff (y), 0, 1f

]

in which •f represents the filtered value of •.Define the prediction error vector as ς = uf − uf , where

uf = ΥTf θ. Then, the prediction error ς related to the

parameter-estimation error is expressed as

ς = ΥTf θ = ΥT

f θ − uf . (34)


Fig. 2. Experimental setup.

The linear regression model (34) is in the standard form towhich various parameter-estimation algorithms can be appliedto estimate θ. For example, when the least-squares-type estima-

tion algorithm [25] is used, ˙θ is updated by the adaptation law

(12) with the adaptation function given by

τ = − 11 + υtr

{ΥT

f ΓΥf

}Υf ς (35)

and the adaptation rate matrix given by

Γ={

κΓ − 1

1+υtr{ΥTf

ΓΥf}ΓΥfΥTf Γ, if λmax (Γ(t))≤ρM

0, otherwise(36)

where κ ≥ 0 is the forgetting factor, ρM is the preset upperbound for ‖Γ(t)‖, and υ ≥ 0 with υ = 0 leading to the unnor-malized algorithm. With these practical modifications, Γ(t) ≤ρMI ∀t. The following lemma summarizes the properties ofsuch estimators [25].

Lemma 1: In the presence of parametric uncertainties only,i.e., d = 0, the projection-type adaptation law (12) with theleast-squares estimator (35) guarantees the following.

1) In general, θ ∈ L∞[0,∞) and ς ∈ L2[0,∞) ∩ L∞[0,∞).2) In addition, if the following persistent excitation (PE)

condition is satisfied: there exist κp > 0 and T > 0such that

t+T∫t

ΥfΥTf dτ ≥ κpIp ∀t (37)

then, the physical parameter estimates θ converge to theirtrue values (i.e., θ → 0 as t → ∞) and θ ∈ L2[0,∞).

Theorem 3: Consider the DIARC control law (25) and theprojection-type adaptation law (12) with the least-squares-typeestimation function (35). After a finite time t0, if there existparametric uncertainties only, i.e., d = 0 ∀t ≥ t0 in (7), andthe PE condition (37) is satisfied, in addition to the robustperformance results stated in Theorem 2, an improved steady-state contouring performance—asymptotic output tracking—isalso achieved, i.e., e → 0 as t → ∞.

Proof of Theorem 3 is presented in Appendix II.

V. EXPERIMENTAL SETUP AND RESULTS

A. Experimental Setup

As shown in Fig. 2, a linear-motor-driven biaxial gantryfrom Rockwell Automation with a dSPACE DS1103 controlsystem is set up as the experimental system. The positionsensors of the gantry are two linear encoders with a resolutionof 0.5 μm after quadrature. The velocity signal is obtainedby the difference of two consecutive position measurements.Standard least-squares identification is performed to obtainthe parameters of the biaxial gantry, and it is found thatnominal values of the gantry system parameters without loadsare M1 = 0.12 V/m/s2, M2 = 0.55 V/m/s2, B1 = 0.35 V/m/s,B2 = 0.5 V/m/s, Af1 = 0.1 V, Af2 = 0.13 V, dN1 = 0 V, anddN2 = 0 V; these values are slightly different from those in ourprevious publications due to the change of the gantry physicalparameters over time with varying lubrication conditions ofbearing. Through varying input voltage and measuring theresultant force when the motor is blocked, the gain from theinput voltage to the actuation force generated is computed askf = 69 N/V. The bounds of the parametric variations arechosen as θmin = [0.06, 0.4, 0.2, 0.3, 0.05, 0.08,−0.5,−0.5]T

and θmax = [0.25, 0.7, 0.6, 0.6, 0.15, 0.25, 0.5, 0.5]T.The following performance indexes are used to quantify the

performance of each control algorithm [23].

1) ‖εc‖rms = ((1/T )∫ T

0 |εc|2dt)1/2, the root-mean-squarevalue of the contouring error, where T represents the totalrunning time of the concerned experimental segment.

2) εcM = maxt{|εc|}, the maximum absolute value ofthe contouring error over the concerned experimentalsegment.

3) ‖ui‖rms = ((1/T )∫ T

0 |ui|2dt)1/2, the average controlinput of each axis over the concerned experimentalsegment.

B. Experimental Results

The control algorithms are implemented by the dSPACEDS1103 system executing programs at a sampling period of


Ts = 0.2 ms, resulting in a velocity measurement resolution of0.0025 m/s. The following control algorithms are compared.C1: GTCF-based DARC contouring control—the control law

presented in Section III.C2: GTCF-based DIARC contouring control—the control law

presented in Section IV.C3: LTCF-based DIARC contouring control—the control law

presented in [23].For a fair comparison, the controller parameter values of

all controllers are chosen the same if they have the samephysical meanings. Specifically, for all controllers, the smoothfunctions Sf (x) and Sf (y) are chosen as (2/π)arctan(9000x)and (2/π)arctan(9000y), respectively, and Λ = diag[100, 30].As in [23], a diagonal large-enough constant total feedback gainmatrix of diag[100, 60] is used for the robust control functionin (21) to ease the implementation of the proposed controllaw. The initial parameter estimates are all chosen as θ(0) =[0.1, 0.55, 0.3, 0.3, 0.1, 0.15, 0, 0]T. The adaptation rates in C1are set as Γ = diag[10, 10, 10, 10, 10, 10, 10000, 10000]. In C2,the bounds of dc in (30) is dcmax = [1, 1]T with γd =diag[10000, 10000]. Second-order transfer functions of damp-ing ratio of 0.7 and natural frequencies of 250 Hz for thex-axis and 150 Hz for the y-axis are used for the filtersin Section IV-B. In (36), the forgetting factor of κ = 0.1and υ = 0.1 are used, with the initial adaptation rates beingΓ(0) = diag[10, 10, 10, 10, 10, 10, 5000, 5000] and ρM = 500.The following tests are performed.Set1: To test the nominal contouring performance of the con-

trollers, experiments are run without payload, which isequivalent to M1 = 0.12 and M2 = 0.55.

Set2: To test the performance robustness of the algorithms toparameter variations, a 5-kg payload is mounted on thegantry, which is equivalent to M1 = 0.19 and M2 = 0.62.

Set3: A large step disturbance (a simulated 0.6-V electricalsignal) is added to the input of the y-axis at t = 2.2 s andremoved at t = 5.2 s to test the performance robustness ofeach controller to disturbances.

1) Experiment I: To test the contouring performance of theproposed algorithms, the biaxial gantry is first commanded totrack an ellipse given by

qd =[xd(t)yd(t)

]=

[0.2sin(4t)

−0.1cos(4t) + 0.1

](38)

which has an angular velocity of ω = 4 rad/s and a speed ofv =

√0.16 + 0.48 cos2(4t) m/s.

The experimental results in terms of performance indexesafter running the gantry for several periods are given in Table I.Overall, all controllers achieve good steady-state contouringperformance during fast elliptical movements with almost thesame amount of control efforts for every test set. Specifically,for both Set1 and Set2, the contouring errors of C1 and C2shown in Fig. 3 are mostly within 9 μm and roughly 70% ofthose in C3, demonstrating the significantly improved perfor-mance of using GTCF and the performance robustness of theproposed controllers to parameter variations. The contouringerrors of Set3 are shown in Fig. 4. As seen from the figures, theadded large disturbances do not affect the contouring perfor-

TABLE ICONTOURING RESULTS OF EXPERIMENT I

Fig. 3. Contouring errors of C1, C2, and C3 for Set1 (no load) and Set2(loaded) in experiment I.

Fig. 4. Contouring errors of C1, C2, and C3 for Set3 (with disturbance) inexperiment I.

mance of the three ARC controllers much except the transientspikes when the sudden changes of the disturbances occur.These results demonstrate the strong performance robustnessof the proposed ARC schemes.

To have a better appreciation of the merits and effectivenessof the proposed GTCF-based ARC controllers, the desired


Fig. 5. Partial contours of C1, C2, C3, and desired qd(t) for all sets inexperiment I.

TABLE IICONTOURING RESULTS OF EXPERIMENT II

contour and the actual contours of three controllers in all threesets are partially shown in Fig. 5 around (0.2 m, 0.1 m) wherethe velocity along the x-axis is near zero and the effect ofCoulomb friction along the x-axis is significant. It can be seenthat, for all sets, the actual contours of C1 and C2 are quite closeto the desired contours, while the actual contour of C3 deviatesfrom the desired contour significantly. All these results clearlydemonstrate the superiority of the proposed GTCF-based ARCcontrollers C1 and C2 over the traditional LTCF-based C3.

2) Experiment II: To test the consistency of the experimen-tal results obtained, the biaxial gantry is also commanded totrack another ellipse described by

qd =[xd(t)yd(t)

]=

[0.2sin(3t)

−0.15cos(3t) + 0.15

](39)

which has an angular velocity of ω = 3 rad/s and a speed ofv =

√0.2025 + 0.1575 cos2(3t) m/s.

The experimental results in Table II show the same trendsas in experiment I—about 40% improvement of contouringaccuracy of C1 and C2 over C3 while using almost the sameamount of control efforts for every test set. The time historiesof contouring errors for Set1 and Set2 are shown in Fig. 6,which further validate the good nominal performances and theperformance robustness of the proposed DIARC controllers toparameter variations. It is also worth noting that the contouringerror of C3 appears to have a constant negative offset whichdoes not exist in those of C1 and C2. This is due to the fact that

Fig. 6. Contouring errors of C1, C2, and C3 for Set1 (no load) and Set2(loaded) in experiment II.

Fig. 7. Contouring errors of C1, C2, and C3 for Set3 (with disturbance) inexperiment II.

the contouring error in C3 is approximately calculated basedon the projection of the actual tracking errors in the LTCFwhich is moving with the desired motion trajectory on thecontour. This further illustrates the superiority of the proposedGTCF-based ARC controllers over the existing LTCF-basedones for high-speed contouring applications. The contouringerrors shown in Fig. 7 for Set3 also show that the added largedisturbances do not affect the contouring performance of theproposed algorithms much except the initial transient when thesudden changes of the disturbances occur. The desired contoursand the actual contours of various controllers for all three setsare partially shown in Fig. 8 around (−0.2 m, 0.15 m). It isseen again that the actual contours of C1 and C2 in all threesets are much more closer to the desired contours than thoseof C3, which further validate the effectiveness of the proposedGTCF-based ARC designs.


Fig. 8. Partial contours of C1, C2, C3, and desired qd(t) for all sets inexperiment II.

Fig. 9. x-axis parameter estimates of C2 for Set1 in experiments I and II(no load).

C. Parameter-Estimation Results

Although the proposed GTCF-based ARC controllers, C1and C2, achieve the same level of excellent contouring perfor-mance, C2 has more accurate parameter estimations. In general,the parameter estimates of C1 either converge slowly or do notconverge at all (omitted due to space limit). In contrast, physicalparameter estimates of C2 consistently approach their offlineestimated values. For example, Fig. 9 shows the time historiesof the x-axis inertia and the Coulomb friction estimates of C2during no-load experiments (Set1), in which C2(I) stands forthe results in contouring experiment I and C2(II) for experimentII. As seen, for both contouring experiments I and II, theinertia and Coulomb friction estimates all approach their offlineestimated values. The same conclusion can be drawn from theresults for loaded experiments (Set2) shown in Fig. 10.

VI. CONCLUSION

In this paper, GTCF-based DARC and DIARC controllershave been developed for the contouring control of a biax-

Fig. 10. x-axis parameter estimates of C2 for Set2 in experiments I and II(loaded).

ial linear-motor-driven gantry system. In contrast to exist-ing LTCF-based controllers, the proposed algorithms utilizethe newly developed orthogonal GTCF when formulating thecontouring control problem. With such a formulation, thecalculation of contouring error is accurate to the first-orderapproximation regardless of the position tracking errors andthe contour curvatures being small enough or not. SpecificARC design methodologies, DARC and DIARC, were thenused to synthesize contouring controllers that can effectivelyhandle the strongly coupled system dynamics in GTCF andthe effect of large parametric uncertainties and disturbances.The existing LTCF-based DIARC and the proposed GTCF-based DARC and DIARC contouring controllers have alsobeen implemented on a high-speed industrial biaxial gantry.Comparative experimental results reveal that both the proposedGTCF-based ARC contouring controllers exhibit much im-proved contouring performances than the existing LTCF-basedDIARC. In addition, the proposed GTCF-based DIARC pos-sesses accurate estimation of physical parameters as well.

APPENDIX I

Proof of Theorem 1: From (P1), we have

12μ1‖s‖2 ≤ V ≤ 1

2μ2‖s‖2. (40)

From (20), (21), condition i of (22), and (40), it follows that

V (t) = − sTKs + sT[us2 − Ψ(r, r, req, req)θ + Δ

]≤ − σmin(K)‖s‖2 + η = −λV + η (41)

which yields (24) and thus proves result A of Theorem 1.Now, consider the situation in B of Theorem 1, i.e., Δ =0 ∀t ≥ t0. Choose a positive definite function Vθ(t) = V (t) +(1/2)θTΓ−1θ. From (41), condition ii of (22), and (P5), itfollows that

Vθ(t) ≤ − sTKs − sTΨ(r, r, req, req)θ + θTΓ−1 ˙θ

= − sTKs + θT[Γ−1 ˙

θ − ΨT(r, r, req, req)s]

≤ − sTKs (42)


which shows that s ∈ L2 ∩ L∞. As s is bounded, s is uniformlycontinuous. By Barbalat’s lemma, s → 0 as t → ∞. �

APPENDIX II

Proof of Theorem 3: Consider a positive function Va as

Va = V +12dc

Tγ−1d dc (43)

where V is given by (15). Noting (26), (29), and Δ = 0, it iseasy to verify that the time derivative of Va is

Va = sT[−Jdc + us − Ψ(r, r, req, req)θ

]+ dc

Tγ−1d

˙dc

= − sTKs + sTus2 − sTΨ(r, r, req, req)θ

+ dc

T[γ−1d

˙dc − JTs]

. (44)

Thus, with the parameter adaptation law (30) for dc, notingcondition ii of (32) and (P5) with treating dc = 0

Va ≤ −sTKs − sTΨ(r, r, req, req)θ. (45)

When the PE condition (37) is satisfied, by Lemma 1, θ ∈L2[0,∞). Moreover, Ψ(r, r, req, req) is bounded as statedin Theorem 2. Thus, it is obvious that Ψ(r, r, req, req)θ ∈L2[0,∞) which leads to s ∈ L2[0,∞). By Barbalat’s lemma,asymptotic output contouring can be proved, which completesthe proof. �

REFERENCES

[1] L. Xu and B. Yao, “Output feedback adaptive robust precision motioncontrol of linear motors,” Automatica, vol. 37, no. 7, pp. 1029–1039,Jul. 2001, the finalist for the Best Student Paper award of ASME DynamicSystem and Control Division in IMECE00.

[2] L. Lu, Z. Chen, B. Yao, and Q. Wang, “Desired compensation adaptiverobust control of a linear motor driven precision industrial gantry withimproved cogging force compensation,” IEEE/ASME Trans. Mechatron-ics, vol. 13, no. 6, pp. 617–624, Dec. 2008.

[3] F. Cupertino, P. Giangrande, G. Pellegrino, and L. Salvatore, “End ef-fects in linear tubular motors and compensated position sensorless controlbased on pulsating voltage injection,” IEEE Trans. Ind. Electron., vol. 58,no. 2, pp. 494–502, Feb. 2011.

[4] Y.-S. Huang and C.-C. Sung, “Function-based controller for linear motorcontrol systems,” IEEE Trans. Ind. Electron., vol. 57, no. 3, pp. 1096–1105, Mar. 2010.

[5] I.-C. Vese, F. Marignetti, and M. M. Radulescu, “Multiphysicsapproach to numerical modeling of a permanent-magnet tubularlinear motor,” IEEE Trans. Ind. Electron., vol. 57, no. 1, pp. 320–326,Jan. 2010.

[6] J. G. Amoros and P. Andrada, “Sensitivity analysis of geometrical param-eters on a double-sided linear switched reluctance motor,” IEEE Trans.Ind. Electron., vol. 57, no. 1, pp. 311–319, Jan. 2010.

[7] G. T.-C. Chiu and M. Tomizuka, “Contouring control of machine tool feeddrive systems: A task coordinate frame approach,” IEEE Trans. ControlSyst. Technol., vol. 9, no. 1, pp. 130–139, Jan. 2001.

[8] C.-L. Chen and K.-C. Lin, “Observer-based contouring controller designof a biaxial stage system subject to friction,” IEEE Trans. Control Syst.Technol., vol. 16, no. 2, pp. 322–329, Mar. 2008.

[9] S.-L. Chen and K.-C. Wu, “Contouring control of smooth paths for multi-axis motion systems based on equivalent errors,” IEEE Trans. ControlSyst. Technol., vol. 15, no. 6, pp. 1151–1158, Nov. 2007.

[10] J. Yang and Z. Li, “A novel contouring error estimation for position loop-based cross-coupled control,” IEEE/ASME Trans. Mechatronics, vol. 16,no. 4, pp. 643–655, Aug. 2011.

[11] Y. Koren, “Cross-coupled biaxial computer control for manufacturingsystems,” Trans. ASME, J. Dyn. Syst. Meas. Control, vol. 102, no. 4,pp. 265–272, Dec. 1980.

[12] B. Yao, S. P. Chan, and D. Wang, “Unified formulation of variable struc-ture control schemes to robot manipulators,” IEEE Trans. Autom. Control,vol. 39, no. 2, pp. 371–376, Feb. 1994.

[13] B. Yao, W. B. Gao, S. P. Chan, and M. Cheng, “VSC coordinated controlof two robot manipulators in the presence of environmental constraints,”IEEE Trans. Autom. Control, vol. 37, no. 11, pp. 1806–1812, Nov. 1992.

[14] M.-Y. Cheng and C.-C. Lee, “Motion controller design for contour-following tasks based on real-time contouring error estimation,” IEEETrans. Ind. Electron., vol. 54, no. 3, pp. 1686–1695, Jun. 2007.

[15] B. Yao, “High performance adaptive robust control of nonlinear systems:A general framework and new schemes,” in Proc. IEEE Conf. DecisionControl, San Diego, CA, 1997, pp. 2489–2494.

[16] B. Yao and M. Tomizuka, “Adaptive robust control of SISO nonlinearsystems in a semi-strict feedback form,” Automatica, vol. 33, no. 5,pp. 893–900, May 1997.

[17] B. Yao, C. Hu, L. Lu, and Q. Wang, “Adaptive robust precision mo-tion control of a high-speed industrial gantry with cogging force com-pensations,” IEEE Trans. Control Syst. Technol., 2011. DOI: 10.1109/TCST.2010.2070504, to be published.

[18] Y. Hong and B. Yao, “A globally stable high performance adaptive robustcontrol algorithm with input saturation for precision motion control oflinear motor drive system,” IEEE/ASME Trans. Mechatronics, vol. 12,no. 2, pp. 198–207, Apr. 2007.

[19] Y. Hong and B. Yao, “A globally stable saturated desired compensationadaptive robust control for linear motor systems with comparative experi-ments,” Automatica, vol. 43, no. 10, pp. 1840–1848, Oct. 2007.

[20] C. Hu, B. Yao, and Q. Wang, “Coordinated adaptive robust contouringcontroller design for an industrial biaxial precision gantry,” IEEE/ASMETrans. Mechatronics, vol. 15, no. 5, pp. 728–735, Oct. 2010.

[21] C. Hu, B. Yao, and Q. Wang, “Coordinated adaptive robust contouringcontrol of an industrial biaxial precision gantry with cogging force com-pensations,” IEEE Trans. Ind. Electron., vol. 57, no. 5, pp. 1746–1754,May 2010.

[22] C. Hu, B. Yao, Z. Chen, and Q. Wang, “Adaptive robust repetitive controlof an industrial biaxial precision gantry for contouring tasks,” IEEE Trans.Control Syst. Technol., 2010. DOI: 10.1109/TCST.2010.2100044, to bepublished.

[23] C. Hu, B. Yao, and Q. Wang, “Integrated direct/indirect adaptive robustcontouring control of a biaxial gantry with accurate parameter estima-tions,” Automatica, vol. 46, no. 4, pp. 701–707, Apr. 2010.

[24] B. Yao, C. Hu, and Q. Wang, “An orthogonal global task coordinate framefor contouring control of biaxial systems,” IEEE/ASME Trans. Mecha-tronics, 2011. DOI: 10.1109/TMECH.2011.2111377, to be published.

[25] B. Yao, “Integrated direct/indirect adaptive robust control of SISO nonlin-ear systems in semi-strict feedback form,” in Proc. Amer. Control Conf.,Denver, CO, Jun. 2003, pp. 3020–3025.

[26] C. Hu, B. Yao, and Q. Wang, “Adaptive robust precision motion control ofsystems with unknown input dead-zones: A case study with comparativeexperiments,” IEEE Trans. Ind. Electron., vol. 58, no. 6, pp. 2454–2464,Jun. 2011.

[27] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adap-tive Control Design. New York: Wiley, 1995.

Chuxiong Hu (S’09–M’11) received the B.Eng.and Ph.D. degrees in mechanical engineering fromZhejiang University, Hangzhou, China, in 2005 and2010, respectively.

From 2007 to 2008, he was a Visiting Scholarin mechanical engineering with Purdue University,West Lafayette, IN. He is currently a PostdoctoralResearcher with the Department of Precision In-struments and Mechanology, Tsinghua University,Beijing, China. His research interests include high-performance multiaxis motion control, precision

mechatronics systems and measurements, adaptive and robust control, andnonlinear systems.


Bin Yao (S’92–M’96–SM’09) received the B.Eng.degree in applied mechanics from Beijing Universityof Aeronautics and Astronautics of China, Beijing,China, in 1987, the M.Eng. degree in electrical en-gineering from Nanyang Technological University,Singapore, Singapore, in 1992, and the Ph.D. degreein mechanical engineering from the University ofCalifornia, Berkeley, in February 1996.

He has been with the School of Mechanical Engi-neering, Purdue University, West Lafayette, IN, since1996 and was promoted to the rank of Professor in

2007. He was honored as a Kuang-piu Professor in 2005 and a ChangjiangChair Professor at Zhejiang University, Hangzhou, China. He has been anAssociate Editor of the ASME Journal of Dynamic Systems, Measurement, andControl since 2006. His research interests include the design and control ofintelligent coordinated control of electromechanical/hydraulic systems, optimaladaptive and robust control, nonlinear observer design and neural networksfor virtual sensing, modeling, fault detection, diagnostics, and adaptive fault-tolerant control, and data fusion.

Dr. Yao was a recipient of a Faculty Early Career Development (CAREER)Award from the National Science Foundation in 1998, a Joint Research Fundfor Outstanding Overseas Chinese Young Scholars from the National NaturalScience Foundation of China in 2005, a Chang Jiang Chair Professorship fromthe Ministry of Education of China in 2010, the O. Hugo Schuck Best Paper(Theory) Award from the American Automatic Control Council in 2004, andthe Outstanding Young Investigator Award of the American Society of Mechan-ical Engineers (ASME) Dynamic Systems and Control Division (DSCD) in2007. He is a member of ASME and has chaired numerous sessions and servedin a number of international program committees of various IEEE, ASME,and International Federation of Automatic Control conferences, including asGeneral Chair of the 2010 IEEE/ASME International Conference on AdvancedIntelligent Mechatronics. From 2000 to 2002, he was the Chair of the Adaptiveand Optimal Control Panel, and from 2001 to 2003, he was the Chair ofthe Fluid Control Panel of the ASME DSCD. He is currently the Chair ofthe ASME DSCD Mechatronics Technical Committee, which he initiated in2005. He was a Technical Editor of the IEEE/ASME TRANSACTIONS ON

MECHATRONICS from 2001 to 2005. More detailed information can be foundat https://engineering.purdue.edu/~byao.

Qingfeng Wang received the M.Eng. and Ph.D.degrees in mechanical engineering from ZhejiangUniversity, Hangzhou, China, in 1988 and 1994,respectively.

He then became a faculty with Zhejiang Univer-sity, where he was promoted to the rank of Professorin 1999 and where he was also the Director of theState Key Laboratory of Fluid Power Transmissionand Control from 2001 to 2005 and currently servesas the Director of the Institute of Mechatronic Con-trol Engineering. His research interests include the

electrohydraulic control components and systems, hybrid power systems andenergy-saving techniques for construction machinery, and system synthesis formechatronic equipments.

global task coordinate frame-based contouring control of linear …byao/papers/tie_11_gtcf.pdf ·...

Documents