Some Issues on Dynamic Control of Parallel Kinematic Machines

1 1 1,2 1,3

Flavien Paccot Omar Ait-Aider Nicolas Andreff Philippe Martinet

Abstract— In this paper, a discussion on parallel kinematic dynamic model of the machine, are widespread for industrial

machine dynamic control shows that such a control has to be

serial manipulators. However, the latter has never been

thought over again. Hence, the well-known computed torque

employed for industrial parallel kinematic machines whereas

control approach is revisited and is shown, when it is performed

a great improvement in dynamic accuracy and workspace

in the Cartesian space and including a Cartesian space dynamic

use could be expected. Indeed, the transposition from serial

model, to be deﬁnitely relevant for parallel kinematic machines.

Moreover, it is shown that greater improvements can be robotics to parallel one is not always straight forward. The

expected with an exteroceptive measure of the end-effector pose.

small amount of experimental results in the literature proves

Finally, experimental results on a complex prototype give a

the troubles in setting up a dynamic control for a parallel

comparison between a linear single-axis control and a computed

kinematic machine and obtaining good performances [9],

torque control to prove our assertions.

[10].

I. INTRODUCTION

Actually, the modeling errors are the main limitation in the

Parallel kinematic machines are spreading in the industry accuracy and stability of a computed torque control [8]. How-

because of their advantages over serial kinematic machines, ever, the dynamic modeling of parallel kinematic machines

such as stiffness, high load and high speed capacities [1]. is quite complex [11]. Therefore, the amount of computation

Nevertheless, these good dynamic performances are not often imposes simpliﬁcations [10], [12]. This leads to non

always achieved [2]. Indeed, improvements are still needed in neglectable modeling errors with regards to accuracy and

design, modeling, identiﬁcation and control to take advantage stability. These errors can be decreased with a kinematic

of parallel kinematic machine performances [3]. In our mind, and dynamic identiﬁcation. The kinematic identiﬁcation es-

the development of adapted control strategies is probably the sentially reduces the inﬂuence of assembly errors [13] while

ﬁeld where remains the largest potential for improving the the dynamic identiﬁcation reduces the inﬂuence of frictions

tracking performances at high speed. and internal torques due to assembly errors [14]. In many

As far as we know, industrial parallel kinematic machines cases, the identiﬁcation is nevertheless not sufﬁcient for

have in most cases a linear single-axis control. This control performing a stable and accurate computed torque control.

strategy seems to be efﬁcient with regards to its large In these cases, robust techniques are generally employed to

presence in machining. However, the dynamic behaviour of cope with the error inﬂuence [10]. Therefore, an industrial

a parallel kinematic machine is strongly nonlinear due to a implementation of such control strategy is not relevant since

dynamic coupling between the kinematic chains linking the the understanding and the mastery of robust techniques

end-effector to the ﬁxed basis, also known as legs. Therefore, require heavy skills and means.

a linear control strategy ensures a good accuracy only at low However, in many cases, the inverse dynamic model of

speed and in a small part of the workspace [4]. Moreover, the a parallel kinematic machine is written in the joint space

efﬁciency is not homogeneous over the workspace since the as a function of the joint variables, like a serial kinematic

dynamic behaviour depends on the end-effector pose [5]. To

machine [11]. Nevertheless, since the kinematics are deﬁned

take into account this heterogeneity, a restricted workspace

by the end-effector conﬁguration, the dynamics should also

can be deﬁned as a space where stiffness, kinematic and

be computed in the Cartesian space (SE ), written as a

3

dynamic properties allow for a good accuracy [1], [6]. In

function of the end-effector pose and its time derivatives and

addition, an optimal path can be computed with regards to the

mapped into the active joints space [9], [15]. In this case,

dynamic behaviour [7]. Therefore, these solutions deal with

the dynamic modeling requires less computation and thus

the weakness of the linear single-axis control by proposing presents less modeling errors than a joint space modeling.

a path with restricted speed in a restricted working space, Nevertheless, the use of a Cartesian space dynamic model is

leading to a suboptimal use of parallel kinematics machine. only relevant with a Cartesian space control as developed in

However, improving the dynamic performance of a ma- further words.

chine by employing a nonlinear dynamic control, such as The motivation of this paper is to develop this discussion

the so-called computed torque control, is a well-known solu- on the dynamic control of parallel kinematic machines. It is

tion [8]–[10]. These control strategies, including the inverse thus originally shown that a Cartesian space control is the

most relevant solution to ensure correct tracking. Experimen-

1

LASMEA - UMR CNRS 6602, Universite ´ Blaise Pascal Cler-

tal results are proposed to emphasize this discussion. Notice

mont Ferrand II, Aubie `re, France

2 that the reader is expected to be familiar with kinematic and

LAMI, Insitut Franc ¸ais de Me ´canique Avance ´e (IFMA),

dynamic modeling as well as with standard control schemes.

Aubie `re, France

3

ISRC, Sungkyunkwan University, Suwon, South Korea Thus, we can focus only on the analysis of the existingb b

˙ ˙

D(Xb,X)

Feedforward

2

d d

Db(Xb)

X dt dt

d b˙

e + Γ q

X

X

d +

e + ω Γ q

Path IKM(X ) PID Machine

d

b˙b ¨

b

Path IKM(X) PID IDM(X,X,X) Machine

+ + d Db(Xb)

+ +

−

+

−

Xb

Fd KM(q)

Fig. 1. Single-axis control with PID controller and feedforward

Fig. 3. Joint space Computed Torque Control for parallel kinematic

2

d d

machines, explicit form

dt dt

X

d

e + ω Γ

q

2

Path Id KM(X) PID IDM(q,q˙,q¨) Machine d d

d

Db(Xb)

dt dt

+

+

b˙

X

−

X

d

+

e ω Γ q

b˙b ¨

Path PID IDM(Xb,X,X) Machine

Fig. 2. Joint space Computed Torque Control for serial kinematic machines

+ +

−

Xb

d

FKM(q)

schemes. The core of this paper is organized as follows:

Section II is devoted to control, Section III presents the

Fig. 4. Cartesian space Computed Torque Control for parallel kinematic

machines

test-bed, namely the Isoglide-4 T3R1 [16], and its dynamic

modeling and Section IV contains experimental results.

II. DYNAMIC CONTROL OF A PARALLEL MACHINE

complexity of joint space computed torque control (see

d

Figure 3 where FKM is a numerical solution to the

As stated above, industrial parallel kinematic machines

b

forward kinematic problem and D is the computed for-

use in most cases a linear single-axis control with a linear

ward instantaneous kinematic matrix). Thus, heavy on-line

feedforward in terms of speed and acceleration (see Figure 1,

computation decreases control speed, accuracy and stability.

where IKM is the Inverse Kinematic Model). However, to

Consequently, a joint space computed torque control for a

ensure a good accuracy, the workspace and speeds should

parallel kinematic machine is rarely met alone but with a

be restricted [1], [6], [7].

robust controller [10].

Indeed, the strongly nonlinear dynamics of a parallel

On the opposite, using a Cartesian space dynamic model

kinematic machine have to be compensated for to increase

implies using a Cartesian space computed torque control, as

attainable workspace, speed and accuracy. The so-called

mentioned by Callegari [9]. We propose here a deepest anal-

computed torque control is a well-know solution for serial

ysis of this assertion. The Cartesian space computed torque

manipulators [8]. It encloses an inverse dynamic model

control is well-known for serial kinematic machines [8].

(IDM) depending on joint positions, speeds and accelerations

d

However, it requires, in this case, more computation than

(see Figure 2). Notice that IKM is a numerical solution

a joint space computed torque control, since the numerical

to the inverse kinematic problem, obtained by a numeri-

inverse instantaneous kinematic matrix is used on-line. It

cal inversion of the closed-form forward kinematic model

may lead to a decrease of control speed, accuracy and

and often performed off-line. This control ensures excellent

stability. Consequently, the Cartesian space computed torque

tracking performances. However, its transposition to parallel

control of serial kinematic machine is rarely used. On

kinematic machines is harder than for the linear single-axis

the opposite, by comparing Figure 3 and Figure 4, which

controller. Let us see why.

represents the Cartesian computed torque control for parallel

Computed torque control of a parallel kinematic machine

kinematic machines, it can be noticed that less numerical

met in the literature is generally performed in the joint

transformations are used. Therefore, a more stable and accu-

space [10]. Nevertheless, in most cases, the inverse dynamic

rate control is performed [18]. Hence, only from the control

model of a parallel kinematics machine depends only on

scheme analysis, a Cartesian space computed torque control

the end-effector pose, velocity and acceleration [9], [15].

is relevant for parallel kinematic machines. Nevertheless, we

Therefore, performing a computed torque control in the joint

can point out some additional practical advantages.

space requires transformations from joint space to Cartesian

space. These forward transformations have a closed-form Firstly, trajectories are most often planned in the Cartesian

expression for most serial kinematic machines. However, space. Thus, a Cartesian space control is more natural since

the duality between serial and parallel kinematic machine the control is performed directly in the task space. In addi-

implies that most parallel kinematic machines have algebraic tion, the desired trajectory is not transformed with the inverse

inverse kinematic models and numerical forward kinematic kinematic model, which can present errors. Consequently,

models [17]. the reference trajectory is not biased by the modeling or

Consequently, the presence of on-line computation in- identiﬁcation errors. Furthermore, the Cartesian space is the

creases the complexity of the control scheme. We propose state space of most parallel kinematic machine since the

an explicit form of this control to emphasize the inherent latter are completely deﬁned by their end-effector pose [19].X

d

Cartesian space control = correct tracking

Cartesian space

joint space control = tracking is lost

joint space

Disturbance shifts the end-effector pose

q

d

q

Fig. 5. Cartesian space ensures correct end-effector reference tracking

contrary to joint space control

Therefore, a Cartesian space control is a state feedback

control, which is known to ensure a better accuracy and

robustness than a control without a state feedback.

Secondly, a better end-effector trajectory tracking is en-

sured with a Cartesian space control than a joint space

one. Indeed, one joint variable conﬁguration leads to several

end-effector poses [20]. In the worst cases, a disturbance

on joint trajectory can thus shift the end-effector position

without changing joint conﬁguration. This can happen es-

Fig. 6. Global view of the Isoglide-4 T3R1

pecially in the neighborhood of singularities (assembling

mode changing trajectory [21]) or cups points (non-singular

be used.

posture changing trajectory [22]). This change of the end-

However, a kinematic model is always biased by the

effector pose is not observed by a joint space control whereas

unavoidable geometrical and assembly errors contrary to a

a Cartesian space one is able to do so (see Figure 5).

Consequently, the Cartesian space control tries to bring direct measure using simpler physics, such as optics. As far

back the end-effector pose to its reference or fails when as we know, the means to measure an object pose (Cartesian

the singularity can not be crossed again. On the contrary, position and orientation) are rare. For example, a laser tracker

a converging joint space control can not tell whether the is an accurate sensor (about 20μm for recent sensors) but

−2

Cartesian reference tracking fails or not. not fast enough (20m.s maximal object acceleration) [26],

[27]. To our knowledge, this sensor has not been integrated in

Last but not least, even on planned path dealing with

a control scheme but it is only used for calibration [27]. On

kinematic and dynamic constraints, the joint position errors

the opposite, the computer vision is a well known solution

are independent from each other when using a joint space

for robot control [28]. However, the accuracy and speed are

control. Therefore, the constraint can not be ensured and

generally quite low. Nevertheless, the technological advances

two types of defects may appear: uncontrolled parasite end-

allows for a fast and accurate vision-based control in a near

effector moves or internal torques on the contrary if these

future [29], [30].

moves are impossible, thus degrading passive joints. Like

two-arm robot control, Cartesian space control can minimize,

III. APPLICATION ON THE ISOGLIDE-4 T3R1

or cancel in the best cases, internal torques [23]. Indeed,

A. Presentation of the test-bed

the regulated errors, which are end-effector pose errors, are

naturally compatible with the end-effector moves.

To validate the above discussion, we propose to apply the

Consequently, Cartesian space control is particularly rel-

Cartesian space dynamic modeling and computed torque con-

evant for parallel kinematic machines. Nevertheless, the

trol to the Isoglide-4 T3R1. This parallel kinematic machine

presence of the forward kinematics in the feedback loop

is a fully-isotropic one with decoupled motion (see Figure 6

can reduce the improvement of a Cartesian space control

and [16]). It is a four degrees of freedom machine with three

over a joint space one. In the general case, this numerical

translations and one rotation. This machine is designed for

transformation can disturb the feedback loop thus leading

high speed machining. Hence, stiffness requirements impose

to stability, accuracy and speed losses and thus imposing

an important weight: 31kg per leg and 14kg for the end-

a robust control [12]. In the author opinion, this issue

effector.

could be improved by using performant forward kinematics

The main advantage of the Isoglide-4 T3R1, as far as

resolution methods [24] or metrological redundancy which

control is concerned, is to have a closed-form expression of

simpliﬁes the forward kinematics [25]. Of course, the ability

the forward kinematic and instantaneous kinematic models:

of employing these methods at high rate should be tested.

Anyhow, the forward kinematics of some parallel kinematic X = q −X

e 1 0

machines have a closed-form expression, like in the Isoglide- Y = q −Y

e 2 0

(1)

4 T3R1 case [16]. Thus, the estimation of the end-effector Z = q −Z

e 3 0

q −q +δZ

4 3

pose is reliable and stable and a Cartesian space control could sinθ =

Land

Parameter CAD values Identiﬁed values Units σ(%)

1 0 0 0

MXR 3.235 5.054 kg.m 0.42

3

2

0 1 0 0

ZZR 1.787 2.443 kg.m 1.29

3

D(X) = (2)

2

ZZR 6.429 8.420 kg.m 0.54

0 0 1 0 2

M 45.011 39.513 kg 0.62

1 1 t

0 0 −

Lcosθ Lcosθ

M 31.4380 39.999 kg 0.40

R1

M X 2.059 0 kg.m

T

P P

where X = [X Y Z θ] is the end-effector pose,

e e e

YY 0.411 0 kg.m

P

X , Y , Z and δZ are constant parameters depending on

0 0 0 Mcomp 45.011 49.180 kg 0.50

3

Mcomp 31.4380 41.005 kg 0.39

4

the actuators position in the reference frame and L is one

Fs 10.907 N 2.76

1

dimension of the end-effector.

Fs 25.558 N 1.25

2

Khalil’s method [31] is preferred over the classical Carte-

Fs 21.044 N 1.71

3

Fs 28.980 N 1.07

4

sian space dynamic modeling [9], [15] since this approach

−1

Fv 36.108 N.s.m 3.81

1

is easy to implement and ensures the known advantages of a

−1

Fv 89.419 N.s.m 2.45

2

Newton-Euler method in a control context. In the Isoglide-4 −1

Fv 35.211 N.s.m 6.34

3

−1

T3R1 case, Khalil’s method leads to a closed-form inverse Fv 64.793 N.s.m 3.10

4

dynamic model depending only on the end-effector pose and

Observation matrix condition number: 355.56

time derivatives. For conciseness concern, the expression of

Number of samples: 65404

the obtained model is not given here.

This test-bed is well suited to the validation of the ap-

TABLE I

proach, since its weight prevents us from neglecting the

Dynamic identiﬁcation results

dynamics. Moreover, its straightforward kinematic models

allow for using a Cartesian control easily and compensate

for the technological lack of reliable and accurate high-speed

sensor of the end-effector pose.

IV. EXPERIMENTAL RESULTS

A. Dynamic identiﬁcation

In order to ﬁt the inverse dynamic model to the real dy-

namics of the machine and ensure the best performances for

computed torque control, dynamic identiﬁcation was realized

(see Table I). The method and notations used here were

proposed by Gue ´gan [32]. Results lead to an observation

matrix condition number of 355.56 which is relatively good.

Inertia parameters (MXR , ZZR , ZZR , M , M ) are

3 3 2 t R1

identiﬁed with a standard deviation from 0.40% to 1.29%,

friction terms (Fs and Fv ) from 1.07% to 6.34%. Let us

i i

remark that some parameters describing the end-effector can

(a) Calibration pattern and inter- (b) Camera and laser

not be identiﬁed because the end-effector is lighter than the

ferometer mounted on the end-

effector

legs, thus having a little inﬂuence on dynamics. Anyhow, the

good results of the identiﬁcation process allows for ensuring

Fig. 7. Straightness measure with an high speed camera and a laser

a stable and accurate computed torque control.

interferometer

B. Dynamic control

The Isoglide-4 T3R1 is designed to be controlled either

in joint space or in Cartesian space thanks to the kinematic

decoupling. Consequently, we ﬁrst propose a comparison be-

tween linear single-axis control and computed torque control

in joint space.

To achieve this comparison, the end-effector trajectory is

measured with a 512×512 camera as exteroceptive measure

running at 250Hz. This provides us with a measure of the real

end-effector trajectory instead of a model biased estimation.

A comparison between the camera and a laser interferometer

(a) Deviation on Y-axis (b) Error between laser interferom-

is performed (see Figure 8) showing that the camera has an eter and camera measures

average accuracy of 26μm and validating further results.

Fig. 8. Comparison between laser interferometer and camera

Both control schemes have the same gain tuning with same

cut-off frequency (ω ) of 5Hz. Nevertheless, derivative gain

cFig. 10. Tracking error on X axis measured with an high speed camera

on a 100mm square

Fig. 9. Comparison between single-axis and CTC controller measured with

an high speed camera on a 100mm XY square

PID CTC

Left edge 0.733mm 0.154mm

Right edge 2.255mm 0.330mm

Bottom edge 3.318mm 0.443mm

Top edge 3.143mm 0.293mm

TABLE II

Measured straightness error on square segment with an high speed camera

in the single-axis controller can not be set at the theoretical

Fig. 11. Orientation tracking error with a computed torque control in the

joint and the Cartesian space, with Forward Kinematics and direct measure

value because the linear actuators we use do not cope with

noise, even ﬁltered. Figure 9 shows a comparison between

single-axis and computed torque controls. The reference

trajectory is a simple 100 mm square in the XY frame.

is applied. The joint sensors have a 1μm accuracy. The

−2

A ﬁfth degree path generation with a 3m.s maximal

direct measure has a 50μm and 0.001rad accuracy. The

acceleration is used. The trajectory is executed segment by

geometrical errors are ﬁxed to 50μm and the dynamic

segment.

parameters errors to 10%. In the Isoglide-4 T3R1 case, the

According to Figure 9, computed torque control in the geometrical errors have a great inﬂuence on the orientation

joint space achieves an accurate tracking while the single- estimation (see Eq 1). Consequently, the comparison is

◦

axis can not. Numerically, the straightness error are divided achieved on a 30 rotation of the end-effector. According

by 7 for X-axis displacement and 10 for Y-axis displacement to the Figure 11, the improvement between joint space and

(see Table II). Furthermore, there is no overshoot at the Cartesian space control with the forward kinematics is small

end of travel with the computed torque control (see Figure since the Cartesian and joint reference are in a very close

10). Thus, using computed torque control instead of a linear relation (see Eq 1). On the opposite, the use of a direct

single-axis control improves tracking, even in the joint space. measure greatly improves the tracking even with a less

Indeed, due to heavy inertia, the dynamic coupling between accurate end-effector pose sensor than the joint one (50μm

−2

legs is not neglectable even at 3m.s . Consequently, the against 1μm). Indeed, the orientation measure leads to a

dynamic behaviour of the machine should be compensated better compensation of the dynamics since the measure is

for to improve accuracy. closer to the real orientation than the estimated one.

At the moment, we are not able to propose experiments On the whole, these simulation and experiment results

on a control with a direct measure of the end-effector pose. show three major points. First, using computed torque con-

Nevertheless, we propose a simulation to compare a joint trol instead of a linear single-axis control improves accu-

space computed torque control, a Cartesian space computed racy especially on heavy parallel kinematic machines like

torque control using the forward kinematics and one using the Isoglide-4 T3R1. Second, the Cartesian space control

a direct measure of the end-effector pose. Realistic noise improves the Cartesian reference tracking. Third, a directmeasure of the end-effector pose, instead of an estimation [15] B. Dasgupta and P. Choudhury. A general strategy based on the

newton-euler approach for the dynamic formulation of parallels ma-

with the Forward Kinematics, leads to better tracking since

nipulators. Mechanism and Machine Theory, 34:801–824, 1999.

the geometrical errors have no inﬂuence on the feeback.

[16] G. Gogu. Fully-isotropic T3R1-type parallel manipulators. In

J. Lennarc ˘ic ˘ and B. Roth, editors, On Advances In Robot Kinematics,

V. CONCLUSION

pages 265–272. Kluwer Academic Publishers, 2004.

[17] K.J. Waldron and K.H. Hunt. Series-parallel dualities in actively

In this paper a discussion on parallel kinematic machine

coordinated mechanisms. International Journal of Robotics Research,

dynamic control was proposed. It showed that performing a

10(2):pp. 473–480, 1991.

[18] F. Paccot, N. Andreff, and P. Martinet. Revisiting the major dynamic

computed torque control in the Cartesian space is relevant

control strategies of parallel robots. In European Control Conference

for parallel kinematic machine. According to the presented

(ECC’07), Kos, Greece, July 2007. To appear.

results, this control improves accuracy by compensating

[19] T. Dallej, N. Andreff, Y. Mezouar, and P. Martinet. 3d pose visual

servoing relieves parallel robot control from joint sensing. In Interna-

for the dynamic behaviour of the machine. However, there

tional Conference on Intelligent Robots and Systems (IROS’06), pages

are two limitations. First, the dynamic model included in

4291–4296, Beijing, China, October 2006.

control has to depend on the end-effector. Second, the end-

[20] M.L. Husty. An algorithm for solving the direct kinematic of the

Gough-Stewart platforms. Technical Report TR-CIMS-94-7, McGill

effector pose and velocity are needed. Generally, the latter are

University, Montre ´al, Canada, 1994.

estimated with numerical models. It results in a lack of ac-

[21] D. Chablat and P. Wenger. Working modes and aspects in fully

curacy, computation time and reliability of the estimation. A

parallel manipulators. In IEEE International Conference on Robotics

and Automation (ICRA’98), pages 1964–1969, Leuven, Belgium, May

ﬁrst solution is metrological redundancy with proprioceptive

1998.

measures to reduce the complexity of forward models and

[22] M. Zein, P. Wenger, and D. Chablat. Singular curves and cusp points

the number of solutions. However, accurate modeling and

in the joint space of 3-rpr parallel manipulators. In IEEE International

Conference on Robotics and Automation (ICRA’06), pages 777–782,

identiﬁcation are still required. A second solution is a direct

Orlando, USA, May 2006.

end-effector pose measure, with a laser tracker or vision for

[23] P. Dauchez, X. Delebarre, and R. Jourdan. Hybrid control of a two-

instance, which seems more promisefull

arm robot handling ﬁrmly a single rigid object. In 2nd International

Workshop on Sensorial Integration for Industrial Robots (SIFIR’89),

REFERENCES pages 67–62, Zaragosa, Spain, November 1989.

[24] J.P. Merlet. Solving the forward kinematics of Gough-type parallel

[1] J.P. Merlet. Parallel robots. Kluwer Academic Publishers, 2000.

manipulator with interval analysis. The International Journal of

[2] J. Tlusty, J. Ziegert, and S. Ridgeway. Fundamental comparison of

Robotics Research, 23:221–235, 2004.

the use of serial and parallel kinematics for machine tools. Annals of

[25] L. Baron and J. Angeles. The direct kinematics of parallel manipu-

the CIRP, 48(1):351–356, 1999.

lators under joint-sensor redundancy. IEEE Transactions on Robotics

[3] J.P. Merlet. Still a long way to go on the road for parallel mecha-

and Automation, 16(1):1–8, 2000.

nisms. In ASME 27th Biennial Mechanisms and Robotics Conference,

[26] http://www.faro.com.

Montreal, Quebec, October 2002.

[27] W.S. Newman, C.E. Birkhimer, R.J. Horning, and A.T. Wilkey. Cal-

[4] B. Denkena and C. Holz. Advanced position and force control

ibration of a motoman p8 robot based on laser tracking. In IEEE

concepts for the linear direct driven hexapod PaLiDa. In The ﬁfth

International Conference on Robotics and Automation (ICRA’00),

Chemnitz Parallel Kinematics Seminar, pages 359–378, Chemnitz,

pages 3597–3602, San Francisco, USA, April 2000.

Germany, April 2006.

[28] S. Hutchinson, G. Hager, and P. Corke. A tutorial on visual servo

[5] C. Brecher, T. Ostermann, and D.A. Friedich. Control concept for

control. IEEE Transactions on Robotics and Automation, 12(5):651–

PKM considering the mechanical coupling between actuators. In The

670, 1996.

ﬁfth Chemnitz Parallel Kinematics Seminar, pages 413–427, Chemnitz,

[29] R. Ginhoux, J. Gangloff, M.F. de Mathelin, L. Soler, M. Are-

Germany, April 2006.

nas Sanchez, and J. Marescaux. Beating heart tracking in robotic

[6] H. Chanal, E. Duc, and P. Ray. A study of the impact of machine tool

surgery using 500 hz visual servoing, model predictive control and

structure on machining processes. International Journal of Machine

an adaptative observer. In IEEE International Conference on Robotics

Tools and Manufacture, 46:98–106, 2006.

and Automation (ICRA’04), pages 274–279, April, New Orleans, USA

[7] H. Abdellatif and B. Heiman. Adapted time-optimal trajectory

2004.

planning for parallel manipulators with full dynamic modeling. In

[30] F. Paccot, N. Andreff, P. Martinet, and W. Khalil. Vision-based

International Conference on Intelligent Robots and Systems (IROS’05),

computed torque control for parallel robots. In The 32nd Annual

pages 411–416, Barcelona, Spain, April 2005.

Conference of the IEEE Industrial Electronics Society (IECON’06),

[8] W. Khalil and E. Dombre. Modeling, identiﬁcation and control of

pages 3851–3856, Paris, France, 7-10 November 2006.

robots. Hermes Penton Science, London-Paris, 2002.

[31] W. Khalil and O. Ibrahim. General solution for the dynamic modeling

[9] M. Callegari, M.C. Palpacelli, and M. Principi. Dynamics modelling

of parallel robots. In IEEE International Conference on Robotics and

and control of the 3-RCC translational platform. Mechatronics,

Automation (ICRA’04), pages 3665–3670, New Orleans, USA, April

16:589–605, 2006.

2004.

[10] A. Vivas. Predictive functionnal control for a parallel robot. IEEE

[32] S. Guegan, W. Khalil, and P. Lemoine. Identiﬁcation of the dynamic

International Conference on Intelligent Robots and Systems (IROS’03),

parameters of the Orthoglide. In IEEE International Conference

pages 2785–2790, October 2003.

on Robotics and Automation (ICRA’03), pages 3272–3277, Taipei,

[11] J. Ko ¨vecses and J.C. Piedboeuf. Methods for dynamic models

Taiwan, September 2003.

of parallel robots and mechanisms. In Workshop on Fundamental

Issues and Future Research Directions for Parallel Mechanisms and

Manipulators, pages 339–347, Quebec, Canada, October 2002.

[12] S.H. Lee, J.B. Song, W.C. Choi, and D. Hong. Position control of

a Stewart platform using inverse dynamics control with approximate

dynamics. Mechatronics, 13:605–619, 2003.

[13] J. Wang and O. Masory. On the accuracy of a stewart platform -

part I: the effect of manufacturing tolerances. In IEEE International

Conference on Robotics and Automation (ICRA’93), pages 114–120,

Atlanta, USA, May 1993.

[14] M.M. Olsen and H.G. Peterson. A new method for estimating

parameters of a dynamic robot model. IEEE Transactions on Robotics

and Automation, 17(1):95–100, 2001.

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