Robotics 2 E Part 7 pdf

loca-Electrical sensors The simplest displacement sensor is a potentiometer: a variable electrical resistor in which the slide arm is mechanically connected to the moving link.. The elec

FIGURE 4.65 Oscillation amplitudes al ofthe mass M versus ratio k2/m (Example).

FIGURE 4.66 Vibration amplitudes of mass M versus ratio

k2/m and time during 5 seconds of the process (Example)

FIGURE 4.67 Layout of a DD device

FIGURE 4.68 Photograph of one of the DDs used

in our experiments

FIGURE 4.69 Layout of an active damper Force P changes

depending upon the free vibrations of the mass m.

FIGURE 4.70 Oscillation amplitude versus frequency of P and

time during the first 5 seconds A "valley" of almost zero

amplitudes at freo^iency about co = 18 I/sec is clearly seen.

FIGURE 4.71 Active damping force generator.

1) Oscillating mass; 2) Core of the magnet;

3) Coil of the magnet; 4) Armature

FIGURE 4.72 Comparison of the free oscillation of mass M

(computation) without (damping takes about 20 sec) and

with actuation of the AD (damping takes about 10 sec)

force is applied to the mass M Obviously, the bigger the mass of the armature 4, thebigger the force

The core 4 is fastened to the arm of the manipulator (or any other object)

An example of a comparison of the vibrations damping processes is shown in Figure4.72 One process, taking about 20 seconds, is calculated for a usual system, withoutany artificial damping means, while the other, taking about 11 seconds, is the result

of AD use

A special control system that carries out all signal transformations must be usedfor this method Its general layout for one control channel is shown in Figure 4.73 Theaccelerometer and the active damper are placed on the end of a robot's arm The signal

FIGURE 4.73 Layout of the proposed AD system

from each accelerometer is doubly integrated and amplified Thereafter, the obtained

power signal enters the active damper where it generates the force P(t) required for

damping

This latter idea of electrically controlled damping being designed to suit differentand various mechanical systems (including manipulators) seems to be a very fruitfulmeans for increasing accuracy of automatic manufacturing machines The main advan-tage of this idea is the possibility to interact between the mechanics and the controlelectronics or computer This kind of interaction recently has been given the name

mechatronics.

Exercise 4E-1

For the mechanisms shown in Figure 4E-1 a) and b), write the motion functions

y = n(.x) and y f = n'(jt), respectively.

For case a) calculate the speed y and the acceleration y of link 2 when x = 0.05 m, x

= 0.1 m/sec, x = 0, and L = 0.15 m, and the force acting on link 1 to overcome force

F= 5N acting on link 2.

For case b) calculate the speed y and the acceleration y of link 3 when 0 = 30°,

0 = 5 rad/sec, 0 = 0, AO = 0.2 m and ACIAB = 2.

Exercise 4E-2

A cam mechanism is shown in Figure 4E-2 The radius of the initial dwelling circle

is r0 = 0.08 m The follower moves along a line passing through the camshaft center O

(i.e., e = 0) The law of motion of the follower y(0) is given by:

FIGURE 4E-1a)

FIGURE 4E-1b)

FIGURE 4E-2

During rotation for 9 = 45°, the cam's profile completes the displacement of the lower for a distance h Calculate the maximum allowed value h which provides the condition where the pressure angle a does not exceed the permitted value amax = 20°;calculate the profile angle 0* at which the pressure angle becomes worse

to them The sensors can be divided into two main groups: analog and digital To thefirst group belong those sensors that respond to changes in the measured value bychanging some other physical value in their output, say, voltage, resistance, pressure,etc In contrast, digital sensors transform the measured value into a sequence of elec-trical pulses Information is carried encoded as the amount of pulses (say, the higherthe number of pulses, the larger the measured dimension), as the frequency of pulses,

or as some other pulse-duration parameter The amplitude of the pulses usually has

no importance in information transmission

5.1 Linear and Angular Displacement Sensors

The most common task of a feedback is to gather information about the real tions of robot or machine links using, for example, sensors that respond to displace-ment or changes in location There are several kinds of these sensors, some of whichwill be considered here

loca-Electrical sensors

The simplest displacement sensor is a potentiometer: a variable electrical resistor

in which the slide arm is mechanically connected to the moving link Thus, the tance changes in accordance with the displacement The electrical displacement orlocation sensors are usually a part of an electrical bridge, the layout of which is shown

resis-175

in Figure 5.la) When a constant voltage V 0 is introduced, the off-balance voltage AV

can be expressed as follows:

There are several methods to use these bridges For instance, keeping the

resis-tances R-L, R 2 , and R± constant so that R l = R 2 = R^ = R and using the resistance R 3 as asensor, i.e., a variable resistor responding to changes in the measured value, we canrewrite Expression (5.1) as

Substituting here R 3 = R + Aft, where AR is a small change of the resistance, so as AR«R we obtain, from (5.2),

In the simplest case, the displacement (or the measurement of some dimension)

is transformed directly into displacement of the slide arm of the resistor Thus, asfollows from Relation (5.3), the change in the output voltage AV across the bridge'sdiagonally opposite pair of terminals a-a is directly proportional to the displacement(for small displacements) However, it is possible to increase the sensitivity of the bridge

by using a so-called differential layout, as shown in Figure 5.1 b For this case, by stituting the following in Expression (5.1),

sub-we obtain

FIGURE 5.1 Layout of an electrical measurementbridge: a) Common circuit; b) Differential circuit

This concept of a bridge feedback can be realized in a design such as that shown

in Figure 5.2 This layout is called a compensating bridge Here resistors R^ and R 3 are

variable The slide arm of resistor R l indicates the location of cutter support 1 driven

by motor 2 via screw drive 3 The slide arm of resistor R 3 is connected to feeler 4 whichtraces the program template 5 (master cam) fastened onto carrier 6, driven by motor

7 via screw drive 8 Thus, when resistance R 3 changes its value due to the template'sdisplacement, the balance of the bridge is disturbed and voltage AV" occurs on theoutput of the circuit This voltage is amplified by amplifier 9 and actuates motor 2,

which moves the cutter so as to change the value of resistance R 1 until the imbalance

of the bridge vanishes Thus, motor 2 compensates for the disturbances in the circuitcaused by motor 7 From Expression (5.1), by substituting J?x = R + AR and R 3 = R-AR

while R 2 = R 4 = R, we obtain

Assuming AR«R this can be rewritten as

The accuracy of such sensors is not high, about 0.5%, and absolute values of about0.25 mm can be measured When the resistors have a circular form, angular displace-ments can be measured

Sometimes a sensor that gives a functional dependence between the rotation andoutput voltage is required Figure 5.3 gives an example Here, bases 1 are wound withhigh resistance wire 2 so that subsequent winds touch one another Arm 3 is able torotate around center 0 The function this device provides is

Figure 5.4 shows a rotating resistance sensor that produces a trapezoidal relation betweenthe angle and the output voltage Here 1 is a resistor, 2 is a conductor, and 3 is a slide

FIGURE 5.2 Electrical bridge used forfeedback in tracking machine

FIGURE 5.3 Resistance sensor formeasuring angular displacementswith a harmonic relation betweenthe measured angle and the outputvoltage.

arm The resistance wire must be wound uniformly to provide linearity during the

appropriate rotation intervals The angles 2a 0 are made of high-conductivity material.Much higher sensitivity can be achieved by using variable-induction sensors (alsocalled variable-reluctance pick-ups) The layout of the simplest of this kind of sensor

is shown in Figure 5.5 It consists of a core 1, coils 2, and armature 3 The coils are fed

by alternating current with a constant frequency CD The alternating-current resistance

Z in this case can be expressed in the form

where R = ohmic resistance, and X L = inductive reactance The latter is described as

where L = inductance of the system For the layout in Figure 5.5 this parameter isdescribed by the following formula:

FIGURE 5.4 Resistance sensor for measuring angular displacements with a

trapezoidal relation between the measured angle and the output voltage

FIGURE 5.5 Layout of an inductiondisplacement sensor.

where

// = magnetic permeability,

Q = cross-sectional area of the core (Q = a • h),

a,h = the dimensions of the cross section of the magnetic circuit,

W= the number of winds,

8 = the width of the gap.

We assume here (to make the formula simple) that the cross-sectional areas of the coreand armature are equal, as are the materials of which they are made Obviously, thegap can be represented as the following sum:

where § 0 = initial gap and x = the measured displacement.

Substituting (5.11) into (5.10) and the latter into (5.9), we see that (5.8) is a tion of jc

func-A more complicated design for an induction sensor is shown in Figure 5.6 Thisdevice consists of housing 1, made of ferromagnetic material with a high magneticpermeability, which constitutes the core of the sensor Two coils 2 and 3 generate the

FIGURE 5.6 Differential induction sensor for displacementmeasurements Cross-sectional view

magnetic flux Armature 4 is mounted on rod 5, which serves as a pick-up for the placement jc (The rod is made of a nonmagnetic material.) The magnetic flux is dividedinto two loops going through the coils and armature 4 The length of the armature'ssections in each loop determines the inductive reactance of each coil Thus, these coils,which are a part of a bridge, change its balance (as in the case presented in Figure5.1b)) Induction sensors are usually limited to measuring ranges not larger than, say,15-20 mm However, the accuracy is on the order of 10 3-104 mm.

dis-Another useful modification of an induction position sensor is shown in Figure 5.7.Here a lead screw 1 with a certain pitch (large enough to suit the design) and profileinteracts with an induction pick-up 2 The alternating current resistance of its coil 3depends on the relative position (see the above explanations) of the thread and thepoles of the magnetic core Thus, fractions of the screw's revolution can be measured.This design is thus made very effective

The next kind of sensor we consider is the variable-capacitance pick-up The bridgelayout of such a sensor is shown in Figure 5.8 The capacitances C of gaps A and B aredescribed by the following expressions, respectively:

where,

s = dielectric permittivity,

S = area of the capacitor's plates,

S = initial gap between the plates, and

Thus, a circuit without a choke (X L = 0) has an alternating current resistance Z:

where the capacitance Cis calculated from (5.12) for each gap, and co - the frequency

of the alternating current

The sensitivity of this layout and sensor is high and can be estimated about 10~4

mm However, the measuring range is small

Specific optical effects can be used as the basis of a very powerful displacementmeasurement method We will briefly describe the principle of a Michelson interfer-ometer that can be applied for accurate displacement determination in industrialsystems where machine elements must move with high precision Interference resultsfrom the algebraic addition of the individual components of two or more light beams

If two of the light beams are of the same frequency, the extent of their interference willdepend on the phase shift between them In Figure 5.9 we show the layout of an inter-ferometer for precision measurement of the location of some machine element 1 Thisdevice consists of a laser light source 2 and two mirrors 3 and 4, which are fastened tothe moving element 1 and the base, respectively There are also a ueain splitter 5, atransparent plate 6, and a signal detector 7 The beam splitter 5 is usually a plane-parallel transparent plate of appreciable thickness, bearing a partially reflecting film

8 on one surface, which divides the light from source 2 into two beams One beam verses the splitter and strikes mirror 3, placed normal to the beam, and then returns tothe splitter where part of it is reflected and enters detector 7 The other beam is reflected

tra-by mirror 4 and part of it is transmitted tra-by the splitter to the detector This latter beamserves as a reference to which the beam reflected from moving mirror 3 is compared(mirror 4 is strictly immobile) Because of the interference due to the phase shift occur-ring between these two beams, the detector obtains (and processes) information aboutthe movement of mirror 3 (and element 1) It is easy to see that the beam striking mirror

3 traverses the thickness of the splitter three times before entering the detector, whereasthe beam reflected from mirror 4 traverses it only once Although this plate does notalter the direction of a ray passing through it, it shifts it laterally and introduces addi-tional path length To correct for this, a second plate 6, identical to 5 except that itbears no partially reflecting film, is placed in the path to mirror 4 and parallel with 5

FIGURE 5.9 Layout of aMichelson interferometer forprecise positioning

This plate is called the compensating plate, and it is easy to see that the paths of thetwo beams are then identical as regards their passage through refracting plates.The device described above is named for its inventor, Michelson Its accuracy isvery high—about 0.0001 mm The detector obtains information as a sequence of brightand dark fringes Thus, the system works in a digital mode, by counting the fringes.This kind of feedback measuring device, because of its high accuracy, is practically theonly solution for automatic robotic machines in manufacturing integrated circuits.Another optical sensor for displacement measurement is based on photosensitiveelements, for example, as shown in Figure 5.10a) The element, the location of which

is to be measured, is provided with a transparent scale 1, beaming a grating of parent and opaque stripes 2 The scale is illuminated with a parallel light beam obtainedvia condenser lens 3 from source 4 The shadow of the scale is projected onto reticle

trans-5, which has an identical grating 6 Obviously, the amount of light going through reticle

5 at any instant is an almost linear function of the position of the scale This light isdetected by photocells 7 and transformed into digital electric signals, which are countedand translated into distances

A problem arises when the direction of movement must be distinguished For thispurpose an auxiliary grating is placed on the same scale This idea is illustrated in Figure

5 lOb) Line 1 is the main grating while line 2 is the auxiliary one When the scale movesrightward, the two gratings produce a sequence of pulses in which an auxiliary pulse

comes atT-r after every pulse from the main grating Conversely, when the scale moves leftward, this time interval equals r Thus, for T- T * r, the system can distinguish the

displacement direction The sensitivity for the described system is about 0.01 mm.The principle of the device shown in Figure 5.10 can easily be transformed for mea-surement of rotation Such rotation encoders are widely used in machine tool andmanipulator designs

Photoelectric cells also permit creating analog-type displacement sensors One sible example is shown in Figure 5.11, the so-called optical wedge This device is a pho-

pos-FIGURE 5.10 Digital optical displacement sensor: a) Layout of the

sensor; b) Layout of the grating used for determining direction

FIGURE 5.11 Analog optical displacement sensor.

toelectric element that has a variable response to illumination along its surface When

a diaphragm 1 with a narrow slit 2 moves along optical wedge 3, the latter's responsecorresponds to the relative position of these elements It is not a highly sensitive device.However, there are situations where it is appropriate

Pneumatic sensors

We now consider pneumatic sensors The basic model we will consider is shown

in Figure 5 12 Its main elements are the nozzles in sections I and II Let us considerthe continuity of flow through these nozzles, which is described in the following form:

Here,

a lt a 2 = coefficient of flow rates in sections I and II, respectively,

fi> fz = cross-sectional areas of the nozzles I and II,

p = density of the gas, assumed to be constant,

V lt V 2 - velocities of gas flow within I and II,

H=working pressure before nozzle I,

h = working pressure before nozzle II.

Now we make some assumptions: first, that the gas density in the two sections is tically equal; second:

prac-FIGURE 5.12 Layout of pneumatic position sensor

Thus, we can express the gas velocity within each nozzle as it is accepted for droppingliquids:

Substituting Expressions (5.16) into Equation (5.14), we obtain

From Equation (5.17) and the second assumption it follows that

The area/j is obviously

We assume that the area^ can be calculated for the model given in Figure 5.12b) in arelatively simple way In this scheme 1 is an enlarged diagram of nozzle II (see Figure5.12a)) and 2 is the surface of an element, the position of which is to be measured The

distance between this surface and the face of the nozzle is s Experiments show that,

if it is true that

or

then for f 2 we can use the formula for the area of the side of a cylinder, namely:

Substituting (5.21) and (5.19) into (5.18) we obtain

The latter formula shows the dependence of the pressure h on the distance s Below

are shown some examples of the use of pneumatic measurements of distances andlinear dimensions The main advantages of this kind of sensor are:

1 The possibility of carrying out the measurements without direct mechanicalcontact between the sensor and the surface of the checked element, if necessary

2 The relatively high sensitivity of this method, which is about 0.001 mm or evenbetter

These advantages permit, for instance, carrying out the checking of dimensionsduring rotation of the measured part, saving time and money