7/26/2019 Simulation of a Cartesian Robot Arm

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SIMULATION

OF

A CAR TESIAN ROBOT ARM

W.

L Nelson and

J .

D

Chang

Bell Laboratories

Murray Hill , New

ABSTRACT

This paperdescribesaprogram or the simulation of

the control logic and ystemdynamics of a robot ar m having three

Cartesian axes of motion plus wo rotat iona laxes of the wrist. T he

program was written in the C languageand includesa fourth-order

Runge-Kuttalgorithmorhentegratio n of the nonlinear

differential equations for the individual axes. Validation of the

program operation and evaluation of the many system param eters are

.also discussed. The sim ulat ed motion is compared and adjusted o fit

experimental data on the actua l arm motion. The results indicate that

th e simulation can be a useful and safe means or off-line testing of

system chang es and new control schemes.

I.

INTRODUCTION

Computerimulations of the ynam ics of robot arm s re found

extensively in th e l i ter at~ re. ' -~ Th e dvantages of having a program

that accu ratel y reflects a robot's d ynam ics are gre at: having a virtual

second robot allows one to try expe rimental control algorithm s without

the danger of hurting the robot

or

personnel.

Th e simulation of the dynamics of the Automatix AID 600 robot was

wri tten in the C anguage, closely ollowing the orm of aprogram

previously written by

M

K Brown for the simulation of aobot

gripper.5

11 DESCRIPTION OF DYNAMICS

T h e A I D

600

robot arm has Cartesian X Y Z axes of motion,plusa

wrist with wo-axis revolute pitch and roll motion. Th e orientation of

these axes with respect to the robot ar m is shown n Fig.

1.

The gross

dynamics of the robot a rm can besimplifiedby assuming the X Y Z

axes uncoupled a nd the wrist a constant-m ass payload with negligible

coupling effects, allowing the individual axeso be simulated

independently of each other.The existingprogramconsiders the X

axis; suitable substitution of constants can yield models for the and

Z

axes.

Figu re 2 is a system block diagra m for one of the robot axes.

In

this

figure, J,

is

the m oment of inertia of th e axis servo-motor, a nd B,

is

the motordamping.The ineardynamics of the robot axis canbe

described by th e two tate variables position, yo, nd velocity yl.

Because the effects of motor gear backlash are o be examined, the

nonlinear effects of th e coupling a nd uncoupling of t he ack nd

pinion system of the arm andmotor gearare described with two

additionalstate variables: gear angle, y2,andgearangular velocity,

y3, with the drive force transmitted hrough he stiffness and friction

of the gear train when the t eeth a re in contact.

Within hevoltage aturation imits, hedriveractsasacontrolled

current source with transcondu ctance Ki, as maintained by a current

feedback loop n the controller circuitry. When the motor reaches the

voltage saturationimit,t an e haracterized by aingle-pole

inductance-res istance model with aback EM F proportional to he

motor angul ar velocity. The m otor is driven by acontroller with an

inpu t for a reference (set-velocity) voltage, and an inverting inpu t for

a tachom eter feedback voltage. Th e controller has aswitchingmode

power drive unit, preced ed by a preamplifier consisting of a ag-lead

212

Iersey 07974

(pole-zero)ompensation etwork, an overall frequency rolloff at

about

1000

Hz, andan outpu t voltage limit, V,. The preamplifier's

two poles and one zero can be described by the linear combination of

twoauxiliary stat e variables,

y4

and y5. The nonlineardynamics of

the motor-controller are represented by the motor current, which is

the state variable, y6.

T h e AID 600 slave control unit computes the set-velocity v oltage for

each movementxisn the high-level comm andoftware, using

informationrom position andpeed-schedulenputs,markeds

'track' data nput inFig. 2 plus eedback data fromoptical encoder

on

the motor shaft.The set-velocity voltage outp ut is modeled asa

sam pled- data device f eriod

5

ms; t he position feedback ptical

encoder in put also has a period of 5 ms. The input of new track data

from themaster controlunitoccurs at 80 ms intervals. The control

logic sed in the slave control nit is a ombination of position,

velocity, andnteg rated position functions, as shownn the lower

portion of Fig. 2. In summary,he dynamics of the closed-loop

control system for a single axis of the robot ar m motion is represented

by the seven state variables:

yo ar m position (m.)

y ,rm velocity (m/s.)

y2 motor anglerad.)

y3 motor velocity (rad /s.)

y4 controller reamp. ariable (volts)

ys

controllerpreamp.variable 2 (volts/s.)

y6motor current (amps.)

Four auxiliaryvariables, which ar e functions of thestate variables,

are: fg, the force coupled throug h the gear drive in Newtons (N), F,,

the tatic friction orce

N),

V,, the motorvoltage, and, V,, the

controller preamp. output voltage.

In terms of these variables, the differential equations of motion are:

o =

Y1

(1)

Y2 = Y3

Y4 = Ys

where B, is the viscous friction coefficient and M the mass of the

arm-rack mechanism; B, is the rot atio nal friction coefficient,

J,

the

inertia,

K,

the torque constant of th e motor, and r the radius of the

pinion gear. The controller param eters re he compensation pole

time constant,

T

the high frequency roll-off time cons tant,

T

nd

CH2008-1/84/0000/0212 01.0001984IEEE

7/26/2019 Simulation of a Cartesian Robot Arm

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the signal and 2 input gains,

K,

and

K,.

The controller nputs are

the set-velocityvoltage,

U(t),

and he achometer eedbackvoltage,

which is

Kt

times the m otor angular velocity, y,(t).

T he controller preamplifier output voltage is given by

Vc = y4 T,YS IVcI

G

VL (2)

As

is indicated

in

Fig. 2 , hemotorvoltag e is proportion al o he

preamplifier output minus the current feedback, i.e.,

V, =

K,(Vc y&i) 9

IV,I

< V,

1

(3)

where K, is the orward gain f the power nit nd Ki is the

conductance of thecurrent eedbackpath. When

K,

is sufficiently

large,nd when lVml

H

f8 = (x+H)k, + (rgy3-yl)bg x < -H 5 )

0

- H < x G H

Th e axis-load rictionhas wo omponents.First, viscous riction

proportional o elocity,which is reasonably well modelled by the

linear force, Bayl; and second, a static friction, F,; which is modelled

as a force doublet

in

the region of zero velocity, i.e.

F , s g n [ ~ , I , l y l l