Model Development and System Identification of a Cartesian Manipulator Using a Laser-Interferometry Based Measurement System

Copyright © IFAC Mechatronic Systems, Sydney, Australia, 2004

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.\ 100EL DEVELOPMENT AND SYSTEM IDENTIFICATION OF A CARTESIAN MANIPULATOR USING A LASER-INTERFEROMETRY BASED MEASUREMENT SYSTEM

Prasan De Waas Tilakaratna, Bijan Shirinzadeh, Gursel Alici l

Robotics and Mechatronics Research LaboratOI)' Department olMechanical Engineering. Monash Unil'ersity Melbourne, Australia BijaI1.Shirin:::.adeh!(!?eng. monash. edu. {//I

A bstract: This paper describes the development and verification of a dynamic model for a Cartesian manipulator. Experiments and simulations are performed in order to determine the parameters of the established model. Optical shaft encoder feedback was proven to be unsatisfactory predominantly due to elasticity of the couplings and ball-nut backlash. As a result, a laser interferometry-based measurement system was used for position data acquisition. Experimental results highlight the occurrence of limit-cycles due to inherent f,'iction by the rails, ball-nuts and bearings. The derived model parameters were determined both by theoretical calculations and experimental response observation. COP.l'I'ight © 200-1 IFA C Keywords: Cartesian manipulators, Systems identification, Robot dynamics, Positioning systems, Position control, Servo systems,

I. INTRODUCTION accuracy and preCISIOn within the entire works pace comparatively to other manipulators. A crucial advantage of a Cartesian manipulator from a mathematical point of view is the decoupling of its position axes (i.e. first three axes). As a consequence, the associated kinematics and dynamics are substantially simple. Numerous studies are available on the design, control, modelling and simulation of Cartesian manipulators in the literature (Vaaler and Seering, 1985; Soroka, 1986; Benjamin et al.. 1985; Ossib and Hollis, 1989; Nelson and Chang, 1984; Callegari, 200 I; Lee, 1998; Meressi. 1998). Although the underlying concept is similar, even among the cantilevered and gantry style Cartesian robots, numerous variants are readily seen. A typical example would be the standard 3-DOF rigid gantry type manipulator and a three dimensional overhead crane (Lee. 1998). This diversity somewhat results in the lack of generality or application restrictions on the studies performed.

The work presented in this paper focuses on system identification for the utilisation in lower pair height ;ldjustment manipulations. A Cartesian manipulator is utili sed to provide a dynamic environment. Identification of its dynamic characteristics has been vital in numerous aspects. C;lrtesian manipulators can be categorised as Canti levered or Gantry-Style depending on the geometry of the manipulator. Their mobility varies from simple 3-DOF to 8-DOF configurations according to manipulation requirements and applications range from high precision multi-axis platforms to very large overhead cranes. They also pl ay a vital role in modern circuit board assembly lines predominantly due to the high accuracies achievable compared to anthropomorphic, SCARA, spherical or cylindrical manipulators. Another vital aspect of a Cartesian manipulator is its uniform

I

Gursel Alici is at University of Wollongong, Australia.

401

Three dimensional overhead cranes with suspension cables have been modelled (Meressi, 1998; Lee, 1(98). In the above investigations dynamic models have been proposed to predict and control the swing of the suspension cable.

Where, 111, p, and e are the load mass, leadscrew pitch and lead screw efficiency respectively . Torque due to friction, i.e. T,r'eI'"" is determined as follows .

r

/,..,1,,,,,

= pFld, .,,,,,,

2m:>

(3) The torque due to gravity, i.e. Tcranll is calculated as follows . T = fl F".,,,,, 21D.!

1.""'-'/1

(4) In the above equations, Fji'i,,,,,,, and F~''''''lIr are friction forces and gravitational forces respectively. Although torque due to the gravitational force is represented in the model, it is only applicable in the vertical axis (i.e. z-axis) provided that z-axis is perfectly vertical. It is also important to note that friction between the lead-screw and the ball-nut is not explicitly shown in the above equations due to the consideration of leadscrew efficiency e.

~

Fig. I. Cartesian Manipulator. In the next section we establish a mathematical model for the 3-DOF Cartesian manipulator which is investigated . We excite the system with step inputs and obtain its behaviour from optical shaft encoders and by a Leica LT500 laser tracker. At this point, the res ults obtained by different means are compared and analysed in order to justity the observed variations. We use the obtained data to determine system dy namic parameters where the obtained model is validated with the aid of simulations.

The manipulator exploited in this investigation can be considered as an inverted gantry robot or a platform as shown in Fig. I. It has three translational degrees of freedom where servo motors are being used in each axis in order to handle a heavy payload and to maintain smooth motion at low velocities. Lead-screws of pitch 5mm are being used in each axis with optical shaft encoders with an output resolution of 4000 counts per revolution in quadrature mode. The result is a system with a resolution of 1.25 microns in each axis. The servo system runs in current mode. AMC BE25AC amplifiers produced by Advance Motion Controls with current output monitoring capabilities are used . The overall setup is shown in Fig. 2.

2. MATHEMATICAL MODEL DEVELOPMENT Dvnamics

of s uch a Cartesian manipulator is between each axis. Static friction as well as viscous friction is included in the developed no nlinear models. A generic form of a second order model is developed as given below and subsequently this generic model is used to infer the models co rresponding to x, y and z axis.

ul~coupled

Sl4'arvi50ry

COfl'lPUtl!rlJ

I

Controller

z;;: r = (J 1• .,.1 •• "' ,,

\7

+ .J m .." ... + .J ,,,,,, ) B( t) + C', B(/)+ ,,', B(i) + T ."."", + T,"".,,, " (I)

\\there . .f'..ad'm''''' J.""",,. inCri'" and J,,,,,, ,ncrl/a are the moment of inertias of the leadscrew, motor and the load respectively. r is the torque applied by the motor. is the instantaneous rotation angle. C and f:,. correspond to rotational viscous damping and st iffness coefficients. T/ri("I,,,,, and r~,,,,,",,, are the torques due to coulomb friction and gravitational force.

e

Mono"

...... Mo1Dr

LJ I'

~' [] .,

c..: Fig. 2. Servo System.

./"'ad",.,." can be calculated or it is often provided by the lead-screw manufacturer. J"'''I'''' is the rotor inertia of the motor which is also a parameter provided by the motor manufacturer. The effective moment of inertia of the load . .f""", is determined as follows. /

Con~1 Cera

]

I

I !

MObr

U

I

Necessary inertial , motor and lead-sc rew parameters are shown in Table I. The efficiency e, of all leadscrew ball nut assemblies is 90%. The parameter C and K,. are not represented due to the inability of determining its value theoretically. However, these parameters corresponding to each axis will be inferred in the subsequent section as an integral part of the system identification process.

mp'

-~JT -:e

.. I,,"d_n·,-"· -

(2)

402

Table I Inertial, Motor and Leadscrew Parameters

X-axis Y-axis Z-axis

Mass (kg)

Gravitational Force (N)

J-Motor (kg_m 2)

Pitch (m)

J-Lead (kg_m 2)

Electrical Time Constant (s)

35

0 0 226

1.3 x 1004 1.5 x 10"4 3.0 x 10'5

0.005 0.005 0.005

8.37 x 10') 8.37 x 10. 5 8.37 x 10,5

0.0054 0.00362 0.0036 7

85 23

By using Eqs.2, 3 and 4 and Table I along with Eq . l, the following differential equations are generated for each axis. One can consider them as partially determ ined models.

The servo drives work in torque mode. Identification of servo drive characteristics which are different to each other was vital. The drives were energised with digital multimeters (DMM) connected at the driver inputs and the current monitor outputs. Motor shafts were shifted from desired positions manually at different levels in order to generate suitable control signals. Input/output voltage measurements were acquired from the DMMs. Fig. 3 depicts the input/output behaviour of the three servo drives . It is evident that the characteristics are linear.

O.0002383B,. + C,., B,+ K,., B = r, - O.0455sgn(B)

(5) IJ()00935B, + C, .,. B,. + K, .. ,.B, =

T,

-0. 111 sgn(B,)

(6) O.OOOI299B,+C,.,B,+ K,.. ,B, =T, -O.O I7 sgn(B,)

45

-01995 ;{

(7)

:;- 3.5

a

The constant term in Eq.7 is due to gravity. The signum function is utilised to characterise coulomb friction in the model. The friction coefficient between lubricated steel is considered to be 0.15 . Since the z-axis is guided through four linear bushings, a friction force of 20N was heuristically selected where coulomb friction is somewhat unsuitable. Parameters er.." C r.y , (','.:, K r.x , K r..I' and A:,..: are heuristically determined in the system identification section with the aid of simulations.

3

~ 2.5 ~

u

2

~

~ 15

x

05

1fllJ! IV)

4 .5

~ 35

Dynamics of the DC servo motor is modelled as first order. Only the electrical time constant and the torque constant are considered in the model. The time constants and torque constants for each axis are provided in Table 1. The generic transfer function for the 1110tor is represented by Eq. 9.

~

3

~ 2.5

(l

2

~

~ 15

,..

0.5

T(S)=~ its) T ,S+ I

lrp.Jt(V)

(9)

Where Eq.9, T, i, KT, and Te are the torque output of the motor, input current, torque constant and electrical time constant, respectively . 3. EXPERIMENTS In this section we establish suitable experimental procedures, present the experimental results, compare results obtained by different sensors and analyse discrepancies for the purpose of system identification. As depicted by Eq.I, it is acceptable to represent the system as a second order model provided that the system is adequately rigid thereby negating the influence of manipulator flexibilities. The primary goal is to establish experiments demonstrating the above fact which will pave the way to evaluate friction, stiffness parameters and modify/adjust other parameters as required.

Inl>Jt (V)

Fig. 3. Input-Output behaviour of the Servo Drives. The following gains are deduced from the above graphs shown in Fig. 3. Gain-X = 1.0223 AN Gain- Y = 0.8419 AN Gain-Z = 0.8434 AN

403

Step input experiments were conducted on all three axes. Position data was acquired via optical shaft encoders and a Laser 1nterferometry based Measurement System (LIMS) (Alici and Shirinzadeh, 2003; Teoh et aI., 2002). The LIMS measures absolute position of the end-effector relatively to the reference frame indicated in Fig. I. Initially. the system was excited by a step input of 0.25111111 (i.e. 200 counts). Encoder results for the x, y and z axes are shown in Fig. 4. The corresponding LI MS results are depicted in Fig. 5. It is also important to note that encoder data is represented as counts rather than mm . Since encoders produce 4000 counts per revolution, a count is equivalent to 1.25 x 10-.1 mm translational travel.

well as absence of stiction . A numerical analysis has been presented to predict experimentally observed limit cycles in a servo system. Townsend and Salisbury ( 1987) have analysed the effects of friction and stiction on force control with integral feedback. Limit cycles due to stiction can be averted by increasing the position gain while ensuring system stability (Townsend and SalisblllY, 1987).

L

300

,.

250

~

.--

---- . .. _-.- -------- ... ----.- . -.--_. .:-,: ';- S imu1 31 ~d Re spons e : : ,

,

I

,

::- f;fj~(h;; \ _~:r\:: -~i-j\j\i\tt::~>~;;t;~~:~~ -J~_ *tJ-V-\JU.----------- -- ---.------_l!_ J~L : V\,_~:ri~:~t~I:R-e:PO!l~i ------ -

~200

3

o

~1~O o

-=

~100

Cl.

w

x

'f x

50

OLI· ____

o

~

__

_200L-______

o

'"

~

____ 04

~

o2

~

_ L_ _ _ _

~

____

~

__

~

t 1

08

______

~

______

~

______

~

15

0.5

o ----------;-----------;---- -- -- -.. :-- ---- ---. -- --

N

_~ L-----~----~----~----~----~

o

0 .2

0 4 1l me (5) 05

Fig_ 5. Experimental Responses (counts)_

(L1MS)

0.8

and

Simulation

Data from the encoders are obtained at a rate of I kHz. The 4000 count encoders provide a resolution of 1.25 ~lm as indicated before. The sampling rate of the LIMS is 1 kHz and has resolution of 1.26 pm . As evident from Fig. 4 and Fig. 5. the L1MS data and corresponding encoder data is dissimilar. Due to similar data acquisition rates and resolution of the L1MS and encoders. it is apparent that the deviations are not due to encoder quantisation and aliasing. Vibrations in LIMS data and the damped nature in encoder data indicate the deviations are predominantly due to structural flexibility between the encoder and the end-effector. Inherent torsional flexibility in the helical couplings utilised is significant compared to lead-screws and is identified as the key source of flexibility in the system which causes response deviations_

Fig. 4 . Step Input Responses from Encoders (counts). Top plot in Fig. 5 clearly indicates the existence of limit cycles due to coulomb friction in the x-axis. This phenomenon has been investigated extensively III the literature (Olsson and Astrom, 2001; Townsend and Salisbury. 1987; Adams and Payandeh , 1995: Adams and Payandeh. 1996; Hensen and Van de Molengraft, 2002). Olsson and Astr0111 (2001) have shown that limit cycles of cl itferent types exist for different controller specifications. Their work illustrates numerous types of limit cycles that occur due to the presence and as

404

SGV

V"locity Convtrt.,

Fig. 6. System Block Diagram for the X-axis In this paper, our focus is limited to the end-effector response . Encoder data as shown in Fig. 4 clearly indicates the time delay of the system and an initial drop in the z-axis due to the gravitational force present. Pure time delay of the system (i.e. all three axes) is in the vicinity of3 ms.

Parameters of the z-axis model were investigated using a strategy similar to that used in the other two axes, although with the inclusion of torque due to the gravitational force, i.e. the axis being vertical. The settling time for the z-axis is 300ms. A steady state error is approximately 110 counts which IS significantly high.

4. SYSTEM IDENTIFICATION The established overall model is given by Eq . 1O. The utilised controller and the developed mathematical models are combined as shown in Fig. G. It is important to note that the torque required to overcome gravity is set to zero. However, this is not the case for the z-axis. A 11 components of the controller are depicted although only proportional control is utilised for the step input experiments.

0 .00028383 [

0 0 .0002935

0 0

o

0

11 .11001200

0 . 15 [

Parameters determined in the mathematical model development section are substituted into the model for each ax is separately. A value for the viscous friction coefficient is heuristically selected and the system is simulated with conditions similar to that of the real step input experiments performed . The simulation results, comparisons with the experimental results and parameter adjustments are comprehensively investigated for each axis in the remainder of this section. It is assumed that the system is continuous due to high servo update and sampling rates used and the pure time delay is negligible.

o o

][0,

o

o

0 .01

o

][0,] [

o o. = 1.19

0,

T,

0, +

[(1.007 0

0,

0 !l .1I1

()

T ,

-0 OI5sgn(0. )

T

-OOlsgn (O )

o ] I)

II ()OS

°0,]+ e.

j

-OOOlsgn(e ) -0 1995

(10) The simulated responses for the three axes in joint space are also shown in Fig. 5 along with experimental responses. It is quite evident by comparing the experimental and the simulated responses that the developed model matches the response very closely. The theoretically determined coulomb friction is different predominantly due to the deviation in the friction coefficient utilised. The x-axis limit cycles evident are seen in the relevant simulated response, although at a much lower scale. 4. DISCUSSION AND CONCLUSION

The ~n value was determined by matching the steady state error (i .e. 25 counts) perceptible in the experimental results. Similarly, an appropriate C .x value was determined in order to attain a settling time of 1000ms. Although a coulomb friction value of 0.0455 Nm was theoretically determined, the value was too high thereby slowing down the response dramatically . Hence, a coulomb friction value of 0.0 I Nm was selected. The most probable cause for this is an imprecise friction coefficient used for lubricated steel on steel contact.

In this paper we have presented the development and the verification of a mathematical model for a 3-DOF Cartesian manipulator. The mathematical model was developed from an abstract level with known parameters such as inertia, coulomb friction and gravitational forces. Simulations were conducted by developing a full model which includes the controller motor architecture, amplifier characteristics, dynamics, and non-linear coulomb friction. Each axis of the system was excited by a step input and simulations were performed under similar conditions . Then, simulation and experimental responses were matched thereby successfully determining the unknown parameters of the model and further fine tuning the known parameters. Limit cycles were observed in the x-axis, The inability to detect the

A similar strategy used for the x-axis was employed for the determination of y-axis model parameters. The mathematically determined torque due to friction was altered as was the case for the x-axis. The settl ing time for the y-axis is 500ms. The steady state error is approximately 4 counts.

405

actual response of the system via the shaft encoders was shown . This is predominantly due to the tlexibility of the helical couplings and backlash of the ball-nuts. The observed time delay is negligible. Steady state errors present in the experimental and simulated responses can be minimised by utilising more appropriate controller strategies. Future work includes the investigation of response variations at the encoder and the end-effector, and implementation of dynamic model-based control. ACKNOLEDGEMENTS This study has been partly funded by Australian Research Council (ARC) and Monash University Grants. REFERENCES Adams, J. and Payandeh. S. (1996) . Methods for low-velocity friction compensation: theory and Journal of Robotic experimental study. .sy.\·tems. Vol. 13, pp. 391-404. Adams . J. D. B. and Payandeh, S. (1995). Experimental evaluation of low velocity friction in robotics . compensation techniques h·oceedinR.I' of the 199j IEEE International Conf'erence on Systems, Man and Cybernetics. Vol. 2, pp. 1705-1710, Vancouver. Alici, G . and Shirinzadeh, B. (2003). Laser Interferometry Based Robot Kinematic Error Modelling and Compensation. Proceedings of the l003 IEEDRSJ International Conference on Il1IelligeJ71 Rohots and Systems, Vol. 3, pp. 35883593. Las Vegas. Benjamin, M., Garcia-Reynoso, A. and Seering, W. (1985). Dynamic and vibration modeling of a cal1esian robot. Proceedings of the 198j IEEE Intemationul Conference on Roholics and ..Jlltomalion, Vol. 2, pp. 990-995, Callegari, M. c., F.; Monti, S.: Santolini, c.; Pagnanell i, P. (200 I). Dynamic models for the reengineering of a high-speed Cartesian robot. P/'IJceedings of the 2(}Ol IEEElASME International Conference on Advanced Intelligenl ;'vfeci7atronics, Vol. 1, pp. 560-565.

406

Hensen, R. H. A. and Van de Molengraft. M. J. G . (2002). Friction induced hunting limit cycles: An event mapping approach. Proceeding of the 2()()2 American Control Conference, Vo!. 3, pp. 22672272, Anchorage. Lee, H.-H. (1998). Modeling and control of a threedimensional overhead crane. Transa ctions ASME, Journal of Dynamic Systems, Measurement and Control. Vo!. 120, pp. 471-476. Meressi, T. (1998). Modeling and control of a three dimensional gantry robot. Proceedings of the 1998 IEEE Conference on Decision and Control, Vo!.2 , pp.1514-1515. Nelson, W. and Chang, J. (1984). Simulation of a cartesian robot arm. Proceedings of the 1984 IEEE International Conference on /?ohotics and Automation, Vo!. I, pp. 212-219. Olsson, H. and Astrom, K. J. (2001). Friction generated limit cycles. IEEE Transactions on Control Systems Technology. Vol. 9, pp . 629-636. Ossib, Z. and Hollis. P. J. (1989). The design and control of a micron-accurate robot arm, Proceedings of the TlIlenty-Firsl Southeastern Symposium on :::'ystem Them),', pp. 553-556 . Soroka, B. M., R. (1986) . Programming and Simulating a Three-armed Cartesian Robot. Proceedings of the 1986 IEEE International Conference on Robotics and Autom(l{ion, Vol. 3, pp. 1766-1771. Teoh. P. L., Shirinzadeh, B., Foong. C. W. and Alici . G. (2002). The measurement uncertainties in the laser interferometry-based sensing and tracking technique. Measllrement : Journal of the International Measurement Confedemtiol1. Vo!. 32, pp. 135-150. Townsend, W. T. and Salisbury, J. K. J. (1987). Effect of Coulomb Friction and Static Friction on Force Control, Proceedings of the 1987 IEEE Internalional Conference on Robotics and AII/omalion. pp. 883-889, Raleigh . Vaaler, E. and Seering, W. P. (1985) . Design of a Cartesian Robot, Proceeding of Robotics and .Manufacturing A ulomalion, Vol. 15. pp. 163-168. Florida.