A Review on Dynamic Control of Parallel Kinematic Machine 37 PAGINI

8/13/2019 A Review on Dynamic Control of Parallel Kinematic Machine 37 PAGINI

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A review on dynamic control of parallel kinematic machine:

theory and experiments

Flavien Paccot1 Nicolas Andreff1,2 Philippe Martinet1,3

1LASMEA - UMR 6602 CNRS, Blaise Pascal University, Clermont-Ferrand, France2LAMI, IFMA, France3ISRC, Sungkyunkwan University, Suwon, South Korea

Abstract

In this article, a review on parallel kinematic ma-chine dynamic control is performed. It is shownthat the classical control strategies from serialrobotics generally used for parallel kinematic ma-chine have to be thought again. Indeed, it is firstshown that the joint space control is not relevantfor these mechanisms for several reasons such asmechanical behaviour or computational efficiency.Consequently, the Cartesian space control should

be preferred over the joint space one. Neverthe-less, some modification to the well-known Cartesianspace control strategies of serial robotics are pro-posed to perfectly suit them to parallel kinematicmachines, particularly a solution using an extero-ceptive measure of the end-effector pose. The ex-pected improvement in terms of accuracy, stabilityand robustness are discussed. A comparison be-tween the main presented strategies is finally per-formed both in simulation and experiments.

1 IntroductionFrom a theoretical point of view, parallel kine-matic machines allow for better dynamic perfor-mances than serial ones, in terms of speed, ac-curacy and stiffness [Merlet, 2000]. Therefore,they seem perfectly suitable for industrial high-speed applications, such as pick-and-place or highspeed machining. However, experiments on parallelkinematic machines point out that these good dy-namic properties are not always established [Wangand Masory, 1993, Tlusty et al., 1999, Pritschow,

2002, Brecher et al., 2006a, Denkena and Holz,2006]. Consequently, the improvement of static anddynamic accuracy is still an up-to-date and pros-perous research field.

On the one hand, recent machines allow for im-pressive maximal acceleration, such as 200m.s2

for the high-speed manipulator PAR4 [Nabat et al.,2005], or 50m.s2 for the Urane SX machinetool [Company and Pierrot, 2002]. Such high accel-eration can not be achievable with serial kinematicmachines. Consequently, the time gain is clearly es-

tablished [Geldart et al., 2003,Terrier et al., 2004].On the other hand, several works deal with issueson the accuracy and stiffness of parallel kinematicmachines. Pritschow [Pritschow, 2002] presents alist of phenomena affecting the accuracy (Figure 1).Two major issues are the object of numerous works.

First major issue, the presence of numerous

Figure 1: Causes of accuracy losses according toPritschow [Pritschow, 2002]

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Nomenclature

X, X, X Any representation of end-effectorpose, velocity and acceleration

q, q,q Joint positions, velocities and accel-erations

F KM Forward kinematic modelIKM Inverse kinematic model

X= Dq Forward instantaneous kinematicsmatrix, abusively called Jacobianmatrix

q= DinvX Inverse instantaneous kinematicsmatrix, abusively called inverse Ja-cobian matrix

D Time derivative of the forward in-stantaneous kinematics matrix

Dinv Time derivative of the inverse in-stantaneous kinematics matrix

F DM Forwards dynamic modelIDM Inverse dynamic model Actuation torquesA Inertia matrixH Vector containing Coriolis, centrifu-

gal and gravity forces

M Inertia matrix of the actuated bod-ies, mapped into the active jointspace, diagonal and constant

I Inertia matrix of the end-effector,diagonal and constant

f Friction forcesFv, Fs Viscous and dry friction parametersKp, Kv, Ki Proportional, derivative and inte-

gral gainuP ID Signal generated by the PID con-

trolleruf f Feedforward compensation termucomp Compensation termvd Desired variable ve= vd v Error signal between the desired

and measured (or estimated) vari-able v

s Laplace variablem Numerical estimation of modelmS(m) Skew matrix associated to the cross-

product by vectorm

Table 1: General notation used in this article

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joints causes kinematic model errors because of

clearances and assembly defects [Wang and Ma-sory, 1993]. Moreover, the complex kinematics of-ten leads to model simplifications decreasing ac-curacy [Pritschow, 2002]. The main solution tothese problems is a performant kinematic identi-fication [Wang and Masory, 1993]. It allows formatching as far as possible the machine model andits real behaviour. The measures used for identifi-cation are performed with various means [Daney,1999, Besnard and Khalil, 1999, Renaud et al.,2006, Chanal et al., 2006b]. Another solution tokinematic errors consists in using adapted model-ing methods to increase the model accuracy while

simplifying the algorithms [Merlet, 2000].Second major issue, the actuators of a parallel

kinematic machine tool do not apply a torque alongthe end-effector motion axis, contrary to a serialone [Tlusty et al., 1999]. It results in a decrease ofstiffness leading to a lack of accuracy during ma-chining process. In this way, a workspace can bedetermined where stiffness allows for sufficient ac-curacy [Chanal et al., 2006a]. Moreover, improve-ments are sought throughout design of new struc-ture [Tlusty et al., 1999,Liu et al., 2000].

In summary, it seems that a parallel kinematic

machine is really faster than a serial one, but gainsin terms of stiffness and accuracy are questionable.Actually, Merlet explains that the advantages ofparallel kinematic machines can only be qualifiedas p otential [Merlet, 2002]. To reach their theo-retical performances, parallel kinematic machinesstill require improvements in design, modeling andcontrol.

Nevertheless, the solutions presented above con-cern only mechanical design, kinematic modelingand identification. To our mind, the control ofparallel kinematic machines is a field where greatpotential remains for improving accuracy. Indeed,most of the work in the literature only reuses theknowledge of serial robotics whereas control strate-gies have to be thought again to improve paral-lel kinematic machines performances. To illustratethis point, a state of the art on control is first pre-sented in this paper, to point out the major issuescurrently met. In this way, the relevant controlstrategies are revisited and novel solutions are pro-posed, in the continuity of previous work [Ait Aideret al., 2006,Paccot et al., 2006,Paccot et al., 2007].Then, the expected improvements from one con-

trol scheme to the other one are presented and

discussed. Finally, some of the presented controlstrategies are applied to a specific test-bed. Withthe prerequisite of an adapted dynamic model anda dynamic identification, a comparison between thedifferent control schemes is proposed with simu-lated and experimental results. Moreover, the com-parison is achieved throughout realistic applicationand relevant measures.

This paper is organised as follows : Section 2 isdevoted to the state of the art on joint space con-trol, Section 3 concerns Cartesian space control andproposed solutions, Section 4 deals with modelingthe test-bed, Section 5 concerns the results.

2 Control of parallel kine-

matic machines in the joint

space

The knowledge on parallel robotics comes directlyfrom serial one. Therefore, parallel kinematic ma-chines are mainly controlled with the same strate-gies as serial ones. Therefore, the main controlmethods met in the literature are linear single-axis

and computed torque control, both in the jointspace.

2.1 Linear single-axis control

In most industrial cases, a linear single-axis con-trol, also called PID control or simple con-trol, is used [Khalil and Dombre, 2002, Brecheret al., 2006a,Denkena and Holz, 2006,Zhiyong andHuang, 2004, Yang and Huang, 2006]. Figure 2 re-minds of this well-known control scheme. This isthe conventional and simplest way to control a sys-tem. It can be used for serial and parallel kinematicmachines. Therefore, controllers can be reused,from serial to parallel kinematic machines with-out major adaptation, making this control strat-egy interesting in a industrial context. In addi-tion, it provides rather good performances withregards to its wide use. The tuning of such acontrol is well known from the classical roboticsmethod [Khalil and Dombre, 2002] to more elab-orated ones, adapted for parallel kinematic ma-chines [Zhiyong and Huang, 2004,Yang and Huang,2006].

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+

++_Path P IDIKM(Xd)

Ms2

Machine

uPID

uff

Xd qde

q

Figure 2: Linear single-axis control with feedfor-ward

+

++_Path P IDIKM(Xd)

s2

MachineM

uPID

uff

Xd qde

q

Figure 3: Linear single-axis control with feedfor-ward, practical version

Let us remind the reader of the conventionalmethod for a theoretical tuning [Khalil and Dom-bre, 2002]. Out of habit, the assumption of simpledynamics only with inertia forces associated witha single constant and diagonal inertia matrix M,and without centrifugal, Coriolis and gravity forcesis made:

=Mq (1)

The PID controller generates directly a torqueinput, uP ID :

upid= Kve+Kpe+Ki

edt (2)

The denominator of the closed loop transfer func-tion can be thus expressed as:

B(s) =

M s3 +Kvs2 +Kps+Ki

(3)

The classical tuning aims at obtaining a third-order negative real poles system:

B(s) =M(s+)3

(4)with chosen with respect to the mechanical res-onance frequency of the controlled machine. Bymatching each term of Equation 3 and Equation 4,the gain values are expressed as follows: Kv = 3M Kp = 3M 2

Ki = M 3

(5)

This tuning gives theoretical values which haveto be adapted in practice. Indeed, the integral gain

Kiis generally increased to compensate for the dry

frictions, while the derivative gain Kv is generallydecreased to cope with measurement noise.In most cases, the linear single-axis control is im-

proved with a feedforward term, uff. The generalformulation of this term is:

uff=Mqd (6)

In this case, the error signal behaviour is fixedby a third-order ordinary differential equation:

M e(3) +Kv e+Kpe+Kie= 0 (7)

where e(3) is the third derivative with respect totime.

The gain values are the same as Equation 5,which allows for a performant error behaviour. Letus notice that the M matrix is often used out ofthe feedforward and PID gain, like in Figure 3. Ityields to the following error signal behaviour:

M(e(3) +Kv e+Kpe+Kie) = 0 (8)

and a gain tuning independent from the machineinertia:

Kv = 3Kp = 32Ki =

3(9)

Furthermore, in an industrial controller, someadditional features are used, such as friction, grav-ity and backlash compensation, to improve accu-racy. Such a control strategy ensures sufficientlygood performances for serial kinematic machinetools to make it still widely used in the industry.Nevertheless, a machine tool is quite slow, veryheavy and stiff. Therefore, a single-axis linear con-trol ensures a efficient compensation of the smalldynamics and the behaviour of the stiff mechanicalstructure ensures a good static accuracy.

On the opposite, the single-axis control is knownto be weak with fast serial manipulators since itdoes not ensure a sufficient compensation of thenonlinear dynamics, leading to a poor dynamic ac-curacy [Khalil and Dombre, 2002]. Actually, re-ported experiments show that such a control cannot either ensure a good accuracy for parallel kine-matic machines [Brecher et al., 2006a, Denkenaand Holz, 2006, Vivas et al., 2003, Ouyang et al.,2002,Honegger et al., 2000]. Indeed, the presence

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+

++_

P ath P IDIKM(Xd) IDM(q, q,q)

s2

Machine

uPID

uff

Xd qd e q

q

q

Figure 4: Joint space Computed Torque Control

of the complex inverse kinematics in the path plan-ning is a first source of static accuracy lack, requir-ing adapted modeling and identification as statedabove. However, the complex dynamic behaviour

is also a very unfavourable phenomenon.As a matter of fact, the dynamic behaviour of

a parallel kinematic machine is strongly non lineardue to a dynamic coupling between legs, which doesnot exists in the serial case. Furthermore, most ofthe parallel kinematic machines have an anisotropicbehaviour. Therefore, the hypothesis of linear dy-namics is only verified at low speed and very lo-cally. Consequently, a linear single-axis control cannot be efficient in the whole workspace with thesame tuning, as established by Brecher [Brecheret al., 2006a]. A first solution is the determina-

tion of a restricted workspace with regard to max-imal accelerations, as initiated by Barrette [Bar-rette and Gosselin, 2005]. This method could beextended with the determination of a workspace,associated with a maximal speed and accelera-tion, where dynamics are fairly homogeneous witha low dynamic coupling. A second method is apath planning with dynamic consideration [Abdel-latif and Heiman, 2005, Oen and Wang, 2006]. Inaddition, the use of an adapted time interpola-tion can smooth the trajectory by limiting jerk orsnap (respectively 4th and 5th order time deriva-tive of the joint position) [Erkorkmaz and Altintas,2001, Fleisig and Spence, 2001, Lambrechts et al.,2005]. It can be noticed that a feedforward com-pensation in terms of jerk and snap can thus beused [Lambrechts et al., 2005]. Such methods aimto ease controller action to ensure the required ac-curacy. However, the limitations of such meth-ods are the decrease of the effective speed andworkspace leading to a low use of the machines ca-pabilities. Moreover, the real machine motion isnot completely mastered since the heavy computa-tion generally imposes an off-line path generation

without any on-line corrections of this path.To improve the real machine motion control,

the control gain tuning can be optimised withdynamic considerations, as proposed by Zhiyong

and Yang [Zhiyong and Huang, 2004, Yang andHuang, 2006]. Other solutions consist in controllaws modification (not always achievable on an in-dustrial controller). Some works deal with nonlin-ear gains [Ouyang et al., 2002], robust control [Kimet al., 2005,Fu and Mills, 2005] and inappreciablephenomena compensation [Brecher et al., 2006b].Nevertheless, these strategies are still based on asingle-axis control with various compensation in anexternal loop. There is no dynamics compensationin the control loop. Yet, a direct and simple way tocompensate for the dynamic behaviour is the wellknown computed torque control.

2.2 Computed Torque Control

The computed torque control is a widespread con-trol strategy for serial manipulators [Khalil andDombre, 2002, Luh et al., 1980]. Figure 4 re-minds of this control scheme. Let us remind thereader of how the classical computed torque con-trol works [Khalil and Dombre, 2002]. The controllaw is based on the Lagrange formulation of themachine inverse dynamic model:

=A(q)q+H(q, q) (10)

By replacing qin Equation 10 by an adapted con-trol signalu, an exact linearisation of the dynamicsis ensured. Indeed, there is only a double integra-tor between control signal and joint variables. Thefollowing control signal is used:

u= qd+Kve+Kde (11)

In this case, the error signal has a second orderbehaviour:

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+

++_

+

+P ath P IDIKM(Xd) IDM(X, X, X)

s2

Machine

F KM

D(X)

D(X)

D(X, X)IDM(q, q, q)

uPID

uff

Xd qd e

q

q

X

X

Figure 5: Joint space Computed Torque Control for parallel kinematic machine, explicit version

e+Kve+Kpe= 0 (12)

The gain tuning is, as it is well known, fixed bya cut-off frequency and a damping:

Kv = 2Kp =

2 (13)

The dampingis generally fixed between 0.9 and1 to avoid overshoot while yielding a good estab-lishing time. The cut-off frequency is fixed tothe highest value with respect to the mechanicalresonance frequency. It can be noticed that the in-

tegral gain is useless because the linearisation ofthe dynamics leads to a double integrator system.However, the integral gain is generally employedin practice. It allows for improving accuracy bycompensating for the lightunmodeledphenomena.Nevertheless, this control strategy can be improvedwith friction and backlash compensation, like thelinear single-axis control.

With such a control scheme, the nonlinear dy-namic behaviour of the machine is compensatedfor in the whole workspace. In this way, the lin-ear controller is associated with an exactly lin-earised system. Therefore, the controller perfor-mances are maximal and homogeneous in the wholeworkspace. Nevertheless, these great performancesare only achievable with a dynamic model reflectingperfectly the real machine behaviour. Indeed, thecomputed torque control does not cope very wellwith modeling errors [Khalil and Dombre, 2002].They create a perturbation on the error behaviourwhich may lead to a lack of stability and accuracy.Since a model almost never reflects exactly the realmachine behaviour, modeling errors are nearly un-avoidable. Consequently, a minimisation of these

modeling errors is required. In this way, dynamic

identification is generally performed [Swevers et al.,1997,Gautier and Poignet, 2001, Olsen and Peter-son, 2001]. Alternately, a more complex model canbe used. Aflexible body dynamic model [Kock andShimacher, 2000b], instead of a rigid body one, al-lows for taking into account deformations, increas-ing model accuracy while increasing on-line com-putation. A model taking into account task in-fluence [Oen and Wang, 2006], instead of a modelof the stand alone mechanical structure, can copewith external torques applied on the end-effector,which are specific for the application (cutting, loadscarrying. . . ). Furthermore, if the influence of themodeling errors is still interfering, robust controltechnique can be employed [Vivas et al., 2003, Leeet al., 2003, Honegger et al., 2000]. Actually, ro-bust techniques are used here for compensating forthe phenomena which can not be modeled, as theywere originally designed for, and not for compen-sating for insufficient modelling, as it is too oftenseen.

2.3 Discussion

The control strategies exposed above are performedin the joint space. Practically, the actuators en-coders are generally the only available measure-ment mean. Theoretically, a serial kinematic ma-chine is completely defined by its joint configura-tion, in terms of kinematics and dynamics [Khaliland Dombre, 2002]. The joint configuration reflectsthus the state of the machine. Consequently, thejoint space control is a state feedback control. Asit is generally admitted, a state feedback control al-lows for ensuring the best accuracy. Therefore, thejoint space control is relevant for serial kinematic

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machine, provided that solving for the inverse kine-

matic problem is accurate enough to translate thedesired Cartesian path into the correct joint refer-ence path.

On the opposite, a parallel kinematic machineis completely defined by its end-effector pose, ex-cept in some rare cases (3RRR for example [Chab-lat and Wenger, 1998]). Actually, this is gen-erally admitted for the kinematics [Waldron andHunt, 1991,Merlet, 2002,Dallej et al., 2006] and itis being extended to the dynamics [Dasgupta andChoudhury, 1999,Khalil and Ibrahim, 2004, Calle-gari et al., 2006]. The end-effector pose can thus beconsidered as the state of a parallel kinematic ma-

chine [Dallej et al., 2006]. Therefore, a joint spacecontrol is not a state feedback control but a biasedobserver feedback control. Consequently, the bestperformances in term of accuracy can not be en-sured which such a control.

Moreover, the instantaneous kinematics and dy-namics depend on the end-effector pose as statedabove. Consequently, a joint space model-basedcontrol, such as the computed torque control,should include the forward kinematic model. Toillustrate this point, we propose an explicit formof the computed torque control in the joint space

which includes these forward transformations (seeFigure 5). In general, the forward kinematics of aparallel kinematic machine do not have a closed-form expression contrary to a serial one. A jointconfiguration can thus lead to several end-effectorposes (namely up to 40 for the Gough-Stewart plat-form [Merlet, 1990, Husty, 1994]). Some solutionscan be removed since they are complex or mechan-ically inadmissible, but the end-effector pose cannot be estimated only from the active joint config-uration with reliability. Indeed, the forward kine-matic problem is a square model since it has ex-actly the same amount of equations and unknowns.Hence, it is sensitive to any measurement noise, noteven to mention the kinematic model and calibra-tion errors. In addition, the on-line computationof the end-effector pose leads to a lack of speed,accuracy and stability. Consequently, the perfor-mances of the control are limited. Furthermore, theimplicit presence of on-line numerical transforma-tions leads, in pratice, to model simplifications thusincreasing modeling errors. As reminded above,the computed torque control has a weak robust-ness with regards to the modeling errors. Thus,

the joint space computed torque control is often

unusable alone. Some solutions are reported suchas simplified dynamics and robust control [Vivaset al., 2003, Lee et al., 2003] or nonlinear feedfor-ward compensation with robust control [Honeggeret al., 2000]. Nevertheless, the mastery of robustcontrol and the perfectible accuracy make the jointspace computed torque control not welcome in anindustrial context.

As a conclusion, the joint space control seems tobe inherently imperfect and unadapted for paral-lel kinematic machines. Since the latter are com-pletely defined by their end-effector pose, improve-ments could be found in the Cartesian space (SE3)

control.

3 Control of the parallel

kinematic machines in the

Cartesian space

To our knowledge, the control of a parallel kine-matic machine in the Cartesian space, or the taskspace, is often mentioned in the literature [Leeet al., 2003, Kock and Shimacher, 2000a, Mar-quet et al., 2001, Beji et al., 1998, Yamane et al.,1998,Callegari et al., 2006,Caccavale et al., 2003].Nevertheless, only few experiments are performed.When it is used, many model simplifications aredone decreasing accuracy and stability [Lee et al.,2003]. The puprose of this section, which the essen-tial theoretical contribution of this paper, is there-fore to propose a complete revisiting of the Carte-sian space control strategies. It is based on theserial-parallel duality [Waldron and Hunt, 1991]and thus requires the following assumption:

Assumption 1 The end-effector pose can be mea-

sured accurately at control frequency.

This assumption will be discussed in Section 3.3.

3.1 Equivalent of single-axis linear

control in the Cartesian space

3.1.1 With the end-effector dynamics only

The Cartesian space equivalent of the linear single-axis control given by Figure 6 is generally used [Cal-legari et al., 2006]. However, it is shown that the

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+

++_

P ath P ID DT(X)I

s2

MachineuPID

uff

Xd e X

Figure 6: Simple PID control in the Cartesianspace, first version

transposition from joint to Cartesian space is notcompletely straightforward. In the control schemeof Figure 6, the simplified dynamicsisexpressed as:

= DT(X)IX (14)

where only the inertia of the end-effectorIis takeninto account. The latter is mapped into the activejoint space with the forward instantaneous kine-matic matrix.

The feedforward compensation term can be thenexpressed as:

uff = Xd (15)

In this case, the error signal behaviour is fixedby the following equation:

DT(X)I(e(3) +Kv e+Kpe+Kie) = 0 (16)

Consequently, the following tuning should beused: Kv = 3Kp = 32

Ki = 3

(17)

Consequently, a similar running between single-axis linear control and this Cartesian space controlstrategy is retrieved here, with similar dynamics,feedforward compensation term and PID tuning.The only difference is the presence of the trans-posed forward instantaneous kinematic matrix inthe control loop. However, we can make some re-marks here. First, the presence of a numerically es-timated model in the control loop can lead to a lackof stability and accuracy, and increases the com-plexity of the control scheme. Second, the dynam-ics in Equation 14 only concerns the end-effectorinertia and the legs inertia is neglected. However,this assumption seems to be too restrictive, par-ticularly in the machine-tool case where legs are

generally heavier than the end-effector. The com-

pensation of the machine dynamic behaviour mighthence not be efficiently achieved. Thus, the accu-racy of this control strategy is questionable. Let ussee wether it is more relevant to take into accountthe legssimplifieddynamics.

3.1.2 With the leg dynamics only

The simplified inverse dynamics in Equation 1 isreused here. The joint acceleration qare expressedas a function of the end-effector pose with the sec-ond order inverse instantaneous kinematics:

=M

Dinv(X)X+ Dinv(X, X) X

(18)

The feedforward term is also expressed in as afunction of the end-effector pose:

uff =M

Dinv(X)Xd+ Dinv(X, X) Xd

(19)


M Dinv(X)e(3) + (Kv+MDinv(X, X))e

+Kpe+Kie= 0 (20)

Consequently, the following tuning has to beused:

Kv = 3M Dinv(X) MDinv(X, X)

Kp = 3M Dinv(X)2

Ki = M Dinv(X)3

(21)

In a first approach, with this formulation, the

tuning is not constant and thus difficult to setup in an industrial context. Nevertheless, we canrearrange each term to propose a lighter controlscheme, as illustrated by Figure 7. In this case, thefeedforward term is:

uf f = Xd (22)

A compensation term is added. It is expressedas:

ucomp= MDinv(X, X) X (23)

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+

++_

+

+P ath P ID MDinv(X)

MDinv(X, X)

s2

Machine

uPID

ucompuff

Xd e

X

X

Figure 7: Simple PID control in the Cartesian space, second version


M Dinv(X)(e(3) +Kv e+Kpe+Kie) = 0 (24)

Consequently, the gain tuning becomes: Kv = 3Kp = 32Ki =

3(25)

This proposed control strategy is the directCartesian space equivalent of the single-axis controlin the joint space. Yet, it can be noticed that thetransposition between Cartesian and joint space is

not as straightforward as it could have seemed atfirst glance. The control scheme complexity hasclearly increased whereas the dynamics compensa-tion issues listed previously are still present sincethe same simplified dynamics is used. Moreover,the end-effector dynamics isneglected.

3.1.3 With the legs and end-effector dy-

namics

Now, the two simplified dynamics used above canbe grouped together to take into account both legs

and end-effector inertia (Figure 8). Thus, a moreelaborated expression of the simplified dynamicscan be:

=Mq+DTIX (26)

The control law is nearly the same as the onepresented above (compare Figure 7 and Figure 8).The tuning is the same as Equation 25. Neverthe-less, a more efficient compensation of the machinedynamic behaviour can be performed here withcomparison to the the two previous cases ( 3.1.1

and 3.1.2) since more phenomena are taken intoaccount. However, the numerical issues of the for-

ward instantaneous kinematic matrix are retrievedhere and thus impose some care.

Let us remark that this formulation is usedby Marquet and Vivas directly in a computedtorque control [Marquet et al., 2001, Vivas et al.,2003]. Nevertheless, since these simplified dynam-ics presents inevitably important modeling errors, apredictive control, asking for heavy computation, isthus employed to ensure good performances at highspeed. Actually, this approach stands on the bor-der between simple control and computed torquecontrol: the control is almost considered as a single-axis control when a simple PID controller is used,

and as a computed torque control when a morecomplex controller is used.

Consequently, the interest of change from thejoint to Cartesian space could be questionable, inthe simple PID control case. Nevertheless, we showin Section 3.3 some mechanical advantages for theCartesian space control. Now, instead of using sim-plified dynamics and time consuming complex con-troller, the use of the complete dynamics in a com-puted torque control can allow for using simplercontroller with lighter computational burden, whileensuring equivalent or better performances.

3.2 Computed Torque Control in the

Cartesian Space

The Cartesian space Computed Torque Control iswell known for serial kinematic machines [Khaliland Dombre, 2002]. The presence of the numerical

inverse instantaneous kinematic matrix Dinv (seeFigure 9) make this control strategy rarely usedfor serial kinematic machines. Indeed, the forwardinstantaneous kinematic matrix of a serial kine-

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+

++_

+

+P ath P ID MDinv(X)

+ DT(X)I

MDinv(X, X)

s2

Machine

uPID

ucompuff

Xd e

X

X

Figure 8: Simple PID control in the Cartesian space, third version

+

++_

+

_

P ath P ID IDM (q, q,q)

s2

MachineDinv(q)

D(q, q)

F KM(q)

uPID

uff

Xd e

q

q

qq

Figure 9: Cartesian space Computed Torque Control for serial kinematic machines

+

++_

P ath P ID IDM (X, X, X)

s2

Machine

uPID

uff

Xd e X

X

X

Figure 10: Cartesian space Computed Torque Control for parallel kinematic machines

matic machine is generally composed of trigono-metric functions, thus making the numerical inver-sion all the more difficult because of the existenceof numerous singularities and nonlineardependence

upon noise.On the opposite, in the parallel kinematic case,

this control scheme is perfectly relevant when it en-closes an inverse dynamic model depending on theend-effector pose and time derivatives [Caccavaleet al., 2003,Callegari et al., 2006]. Indeed, there is aminimal use of numerical transformations when theend-effector pose and speed are measured (see Fig-ure 10). Actually, the only used numerical trans-formation is the transposed forward instantaneouskinematics matrix used to map the Cartesian space

dynamics into the active joint space [Dasgupta andChoudhury, 1999,Khalil and Ibrahim, 2004, Calle-gari et al., 2006]. Since inverting this matrix con-sists only of a numerical inversion of a quite sim-

ple matrix, the computational burden is less im-portant than for solving for the forward kinematicsproblem. Moreover, the Cartesian space ComputedTorque Control for parallel kinematic machines isdual with the joint space Computed Torque Con-trol for the serial kinematic machines (see Figure 4and Figure 10). Consequently, the behaviour ofthe joint space computed torque control describedabove is retrieved here, namely the error behaviourin Equation 12 and the tuning in Equation 13,which we recall here:

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e+Kve+Kpe= 0 (27)and

Kv = 2Kp =

2 (28)

Consequently, the known performances of thecomputed torque control could be thus expected,with the prerequisite of a good dynamic modeling,a good dynamic identification and a good algorithmfor the remaining numerical transformation.

3.3 Discussion

3.3.1 Joint space or Cartesian space control?

The Cartesian space control is particularly rele-vant for parallel kinematic machines. Theoretically,since the end-effector pose is the state of a paral-lel kinematic machine, the Cartesian space controlensures a state feedback control leading to a bet-ter accuracy than a joint space control which is nota state feedback control any more. Moreover, byusing a Cartesian space inverse dynamic model ina Cartesian space computed torque control, a min-imal use of numerical transformations is required

leading to a fast, stable and accurate control, whena good model is used and a good dynamic iden-tification is performed. Furthermore, some addi-tional advantages can be noticed when a fast andaccurate end-effector pose measure is available (As-sumption 1).

First of all, in a joint space control, the regu-lated error is the error between a transformed de-sired trajectory, thus biased by the modeling errors,and a measure not reflecting the real end-effectorpose, insensitive to backlashes or deformation. Onthe opposite, in the Cartesian space control case,the regulated error is the error between the mea-sured and desired end-effector trajectories. Conse-quently, a Cartesian space control ensures a directtask control and thus can be more accurate than ajoint space one.

Secondly, since the inverse kinematic model isnot used to compute the joint reference path (seeFigure 5 and Figure 10), the constraints on kine-matic identification could be released. Indeed,without any kinematic identification, the Cartesiancontrol performs an accurate positioning of the end-effector, when a point to point task is desired, since

Disturbance shifts the end-effector pose

Xd

Cartesian space control= correct tracking

joint space control= tracking is lost

qd

q

Cartesian space

joint space

Figure 11: Cartesian space ensures correct end-effector reference tracking contrary to joint spacecontrol

the reference trajectory is not biased by the inversekinematic model errors. Furthermore, as far as thetrajectory tracking is concerned, the dynamic iden-tification prevails against the kinematic one in theminimisation of the dynamic modeling errors. Inaddition, a dynamic identification, which is linear,is easier to set up than a kinematic one, which isnonlinear.

Thirdly, a Cartesian space control is more in-teresting in the neighbourhood of singularities.Indeed, one joint variable configuration leads toseveral end-effector poses [Husty, 1994]. In theworst cases, a disturbance on joint trajectory canthus shift the end-effector pose without changingjoint configuration. This can happen especially inthe neighborhood of singularities (assembling modechanging trajectory [Chablat and Wenger, 1998]) orin cups points (non-singular posture changing tra-jectory [Zein et al., 2006]). This change of the end-effector pose is not observed by a joint space con-

trol whereas a Cartesian space one is able to do so(see Figure 11). Consequently, the Cartesian spacecontrol tries to bring back the end-effector pose toits reference or fails when the singularity can notbe crossed again. On the contrary, a convergingjoint space control can not tell whether the Carte-sian reference tracking fails or not. Consequently, aCartesian space control can ensure a more reliabletracking than a joint space control.

Last but not least, even on planned path takinginto account kinematic and dynamic constraints,

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X= [Xe Ye Ze ]T Cartesian variables de-

scribing the end-effectorpose X

X0, Y0, Z0, Z Constant parameters de-scribing the reference po-sition of each leg in thefixed basis

q1i = qi, q2i, q3i Joint variables describingthe leg i

qjiki = qji +qki Joint variables sumsji = sin qji , cji =cos qji

Trigonometric operationson joint variables

=[0 0]T Angular velocity of theend-effector

L Length of the end-effectord3, d4 Length of arm and forarm

of a legM R1 Mass of a legM XR2, M Y G2 First moments of the

arm of each leg aroundX (grouped with otherterms) and Y axis

M XG3, M Y G3 First moments of theforarm of each leg aroundX and Y axis

ZZ2, Z Z R3 Inertia term of the forarmand the arm (groupedwith other terms) of eachleg around Z axis

Mcomp3, Mcomp4 Equivalent mass of theforces applied by thecompensator mounted onthe vertical legs

MP Mass of the end-effectorMt =M R1+MP Mass of the end-effector

and a legM SP =[MPXP MPYP MPZP]

Vector of the first mo-ments of the end-effectoraround the fixed basisframe

IP Inertia matrix of the end-effector

Y YP Inertia term of the end-effector around Z-axis

g Acceleration of gravity

Table 2: Notation for the modeling of the Isoglide-4 T3R1

the joint position errors are independent from eachother when using a joint space control. Therefore,the kinematic constraints can not be ensured andtwo types of defects may appear: uncontrolled par-asite end-effector moves or internal torques if thesemoves are impossible, thus degrading passive joints.Like two-arm robot control [Dauchez et al., 1989],Cartesian space control can minimize, or cancel inthe best cases, internal torques [Marquet et al.,2001]. Indeed, the regulated errors, which are end-effector pose errors, are naturally compatible withthe end-effector moves.

Consequently, the theoretical advantages of theCartesian space control over the joint space one arenow undoubtful. Therefore, the Cartesian spacecontrol seems perfectly relevant for the parallelkinematic machines and should always be used.However, the discussion made above assumes tohave an available fast and accurate observation ofthe end-effector pose. This point remains the mainissue making the Cartesian space control use occa-

sional. Indeed, the measure of the end-effector poseis not an easy deal.

3.3.2 Comments on Assumption 1

In the literature, the observation is generally indi-rect: the end-effector pose is estimated throughoutthe forward kinematics problem solving [Lee et al.,2003, Kock and Shimacher, 2000a, Marquet et al.,2001, Beji et al., 1998, Yamane et al., 1998, Cal-legari et al., 2006, Caccavale et al., 2003]. Thus,the numerical estimation issues, such as compu-tation time, stability, reliability and accuracy, areretrieved here. In such a case, the property of astable and accurate control is called into questionand should be investigated. Nevertheless, adaptedalgorithms [Merlet, 2004] or metrological redun-dancy [Baron and Angeles, 2000, Marquet et al.,2002] can decrease the forward kinematics complex-ity and computation. Thus, it can improve the ac-curacy and stability of the control. However, the

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use of a kinematic model imposes a heavy modeling

and an accurate kinematic identification since themeasure is biased by the kinematic errors.On the opposite, instead of using a mechanical

model, a direct measure can be used. To our knowl-edge, the means to measure the end-effector poseare few and far between. The laser-tracker andcomputer vision are the main. On the one hand,the laser tracker allows for a very accurate and fastCartesian position measure (20m and 3kH z [Far,]). Nevertheless, it is very expensive and hard touse. In addition, the orientation measure is notmastered. To our knowledge, it is only used forkinematic identification [Newman et al., 2000] and

was never used in the control loop.On the other hand, computer vision is no as accu-

rate and fast but is very easy to implement in a con-trol scheme. It is a well known solution for the kine-matic control of serial kinematic machine, namelyvisual servoing [Weiss et al., 1987, Espiau et al.,1992, Hutchinson et al., 1996]. Recent works dealwith visual servoing of parallel kinematic machineand show good properties [Kino et al., 1999,Dallejet al., 2006]. However, only kinematic control isconcerned. On the opposite, Ginhoux and Gan-gloff proposed a fast visual servoing of serial kine-

matic machine [Ginhoux et al., 2004]. However,the dynamics are compensated for with a robustcontroller and not with a computed torque con-trol. To our mind, the application of such a con-trol to parallel kinematic machines is not relevantaccording to what we stated above. A more rel-evant solution could be a visual computed torquecontrol as initiated by Fakhry for serial kinematicmachine [Fakhry and Wilson, 1996]. To our knowl-edge, there is no work on the fast visual servo-ing of parallel kinematic machine whereas goodperformances could be expected [Ait-Aider et al.,2006,Paccot et al., 2006].

To conclude, the Cartesian space control of par-allel kinematic machines seems to be a relevant so-lution improving accuracy, stability, speed, reliabil-ity and mechanical behaviour. Let us now validateexperimentally the theoretical discussion above.

4 Modeling of the test-bed

The notation used is this section is described inTable 2.

Figure 12: Global view of the Isoglide-4 T3R1

End-effector

Fixed

basis

Leg

P

P

P

P

RR

RR

RR

RR

U

U

U

U

Figure 13: Kinematic scheme of the Isoglide-4T3R1

q1i

q2i

q3i

d3d3

d4

Link between leg and end-effector

Figure 14: Detail of one leg

4.1 Presentation of the test-bed

The test-bed is the Isoglide-4 T3R1 (see Figure 12and [Gogu, 2004]). This parallel kinematic machine

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is a fully-isotropic one with decoupled motion. It is

a four degrees of freedom machine with three trans-lations and one rotation. This machine is composedof four identical legs. Each leg contains one actu-ated prismatic joint and two passive revolute joint,linked to the end-effector by one universal joint (seeFigure 13 and Figure 14). The actuation is per-formed with linear motors.

This machine is designed for high speed machin-ing. Hence, the structure weight is important tomeet the stiffness requirements: 31kg for each legand 14kg for the end-effector. Therefore, the legdynamics have a great influence and can thus not beneglected, contrary to common light parallel kine-

matic machines for pick-and-place. Consequently,a complete dynamic modeling and a performant dy-namic control is required to ensure the high accu-racy required in machining. This test-bed is thusrelevant for validating the assumption made on theweakness of single-axis linear control.

The main advantage of the Isoglide-4 T3R1, asfar as control is concerned, is to have a closed-formexpression of the forward kinematic and instanta-neous kinematic models:

Xe = q1 X0

Ye = q2

Y0Ze = q3 Z0sin = q4q3+Z

L

(29)

and

D(X) =

1 0 0 00 1 0 00 0 1 00 0 1

Lcos1

Lcos

(30)Therefore, the simple and closed-form expres-

sion of the forward kinematics are interesting forvalidating the proposed Cartesian space controlschemes. Indeed, it removes from the numeri-cal estimation issues and the lack of fast and ac-curate end-effector pose measure. Therefore, amore fairly comparison between forward kinematicmodel based control and exteroceptive measurebased control could be achieved. Actually, the nu-merical estimations issues influence on control be-haviour, which are hard to quantify, are removed.The comparison can only be achieved in term ofsensor and identification accuracy with regards tothe control accuracy.

4.2 Dynamic modeling

Achieving a performant computed torque con-trol requires a Cartesian space dynamic modelingmethod with an easy implementation, a low com-putation cost and minimal simplifications. In thisway, Khalils method [Khalil and Ibrahim, 2004] ispreferred on other known methods [Dasgupta andChoudhury, 1999,Tsai, 2000,Callegari et al., 2006].Indeed, it is based on the Newton-Euler algorithm,known to be relevant in a control context. More-over, the application of the method is easy due avery simple formulation.

According to Khalil, the inverse dynamic model

can be simply expressed as [Khalil and Ibrahim,2004]:

=DTFp+ ni=1

JTpiDTi (Hi+ Gi)

+ f (31)

where:

Fp are the dynamics of the end-effector

n is the number of legs

Di is the inverse instantaneous kinematic ma-

trix of leg i Jpi is a Jacobian matrix linking the Cartesian

coordinates of the end of the legi to the Carte-sian coordinates of the end-effector.

Hi are the dynamics of the leg i.

Gi is the gravity vector of the leg i.

This modeling method is thus achieved trough acomplete modeling of each leg and a determinationof the end-effector dynamics.

4.2.1 Modeling of each leg

A leg can be seen as a stand-alone 3-PRR serialkinematic machine. The modeling of such a serialkinematic machine is well known. Consequently,we only give the obtained models without detailson the method. The kinematics are determinedwith Khalil-Kleinfinger notation and the dynamicswith the Newton-Euler algorithm and the notationin [Khalil and Dombre, 2002].

The inverse instantaneous kinematic matrix ofleg 1 expresses as:

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D1= 1 0 00 s2131

d3s31

c2131d3s31

0 d4s2131+d3s21d4d3s31

d4s2131+d3s21d4d3s31

(32)Each term of the inverse dynamics of the first

leg, H1= [H11 H12 H13]T are detailled below:

H11= M R1q11 (33)

H12= (ZZR2 +2d3MXG3c312d3MYG3s31)q21+ (ZZ3 +d3MXG3c31d3MYG3s31)q31

+ (d3MXG3s31+d3MYG3c31) q221

(d3MXG3s31+d3MYG3c31) q22131

(34)

H13= (ZZ3 +d3MXG3c31d3MYG3s31)q21+ (ZZ3) q31+ (d3MXG3s31+d3MYG3c31) q

221

(35)The passive joint variables are expressed as func-

tion of the end-effector pose with simple trigono-metric relations. Other legs have similar models.The changecomesfrom the position of the legs in

the Cartesian space, modifying the gravity termsand Jacobian matrix organisation. The gravity vec-tor, Gi, for each leg, are detailled below:

G1 =

0

(MXR2s21+MYG2c21+MXG3s2131+MYG3c2131)g(MXG3s2131+MYG3c2131)g

(36)

G2 =

0

(MXR2c22+MYG2s22+MXG3c2232+MYG3s2232)g(MXG3c2232+MYG3s2232)g

(37)

G3 =

(MR1+Mcomp3)g

00

(38)

G4= (MR1+Mcomp4)g00

(39)Two gravity compensators are used for the ver-

tical legs, namely the third and fourth, explainingthe presence of termsMcomp3and Mcomp4. Theirinfluence is approximated as constant.

The Jacobian matrices Jpi have a very simpleexpression. Indeed, the end of each leg is definedby its Cartesian position. The end-effector is de-fined by four Cartesian variables [Xe Ye Ze ]

T.For the first and third legs, there is only an offset

between the Cartesian position of the legs and the

end-effector and the orientation has no influence.For the two other legs, the orientation has aninfluence which is easily determined with trigono-metric relations. Consequently, the matrices havethe following expressions:

Jp1= Jp3 =

1 0 0 00 1 0 00 0 1 0

(40)Jp2= Jp4 =

1 0 0 Ls0 1 0 00 0 1 Lc

(41)

4.2.2 Dynamics of the end-effector

The dynamics of the end-effector are determinedwith the Newton-Euler equation [Khalil and Dom-bre, 2002,Khalil and Ibrahim, 2004]:

Fp = PX+

( M Sp)

MpI3M Sp

g (42)

where P is the inertia tensor of the end-effectorexpressed as:

P = MPI3 S(M SP)S(M SP) IP

(43)Only the terms along the end-effector degrees of

freedom are retained which yields the final expres-sion of the end-effector dynamics in Equation 44.

4.2.3 Inverse dynamic model of the

Isoglide-4 T3R1

The inverse dynamic model is obtained by theEquation 31. For conciseness sake, the global ex-pression is not mentioned here. Nevertheless, theobtained model has a closed-form expression allow-ing for an interpretation of each term. The mainterm is the inertia of the legs and the end-effectorin translation, this term thus has a great influenceon the tracking performances. Then the other mainterms concern the coupling between legs due to theheavy inertia. Even if a kinematic decoupling is en-sured, the dynamic coupling stays present and cannot be compensated for with a linear control. Thelast term concerns the rotation inertia of the end-effector, this term is not preponderant and can beneglected if necessary.

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FP = MPXe+ (MPXPs+ MPZPc)+ (MPXPc+ MPZPs)

2

MPYeMP(Ze g) + (MPXPc+ MPZPs) (MPXPs+ MPZPc)

2

Y YP+ (MPXPs+ MPZPc)Xe+ (MPXPc+MPZPs)(Ze g)

(44)

+

+

F DM R R

X X q

IK M

Machine

10%

error

ondyna

mic

param

eters

50m

error

ongeom

etric

al

param

eters

1m

accura

cy

onjoint

sens

ors

Figure 15: Dynamic model of the machine used insimulation

A simple friction forces model is implemented tocompensate for the latter and improve accuracy:

f = (Fvq+Fs)sign(q) (45)

5 Results

5.1 Simulation

First, the improvement of the joint space computedtorque control (Figure 5) over the single-axis (Fig-ure 2) control will be shown. A comparison will beachieved in term of straightness error and trackingerror on a relevant trajectory. Second, a compar-ison between computed torque control in the jointspace and in the Cartesian space (Figure 5 and Fig-ure 10) will be performed. The Cartesian spacecomputed torque control will be performed withthe forward kinematics and with a direct measureto emphasis the improvement when using the directmeasure rather than the forward kinematics.

The machine behaviour is simulated with the for-ward dynamic model obtaining by inverting Equa-tion 31 which allows for computing the end-effectoracceleration (see Figure 15). Realistic noises anderrors are used, such as a 10% error on dynamic pa-rameters, 50maccuracy for geometric parameters(required manufacturing and assembly tolerance forthe Isoglide-4 T3R1). Let us stress out that neither

Figure 16: Reference trajectory in Cartesian space

deformations nor assembly errors (such as non per-pendicular axis) are simulated. A 1m accuracy is

fixed for the joint sensors, and a 20mand 104

radaccuracy for the direct measure (laser tracker per-formances). The tuning is done with = 5Hz andthe control and sensors have a 1kH zsampling rate.

The reference trajectory, Figure 16, is composedof one translation along the X axis, one transla-tion along X, Y and Z axes and one translationalong the three axes with a rotation. The firstpart of the trajectory allows for pointing out theability of the control strategy to compensate forthe dynamic coupling between legs. The secondpart allows for comparing the compensation of theinertia forces. And the last part allows for com-paring joint space control and Cartesian space oneperformed with forward kinematics and direct end-effector pose measure. Indeed, there is only a dif-ference, between forward and inverse kinematics,on the rotation since the Isoglide-4 T3R1 has de-coupled translations (see Equation 29).

A fifth degree polynomial point to point inter-polation is used to have a smooth trajectory. Themaximal acceleration is fixed to 3m.s2 to simulatea machining operation. This low speed is far fromhigh-speed pick-and-place one.

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(a) XY trajectory

(b) Sharp corner crossing

Figure 17: Comparison, in the XYplane, betweensingle-axis linear control and the joint space com-puted torque control

5.1.1 Joint space computed torque control

versus single axis control

Figure 17 shows the trajectory in the XY planeperformed by the two control strategies, single-axis

Single-axis Joint space

CTC

First seg-ment

0.759mm 0.026mm

Secondsegment

1.900mm 0.089mm

Thirdsegment

3.758mm 0.180mm

Table 3: Straightness error on each segment mea-sured in simulation

Figure 18: Tracking error on X axis for the single-axis linear control and joint space computed torquecontrol, and bias between reference and performedtrajectory

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control and the computed torque control in the

joint space, and the reference. On Figure 17(a), thetwo control strategies are biased with regards to thereference trajectory. Figure 17(b) shows the detailsof a sharp corner crossing, with zero speed, and re-veals a difference between single-axis and computedtorque control. Indeed, the trajectory followed withthe single-axis control is not completely straightand presents some oscillations in the corner, con-trary to the joint space computed torque control.This is numerically verified in Table 3. The com-puted torque control allows for straightness errorsabout twenty times less important than single-axiscontrol does. Figure 18 shows the tracking error

on X axis for the same two control strategies. Thesingle-axis control presents important tracking er-rors with a maximum along the four axes displace-ment (2.5mm peak to peak). On the opposite, thecomputed torque control allows for small trackingerrors. These errors are distributed around the con-stant bias, 318m, between the performed trajec-tory and the reference trajectory.

Thus, using a computed torque control instead ofsimple control improves dynamic accuracy of themachine. Indeed, the high dynamic coupling be-tween legs, due to important masses, is clearly not

neglectable, even at machining speed often consid-ered as quasi-static. Therefore, the complete ma-chine dynamics should be taken into account inthe control loop. Moreover, the strong influence ofthe kinematic identification is retrieved here. Ac-tually, there is a bias between the Cartesian ref-erence and the joint reference, due to remainingkinematic errors. In the Isoglide-4 T3R1 case, thisbias is constant and can be seen as an adjustableoffset. However, in more complex cases, this biasis not constant along the workspace and can thusnot be compensated for without a more accuratekinematic identification asking for more accuratemeasure and heavier computation.

5.1.2 Joint space computed torque versus

Cartesian one

Figure 19 show the performed trajectory in the XYplaneby the joint space computed torque control,the Cartesian space computed torque control withforward kinematics and the Cartesian space com-puted torque control with direct end-effector posemeasure. The first two control strategies have ex-

Figure 19: Comparison, in the XY plane, be-tween joint and Cartesian space computed torquecontrol, with forward kinematics and direct end-effector pose measure

actly the same behaviour and bias with regards

to the reference trajectory. On the opposite, thetrajectory performed by the computed torque con-trol with direct end-effector pose measure does notpresent a bias and is mixed with the reference tra-jectory. Figure 20 shows the orientation trackingerror for the computed torque control in the jointand the Cartesian space. Joint space and Cartesianspace, with forwards kinematics, computed torquecontrol have exactly the same behaviour. The com-puted torque control with direct end-effector posemeasure allows for better accuracy than the twoother control strategies, since there is no bias andthere are better tracking performances during therotation at the end of the trajectory, between times1.3 and 1.8 seconds. The tracking performances arenumerically summarized in Table 4 and Table 5.The joint space computed torque and the Cartesianspace computed torque control with forward kine-matics have strictly the same performances. Onthe opposite, the use of a direct end-effector mea-sure allows for better mean errors (13m versus322m) and similar standard deviation for X-axisand straighness errors. Concerning the orientation,the direct measure allows for a tracking error thirty

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Figure 20: Comparison, on orientation error track-ing, between joint space and Cartesian space con-trol, with forward kinematics and direct measure ofthe end-effector pose

jointspaceCTC

CTCwithFKM

CTCwithdirectmeasure

First seg-ment

0.026mm 0.026mm 0.040mm

Secondsegment

0.089mm 0.089mm 0.083mm

Thirdsegment

0.180mm 0.180mm 0.147mm

Table 4: Straightness error on each segment mea-sured in simulation

times less important than the forward kinematicsdoes.

The use of the Cartesian space control witha direct measure of the end-effector pose insteadof a joint space control does improves static anddynamic accuracy, whereas a forward kinematicsbased control only allows for good dynamic perfor-mances but reduced geometric accuracy. Indeed,the use of a direct measure allows for compensatingfor the kinematic errors without extremely accuratekinematic identification concern. The performedtrajectory is not shifted with respect to the refer-ence one. Furthermore, it needs to be underlinedthat even though the direct measure is less accu-rate than the joint sensors, using it for control en-

Control

Mean of

trackingerror onX

Standarddevia-

tion oftrackingerror onX

Mean of

trackingerror on

Standarddevia-

tion oftrackingerror on

JointspaceCTC

0.322mm 0.024mm 2 0.079

CTCwithFKM

0.322mm 0.024mm 2 0.079

CTCwithdirectmeasure

0.013mm 0.026mm 0 0.006

Table 5: Tracking errors along the trajectory

sures equivalent translation tracking performancesand better orientation ones. Actually, these perfor-mances are closed to the direct measureaccuracy.In this case, the sensor accuracy has thus more in-fluence on the control accuracy than the modelingerrors. Finally, the numerical results shows that a100m accuracy, which is the minimum requiredin machining, can be achieved with the computedtorque control without particular care. Neverthe-less, this can be still improved with less error onparameters (in other words a better identification),a more elaborated gain tuning and a more accurate

sensor.

5.2 Experiments

5.2.1 Dynamic identification

In order to fit the inverse dynamic model to the realdynamics of the machine and ensure the best per-formances for computed torque control, dynamicidentification was realized (see Table 6). Themethod used here was proposed by Guegan [Gue-gan et al., 2003]. The chosen exciting trajectoryis composed of axis-by-axis displacements with ac-celeration ranging from 0.5 to 3 m.s2. A simu-lation on different trajectories (circles, axis-by-axisdisplacements, coupled axis displacement and ran-dom trajectory) shows that the axis-by-axis dis-placements obtained the best condition number.Indeed, it allows for having both free and con-strained moves on each axis. The parameters con-cerning the dynamic coupling are thus determinedwith the torques recorded during free moves. Thoseconcerning the inertia terms are determined duringconstrained moves.

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Parameter CAD

values

Identi-fiedvalues

Units (%)

M XR3 3.235 5.054 kg.m 0.42M Y G3 0 0 kg.mM XR2 7.971 0 kg.mM Y G2 0 0 kg.mZZ3 1.787 2.443 kg.m

2 1.29ZZ R2 6.429 8.420 kg.m

2 0.54Mt 45.011 39.513 kg 0.62MR1 31.4380 39.999 kg 0.40MPXP 2.059 0 kg.mMPZP 0 0 kg.mY YP 0.411 0 kg.mMcomp3 45.011 49.180 kg 0.50Mcomp4 31.4380 41.005 kg 0.39F s1 10.907 N 2.76F s2 25.558 N 1.25F s3 21.044 N 1.71F s4 28.980 N 1.07F v1 36.108 N.s.m

1 3.81F v2 89.419 N.s.m

1 2.45F v3 35.211 N.s.m

1 6.34F v4 64.793 N.s.m

1 3.10

Observation matrix condition number: 355.56Number of samples: 65404

Table 6: Dynamic identification results

Results lead to an observation matrix condition

number of 355.56 which is relatively good. Inertiaparameters (M XR3, ZZ R3, ZZ R2, Mt, MR1) areidentified with a standard deviation from 0.40% to1.29%, friction terms (F si and F vi) from 1.07% to6.34%. Let us remark that some parameters de-scribing the end-effector can not be identified be-cause the end-effector is lighter than the legs, thushaving a little influence on dynamics. Anyhow, thegood results of the identification process allows forthe fulfillement of the small modeling errors con-dition, necessary to ensure a stable and accuratecomputed torque control.

5.2.2 Experiments

The simulation showed that a Cartesian space com-puted torque control with forward kinematics and ajoint space computed torque control have the samebehaviour. The expected improvements could onlybe established with a control using a direct measureof the end-effector pose. At the moment, the com-puter vision is not accurate and fast enough to setup relevant experiments and a laser tracker is tooexpensive. Consequently, we can only propose anexperimental comparison between single-axis con-

trol and computed torque control in the Cartesianspace with the forward kinematics. To achieve thiscomparison, the end-effector trajectory is measuredwith a 512 512 camera as an exteroceptive mea-sure running at 250Hz. This provides us with ameasure of the real end-effector trajectory insteadof a model biased estimation. A comparison be-tween the camera and a laser interferometer is per-formed (see Figure 21). Figure 22 shows that thecamera has an average accuracy of 26mcomparedto the interferometer measure thus validating fur-ther results.

Both control schemes have the same gain tuningwith same cut-off frequency (c) of 5Hz. Never-theless, the derivative gain in the single-axis con-troller can not be set at its theoretical value becausethe linear actuators we use do not cope with noise,even filtered. The reference trajectory is a simple100 mm square in the XY frame. A fifth degreepath generation with a 3m.s2 maximal accelera-tion is used. The trajectory is executed segment bysegment.

Figure 23 shows the performed trajectory in theXY plane for the two control strategies, single-

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(a) Calibration pattern andinterferometer mounted onthe end-effector

(b) Camera and laser

Figure 21: Straightness measure with an high speedcamera and a laser interferometer

axis and computed torque control, compared withthe reference trajectory. Let us stress out thatthe Figure 23 only represents the relative displace-ments. Thus, the bias due to the geometrical er-rors is not measured. The computed torque con-trol achieves an accurate tracking while the single-

axis can not. Indeed, the computed torque controlperforms straight displacements whereas the single-axis control presents some oscillations around thereference. Numerically, the straightness error aredivided by 7 for X-axis displacement and 10 for Y-axis displacement (see Table 7). Figure 24 showsthe time evolution of the end-effector position alongthe X axis for the reference trajectory, the singleaxis control and the computed torque control. Itcan be noticed that the computed torque controlpresents good tracking performances whereas thesingle-axis control presents important tracking er-rors and overshoot.

Thus, these experiments validate the simulationresults above. In other words, using computedtorque control instead of a linear single-axis con-trol improves tracking. It can be noticed that theobtained results are worse than those expected. In-deed, simulations do not take into account assem-bly defects, making the difference between the sim-ulated model and the control model smaller thanthe difference between the real machine and thecontrol model. The assembly defects are treated inthe kinematic models [Rizk et al., 2006] and should

(a) Deviation on Y-axis

(b) Error between laser interferometer and camera measures

Figure 22: Comparison between laser interferome-ter and camera

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Figure 23: Comparison between single-axis andCTC controller measured with a high speed cameraon a 100mm XY square

PID CTC

Left edge 0.733mm 0.154mmRight edge 2.255mm 0.330mm

Bottom edge 3.318mm 0.443mmTop edge 3.143mm 0.293mm

Table 7: Measured straightness error on the squaresegments with a high speed camera

be extended to dynamics. In addition, a more accu-rate identification with exteroceptive measure, suchas computer vision [Renaud et al., 2006], could beused to improve control accuracy.

Conclusion

In this article, we aimed at showing that controlof parallel kinematic machine should be thoughtagain. To our mind, a computed torque controlin the Cartesian space, with an exteroceptive end-effector pose measure and a Cartesian space dy-namic model, is the relevant solution to ensure thebest performances and machine capabilities use.

Indeed, the inherent complexity of the closed me-chanical structure leads to highly nonlinear dynam-ics with dynamic leg coupling. Therefore a single-axis control can not ensure correct performances

(a) End-effector pose along X axis

(b) Tracking error along X axis

Figure 24: End-effector pose along X axis measured

with a high speed camera on a 100mm square

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while using the whole workspace and the machine

maximal speed capabilities. In this way, the com-puted torque control is known to be a relevant so-lution for serial kinematic machines. However, thecomputed torque control is often forsaken by theparallel kinematic machine community since it of-ten requires robust control. Nevertheless, thesecontrol schemes of serial robotics are classically per-formed in the joint space and thus should not bereused directly for parallel kinematic machines.

Actually, since a parallel kinematics machine isdefined by its end-effector configuration, using aCartesian space control is more relevant than us-ing the classical joint space one. In addition, whenit is performed with an exteroceptive end-effectorpose measure, the modeling errors sources are min-imised leading to a more stable control than thejoint space one, making the robust control uselessor, at least, used only for unknown disturbancesrejection. Furthermore, we showed that a Carte-sian space control allows for a better mechanicalstructure handling than the joint space one.

Simulations were performed to validate the abovediscussion and some of them were validated experi-mentally. Nevertheless, further experiments shouldbe performed to validate the improvement of the

use of an exteroceptive measure of the end-effector,such as computer vision or laser tracker. Theseexperiments should be preceded by a more perfor-mant identification. The test-bed is very particu-lar, therefore experiments on other structures, suchas Gough-Stewart platform, should be done to val-idate the genericity of the approach and validatethe internal torque minimisation and the behaviourin the neighborhood of singularities. Last but notleast, a theoretical demonstration of the controlaccuracy, stability and robustness in regards withmeasure and modeling errors could be performed

as initiated in [Paccot et al., 2007].

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A Review on Dynamic Control of Parallel Kinematic Machine 37 PAGINI