Grenoble, France, May 17-19, 2006

IDMME 2006

PRECISION STUDY OF A DECOUPLED FOUR DEGREES OF FREEDOM PARALLEL ROBOT INCLUDING MANUFACTURING AND ASSEMBLING ERRORS R. RIZKa,b, N. ANDREFFa,b, J. C FAUROUXa, J. M. LAVESTb,G. GOGUa a

:Mechanical Engineering Research Group (LaMI) French Institute of Advanced Mechanics and Blaise Pascal University Campus de Clermont-Ferrand/ Les Cézeaux, BP 265, Aubiere Cedex, France b Laboratory of Sciences and Materials for Electronics and of Automatic Blaise Pascal University /CNRS, Clermont-Ferrand, France, rrizk/fauroux/andreff/gogu/@ifma.fr, [email protected]

Abstract: This paper presents a study of geometric errors due to assembly and manufacturing tolerance on the kinematic accuracy of a parallel robot with four degrees of freedom with decoupled motions and fully isotropic in translations- ISOGLIDE4-T3R1 The effects of geometric errors on the kinematic accuracy are analytically calculated under the hypothesis that the components are rigid bodies by using the ISOGLIDE4-T3R1 forward jacobian. Analytic calculations give a general form for the pose inaccuracy in the workspace. They give both a form for the ISOGLIDE4-T3R1 sensitivity to geometric errors what allows to quantify which defect is more harmful, and how the sensitivity varies in the workspace. In a second part in this paper, geometric error measurements on the robot prototype are presented. They are carried out with vision based metrology. This system not only has the required capabilities but also has the advantage to measure six degrees of freedom in the same coordinate system without contact. The metrology results and analytic calculations allow quantifying the hypothesis of rigid bodies and giving prospects for the error influences on the robot stiffness and accuracy with deformable bodies.

Keywords: Parallel robot, decoupled motions, accuracy, sensitivity, Isoglide4-T3R1, Metrology by vision. 1 Introduction Parallel mechanisms are emerging in the industry (machine-tools, high-speed pick-andplace robots, flight simulators, medical robots, for instance). A parallel mechanism can be defined as a mechanism with closed kinematic chain, made up of an end-effector with N degrees of freedom and a fixed base, connected to each other by at least two kinematic chains, the motorization being carried out by N actuators [1]. This allows parallel mechanism to bear higher loads, at higher speed and often with a higher repeatability [1]. This paper presents a study of the kinematic accuracy of a parallel robot with four degrees of mobility and decoupled motions called ISOGLIDE4-T3R1, isotropic in translation and have three translations and one rotation for the end-effector. The mechanism is proposed in [2]. Each of the four kinematic chains of ISOGLIDE4-T3R1 is also called "leg", the body of the leg connected to the actuator is called "arm" the second body "forearm". An essential property of 1

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the parallel mechanisms is the existence of not actuated joints known as "passive". For ISOGLIDE4-T3R1, they are revolute joints between the legs and the linear actuators, between the arm and the forearm of each leg, and between the leg and the end-effector formed by a platform (Figure 1). A parallel robot is characterized by its differential kinematic model which establishes a relation between the infinitesimal variation of articular coordinates [dq ] and the infinitesimal variation of the operational coordinates [dX ] . This relation can be expressed in matrix form as:

[A ][dX ] = [B][dq ] .

(1)

Matrix [A] is known as the parallel matrix and matrix [B] as the serial matrix [3] and this relation can be also expressed in this form if [A] is reversible:

[dX] = [h J P ][dq]

(2)

where P is the robot end-effector mobile platform characteristic point, h the reference coordinates system in which the operational variables are expressed. is known as the mechanism forward jacobian matrix. In the axiomatic design, this matrix is called "design matrix" [4]. The mechanism has a fully isotropic parallel manipulator if is proportional to the identity matrix. It is with uncoupled motions if is diagonal and decoupled motions if is triangular. In the other cases, the motions of the mechanism are coupled [5]. Several parallel robots of type T3R1 (three translations and one rotation of the end-effector) with coupled motions are proposed in the literature [1, 6-12]. ISOGLIDE4-T3R1 is the first parallel robot with four decoupled motions[20]. The development of the forward jacobian is based on the closure of the loops which form the various kinematic chains. With the coupling, it is difficult to express the relations between the articular coordinates and the operational coordinates. In a parallel robot with decoupled motions, the forward jacobian is simpler. To have motion decoupling, high accuracy of perpendicularity and parallelism between the joints will be imposed. However, the large number of links and passive joints often limit robot performance in terms of accuracy [13]. Industrial tolerances can influence decoupling and isotropy and consequently accuracy. The inaccuracies of the kinematic behavior of the robot as well as the pose (position and orientation) of the moving platform can be calculated by integrating the geometric errors in the forward jacobian matrix or they can be measured on the robot prototype. Several methods to perform kinematic identification of parallel mechanisms are proposed in the literature [14-17]. In this paper, measurements are made by artificial vision. This system not only has the required accuracy but also has the advantage of being contactless, measuring six degrees of freedom (position and orientation) in the same coordinate system and finally being low cost compared to other metrology systems. The goals of this study are to answer the following questions: which defect is more harmful? How varies the sensitivity of the endeffector pose to the geometric defects in the workspace? And finally to what extent the rigid bodies hypothesis is correct?

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2 Kinematic structure and properties For the majority of parallel mechanisms, the forward jacobian does not exist in analytical form. Its unicity is not in addition assured. This difficulty is due often to the coupling with which, in the majority of the cases, it will be impossible to obtain simple linear relations, since the equations of closure give non-linear equations. In the case of ISOGLIDE4-T3R1 since it is decoupled, the forward jacobian is simple. In the ideal case (no assembly and manufacturing errors), equation (2) becomes: 1 0 0 0 dx dq P

0

dy P

1

= 0

0 1 0 − r. cos ϕ p

dz P dϕ P LP=r

0

0

1

0 1 r. cos ϕ p

0

B4

C2

dq 4

q2

D4

C4 Y

D2

X

L

Z

p

q3

M3

(3)

A2

q4

B3

.

dq 3

B2

A4

A3

dq 2

M2

M4

O

1

P D1

D3 C3

C1

Q B1

q1

A1

M1

Figure 1. ISOGLIDE4-T3R1 kinematic diagram and prototype The forward jacobian matrix expressed in the reference coordinate system (Oxyz) is triangular, therefore ISOGLIDE4-T3R1 has decoupled motions. The submatrix which connects infinitesimal displacements of the first three actuators, with the end-effector infinitesimal translations is the identity matrix, therefore ISOGLIDE4-T3R1 is isotropic in translation. Subsequent calculations will take place in the reference coordinate system (OXYZ) related to the frame, having the Y axis according to the ascending vertical and axes X and Z in the horizontal plane. In the ideal case, the M1, M2 and M3 actuators axes are following axes X, Y and Z respectively, the axis of M4 is parallel to that of M2.

3 Assembly tolerances and errors It is known that this type of machine is not infinitely rigid and does not have the same rigidity in all its workspace [18]. Moreover, materials are not perfectly homogeneous and during the realization, tool wear and consequently manufacturing quality are not the same during all the stages of realization. Because of these phenomena, realization is not perfect and errors exist inevitably in the components. Since the paper deals with kinematic accuracy, the study is limited to two particular cases for which the motion is always possible even with infinitely rigid components. The effect of these errors on the kinematic characteristics (isotropy in translation and motion decoupling) was presented in [19] on Version 2 of the ISOGLIDE4-T3R1. In this paper we present the effects on the pose accuracy. Experiment was implemented on the version 1 of ISOGLIDE4-T3R1 (Figure 1). Version 2 has an 3

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hyperstaticity degree equal to 4 and version 1 has an hyperstaticity degree equal to 2 [20]. Both versions have the same differential kinematic model so analytical calculations presented in[19] can be applied.

3.1 Error of perpendicularity between actuators Let us consider the case where the axes of actuators, M1, M2, M3 and M4 form angles m, m, m and m with axes X, Y, Z and Y respectively (Figure 2). We obtain the following forward jacobian: dx P dy P = dz P dϕ P

cos α m 0 0 0

−

0 cos β m 0 cos β m

0 0 cos γ m

q . cos δ m − q 2 . cos β m r. 1 − 4 r

0 0 0 cos δ m

0

2

q 4 . cos δ m − q 2 . cos β m r

r. 1 −

dq 1

2

dq 2 dq 3 dq 4 (4)

LP=r

M2

M4 B2

B4

A2

A4 C2 q4

q2

D4

C4 Y

O

D2

X

L

y

Z

p

q3

A3

P D1

D3

B3

M3

C3

C1 C'1

Q B1

q1 m

A1

M1

B'1

A '1

Figure 2. Kinematic diagram of the ISOGLIDE4-T3R1 with orientation defects of the actuators The inaccuracy can be calculated as the difference with same articular coordinates between the ideal pose and the pose with orientation errors of actuators e dx p

e dy p = e dz p e dϕ p

1 − cos α m 0 0 0

−

0 1 − cos β m 0 1 − cos β m

0 0 1 − cos γ m

q . cos δ m − q 2 . cos β m r. 1 − 4 r

2

0

0 0 0 1 − cos δ m q . cos δ m − q 2 . cos β m r. 1 − 4 r

2

(5)

dq 1 dq 2 dq 3 dq 4

Figure 3 show the position errors along X caused by M1 orientation error. This effect is the same for Y and Z position. Equation (5) and Figure 3 show that the error does not depend 4

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m =0º

m =0.125º

m =0.25º

m =0.375º

am=0.5º

0

-0.005

Erroor exp (mm)

-0.01

-0.015

-0.02

-0.025

-0.03 0

100

200

300 400 500 Distance covered by M1

600

700

800

Figure3. Position errors along X caused by motor M1 orientation error (units are in mm)

Figure 4. Rotation errors with actuators orientation errors

only on the value of the defect but also on infinitesimal displacement dq1.we have to integrate (5) to have position errors (Figure 3). Figure 4 and Equation (5) present several important points. The rotation error does not depend linearly on the infinitesimal articular coordinate displacements dq2 and dq4. It depends on q2 and q4 as well as on m and m. The errors graphs (Figure 4) show that the rotation error is cancelled when m and m are equal. Rotation increases when the difference between the Y coordinate of points L (see the kinematic diagram in figures 1 and 2) and P increases. When m> m, Y coordinate of point decreases more than Y coordinate of point L. That leads to a positive supplement of rotation, therefore a negative error. When both defects are of the same value, both Y coordinate components decrease by keeping the same proportion that preserves the rotation angle. The influence of vertical position q2 is also significant. In Figure 4, endeffector rotation errors with same m and m defects and desired rotation p are not the same when q2 changes (see Figure 4-a and Figure 4-c). When q2 increases, the end-effector vertical position error increases linearly such as X position error for actuator M1 in Figure 3, so the difference between the two vertical leg extremities changes and the rotation angle changes too. In addition, the rotation error is not the same for orientations p and – p (see the zoom in Figure 4-d). For p, q4 is bigger than q2 for a certain value Y. For- p, q4 is smaller than q2 for the same Y. The effect of m error is stronger when q4 is big such as the effect of m on XP (Figure 3)

3.2 Defect in the orientation of the leg revolute joints In this part, parallelism between revolute joints named Bi, Ci and Di of the leg Li is assumed (Figure 5). We focused on orientation defect of the whole leg with respect to the actuator on point Bi. This time the defect is not the angle between axis of the leg and the reference axis [19]. If rotation is not around one of the principal axes of the reference coordinate system, to establish the forward jacobian it is necessary to break up the defect into two consecutive rotations around the principal axes of the reference coordinate system[21]. All the legs have defects in their orientation. Defect angles will be named , , for leg L1 around axes Y and Z; , , for leg L2 around axes X and Z; , , for leg L3 around axes X and Y; , for leg L4 around axes X and Z. The loop closures give:

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LP=r

M2

M4 B2

B4

A2

A4 C2 q4

q2

D4

C4 Y

O

D2

X

L

Z

p

q3

A3

B3

M3

D3

P

D1

D'1

C3

C'1

Q B'1

q1

M1

A1

Figure 5: ISOGLIDE4-T3R1 with defects of orientation for leg 1. Kinematic diagram and CAD model 1 0 0 1 0 0 0 −1

0 0 1 0

0 0 0 1

dq 1 dq 2 dq 3 dq 4

=

1 − tan δ

tan β 1

− tan α tan γ

0 0

tan ϕ − tan ε 1 0 tan δ − tan ς 0 tan η − tan γ r. cos ϕ P

dx p dy P

(6)

dz P dϕ P

there, 1 + tan ε tan γ − tan β − tan ε tan α tan β. tan γ + tan α dx p − tan γ + tan α tan δ tan δ + tan ϕ tan γ 1 + tan α tan ϕ r cos ϕ p dy p = tan δ tan ε − tan ϕ tan ε + tan β tan ϕ 1 + tan δ tan β dz p D A B C dϕ p r cos ϕ p r cos ϕ p r cos ϕ p

0 0 0 D r 2 cos 2 ϕ p

dq 1 dq 2 .(7) dq 3 dq 4

where D is the determinant of the serial matrix established in (6). A, B, C are functions of the errors and the rotation angle

D = r cos ϕ p [1 + tan γ (tan ε + tan ϕ tan β ) + tan δ(tan β − tan δ tan ς ) + tan ϕ tan α ] A = − tan δ tan ε tan η + tan ϕ tan η − tan ϕ tan γ − tan δ + tan ζ + tan ζ tan ε tan γ B = tan ε tan η + tan ε tan γ − tan ϕ tan β tan η + tan ϕ tan β tan γ + tan δ tan β

− tan δ tan ε tan α − tan ς tan β + tan ς tan ε tan α C = − tan η + tan γ − tan δ tan β tan η − tan δ tan α + tan ς tan β tan γ + tan ς tan α Figures 6 and 7 present error on the end effector rotation when both vertical actuators are animated for various rotation angles of the platform. Calculations where performed for an example of legs defect orientation ( , ). Both figures show that the error is increasing with the articular coordinate q3. Other interesting point is that errors are quite different for two opposite desired rotations same as in the previous case. In fact, error is the result of motions coupling is not the angle between the ideal and the real orientation [19]. In spite of the motion decoupling, the pose accuracy depends on xP yp zp and p. In Figure 6 errors are null when q3 is between 200mm and 300mm but in Figure 7 when q3=0. In fact in Figure 6 for non zero 6

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values,

the

3

5 p=-45º -22.5 º 0º 22.5 º 45 º

2.5

p=-45º -22.5º 0º 22.5º 45º

4

2

3

Error e p (deg)

Error e p (deg)

1.5

1

0.5

0

-0.5

2

1

0 -1

-1.5 0

100

200

300

400 q3 in mm

500

600

700

-1 0

800

200

300

400 q3 (mm)

500

600

700

800

Figure 7. Errors on rotation according to q3 for q1=q2=250mm =1° = = = = = = =0°

Figure 6. Errors on rotation according to q3 for q1=q2=250mm = =1° = = = = = = 0°

defect and non zero.

100

are reciprocally compensated. No compensation exist when just one of then is

4 Sensitivity Analysis The sensitivity analysis of the ISOGLIDE4-T3R1 to an error is the analysis of the influence of the error variation on each operational coordinate of the robot. Mathematically, the sensitivity of an operational coordinate is its partial derivative with respect to the defect. For actuator orientation errors we obtain: ∂dx p ∂α m ∂dy p ∂β m

∂dz p ∂γ m ∂dϕ p ∂β m ∂dϕ p ∂δ m with

= dq1 . sin α m

(8)

= dq 2 . sin β m

(9)

= dq 3 . sin γ m

(10)

= B(a dq 2 + b dq 4 )

(11)

= D[c dq 2 + d dq 4 ]

(12)

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(

)

a = − r 2 + q 4 cos δ m (q 4 cos δ m − q 2 cos β m ) c = −q 4 cos β m (q 4 cos δ m − q 2 cos β m ) sin β m

B= r3 1−

(q 4 cos δ m − q 2 cos β m ) r

2

3 2

b = −q 2 cos δ m (q 4 cos δ m − q 2 cos β m )

(

)

d = r 2 − q 2 cos β m (q 4 cos δ m − q 2 cos β m ) sin δ m

D= r3 1−

2

(q 4 cos δ m − q 2 cos β m )

2

3 2

r2

The sensitivity of one operational coordinate to an actuator orientation defect depends on the corresponding articular coordinate. Equations (11) and (12) show that orientation sensitivity depends on actuator orientations defects m and m and their infinitesimal displacements dq2 and dq4. Graph 8-a shows the factor of dependence on dq2 of the sensitivity to m. Graph 8- shows the factor of dependence on dq4 of the sensitivity to m. Graph 8-c shows the factor of dependence on dq2 of the sensitivity to m. Graph 8-d shows the factor of dependence on dq4 of the sensitivity to m. If q 2 cos m = q 4 cos m . Error on dϕ p is null [19], the sensitivity of the rotation to

m

depends only on dq2 and to

m only

on dq4.

bB (rad/rad)

aB (rad/rad)

cD (rad/rad)

dD (rad/rad)

Figure 8. Sensitivity factors of rotation to M2 and M4 actuator orientation

m= m=0.1º

5 Discussion Above results show that the defects of actuator orientation influence linearly according to the articular coordinate and not linearly to the defect amplitude on the ISOGLIDE4-T3R1 operational translations precision. On the other hand, defect effects are much more 8

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IDMME 2006

complicated on rotation, and for two defects of equal values of m m, inaccuracies are cancelled. This phenomenon can be explained by the coupling of both vertical actuators to define the variation of the rotation angle (dϕ P ) (equation 4). In addition, the inaccuracies due to leg orientation defect are much more significant than the inaccuracies due to actuators orientation. This is due to the coupling induced by the legs perpendicularity defects. Moreover it is seen that the legs orientation defects are coupled (equation 7) what complicates also the correction by the command. Another very interesting point is that the inaccuracies in absolute value were increasing with the articular coordinates.

6 Experimental validation This section presents an experimental validation by artificial vision for the study presented above. Manufacturing and assembly errors are measured. Distances covered by actuators that represent articular coordinates are measured in the same time with the end effector pose (position and orientation) in a same global coordinate system. The difference between the ideal and the real pose gives errors. In this paper the parallelism defects between the passive joints of the same leg are neglected.

6.1 Experiment The experiment is divided in three parts: -Actuator orientations measurement what is realised with stereoscopic heads. -Leg rotation axis orientation measurement is realised either with stereoscopic heads. -End effector pose with actuators displacements measurement what require using simultaneously stereo heads and a calibrated monocular camera mounted on the end effector. 6.1.1

Actuators translational axis orientation

On each actuator a photoreflective mark is stuck (Figure 9) to realise measurements about ten positions are taken. For each position the stereoscopic head takes 100 images. Results given by the stereoscopic system are stored in spreadsheet. With a numerical program images are filtered and their mean is calculated. Normally, these points are distributed on a perfect line but because the incertitude due to the actuator profile, an approximate line is flitted in least square sense. The accuracy of this system and the experiment set up have been presented in [22]. 6.1.2

Figure 9: Actuator measurment

Mark trajectory Mark Arm Li

Legs rotational axis

Considering a mark stuck on an arm. When the arm of the leg Li turns around the revolute joint Bi, the trajectory of every point in the arm, in particular the mark can be seen as a circle located in a plane perpendicular to the revolute joint Bi axis. To calculate the Bi

Bi Rotational axis Figure 10: Leg rotational axis estimation

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axis, the equation of the plane in witch the mark exists should be estimated. Three points are enough to find the plane. In an experiment, we used about 10 points and the best plane is fitted in least square sense, and the normal to the plane can be lead to the revolute joint axis Bi. (Figure 10)

Calibration target

Figure 11: Monocular camera fixed on the end effector 6.1.3

Figure 12: experimental device

End effector pose measurement

A camera was fixed on the end-effector (Figure 11) using a known object (calibration target) (Figure 12). The camera provides for all images the end effector pose (position and orientation) in the same global coordinate system. Using simultaneously the stereo heads for actuators and the camera for the end effector, the real and theoretical end effector pose errors are able to be estimated. The comparison of theoretical errors and real errors allows us to quantify the influence of the hypothesis of rigid bodies.

6.2 Results Table 1 gives the measured orientation actuator m, m and m defects and legs orientation defects , , , , , , , . These values are calculated in a coordinate system having axis X parallel with the axis of M1 ( m=0) and axis Z such as plan XOZ is parallel with the M1 and M3 axes. This coordinate system was chosen because the horizontal plane and the vertical direction cannot be easily determinated and allows

Component M2 M3 M4 L1 L2 L3 L4

eliminating one defect. (See kinematic diagram in Figure 1 for notations)

error m m m

/ / / /

Value (deg) 0.215 0.815 0.0917 1.760/1.128 1.985/1.124 0.510/0.376 0.821/0.3270

Table 1: Geometric errors

Results presented in Figure 13 have been realized with both vision systems (stereoscopic heads and monocular camera). These results give several informations. The first and most important conclusion is that the end-effector parasites rotations are very small with respect to

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mm

mm 2.5

mm

1.2

20

1 2

0.4 0.2 0

z position errors

15

0.6

y position errors

q3=0 q3=70 q3=140 q3=210 q3=280

x position errors

0.8

10

5

-0.2

1.5

1

0.5

-0.4 0

100

200 300 q1 (mm)

400

500

deg

100

200 300 q1(mm)

400

0.05

0

-0.05

-0.1

0

100

200 300 q1 (mm)

400

500

0

500

0

100

200 300 q1(mm)

400

500

0

100

200 300 q1 (mm)

400

500

deg-12.8

0.04 EE parasit rotation arround y axis

EE parasit rotation arround x axis

0

deg

0.1

-0.15

0

0.02 EE orientation fluctuation (deg)

-0.6

0 -0.02 -0.04 -0.06 -0.08

-13

-13.2

-13.4

-13.6

-0.1 -0.12

0

100

200 300 q1(mm)

400

500

-13.8

Figure 13. End Effector pose errors

the position errors and to p fluctuations. Thus it means that the robot conserves his 4 degrees of mobility. On the other hand, positions errors are bigger in the Y direction. End-effector errors are essentially due to deformations and geometric errors. In this paper only the self weight is applied as load. Errors in horizontal plane are due to geometric errors and pre-stress due to the hyperstaticity of the robot. Errors in the Y direction are caused by deflexions of legs L2 and L4 and assembly errors. But deflexion on L2 and L4 causes a localised deformation in the revolute joints between legs and actuators M2 and M4, so it introduces a new geometric error that amplifies the vertical position error. The p fluctuation is the result of the difference between both vertical legs deflections. Finally it is clear that the effect of the self weight is very important but it can be assumed as an amplification of the vertical legs orientation errors. Note these measurements were done on a prototype not totally achieved and before accurate adjustments.

7 Conclusions According to previous considerations we can get some answers to the questions addressed in introduction. The study showed that the robot is more sensitive to legs orientation then to actuators orientation. Moreover the dominating sensitivity is achived with respect to orientation errors of actuators M1 and M4 and legs L1 and L4. In addition, uncertainty increases with the articular coordinates. Finally in a kinematic study under just self weight loading, the rigid bodies hypothesis can be applied and deformations will appear as additional defect for the orientations of leg L2 and L4. Manufacturing and assembly errors will create pre-stress in the robot deformable structure. Their effect analysis in a stiffness study will require a second order finite elements analysis what will be presented in a next paper.

Acknowledgment: This work was supported by the TIMS Research federation (FR TIMS/CNRS2586) in the frame of auverfiabilis project of Regional Council of Auvergne.

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Grenoble, France, May 17-19, 2006

IDMME 2006

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