AN ANALYSIS OF TEMPERATUREINDUCED ERRORS FOR AN

ULTRASOUND DISTANCE

MEASURING SYSTEM

by

David Paul Wenger

Rensselaer Polytechnic Institute

Electrical, Computer, and Systems Engineering Department

Troy, New York 12180-3590

December 1991

CIRSSE REPORT #106
AN ANALYSIS OF TEMPERATURE-INDUCED ERRORS

FOR AN ULTRASOUND DISTANCE MEASURING SYSTEM

ty

David Paul Wenger

A Thesis Submitted to the Graduate

Faculty of Rensselaer Polytechnic Institute

in Partial Fulfillment of the

Requirements for the Degree of

MASTER OF SCIENCE

Approved:

Dr. Robert B. Kelley

Thesis Advisor

Rensselaer Polytechnic Institute

Troy, New York

December 1991
CONTENTS

Page

LIST OF TABLES .................................................... iv

LIST OF FIGURES ................................................... v

ACKNOWLEDGMENTS ............................................. vi

1. INTRODUCTION ................................................. 1

1.1 Motivation ................................................ 1

1.2 General Problem Solution .................................. 1

1.3 Outline of Thesis ........................................... 3

2. BACKGROUND .................................................. 4

2.1 Ultrasound ................................................ 4

2.2 Speed of Sound in a Medium ................................ 5

3. SOLUTION AND ANALYSIS ...................................... 9

3.1 Problem Solution ........................................... 9

3.2 Sensitivity Analysis of Solution ............................. 11

3.3 Reducing Slant Range Measurement Errors .................. 18

4. EXPERIMENTAL RESULTS ....................................... 21

4.1 Description of Experiment .................................. 21

4.2 Results of Experiment ...................................... 22

5. PROTOTYPE COLLISION AVOIDANCE SYSTEM .................... 24

5.1 Prototype System Design .................................... 24

5.2 Prototype System Accuracy .................................. 26

ii
.

5.3 Improving System Design .................................. 30

CONCLUSIONS .................................................. 32

6.1

6.2

Summary and Conclusions ................................. 32

Suggestions for Further Research ............................ 33

7. LITERATURE CITED .............................................. 35

iii
LIST OF TABLES

Table 2.1

Table 3.1

Table 5.1

Page

Vapor Pressure of Water ................................ 7

Relation Between ciT and dL Showing the % Error in L ..... 16

Reverberation Time For Rooms of Different Sizes .......... 29

iv
Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 5.1

LIST OF FIGURES

Page

Geometrical Setup of Solution ........................... 10

Relation Between dc and dT ............................. 12

Relation between Time of Flight and Speed of Sound ....... 14

Setup to Calibrate Speed of Sound Value in 2- and 3-Space ... 19

Minimal Setup of Ultrasound Transmitters and Receivers .,. 25

V
ACKNOWLEDGMENTS

As I complete my work on this project and review the past year spent

on it I begin realize the number of people who have supported me in so

many ways. I would firstlike to thank Dr. Robert Kelley for his guidance and

analytical technique which led me to this project and taught me the

importance of thoroughness. I also want to thank Andrew Silverthorne for

his interest in the work and his willingness to go out of his way to help collect

the ultrasound data. I would also like to thank my very good friends Keith

Fitch and Brian Mitchell who both played key roles in motivating me to

continue on with graduate studies. Additionally, I give my thanks and love

to my entire family who, throughout my life,have encouraged me to be the

best I can; especially my parents who, through their own lives,have taught

me to respect, as well as utilizeto its fullest,the mind which God has given

me. Finally, I need to express my fullest and sincerest love to my wife,

Monica, who was so giving, supportive, and understanding through itall.

vi
CHAPTER 1

INTRODUCTION

Motivation

As an aid to the Center for Intelligent Robotic Systems for Space

Exploration's (CIRSSE's) dual PUMA 560 robot arm testbed at Rensselaer,

research concerning the amount of error in ultrasound distance

measurements has been carried out to enable further research and design of

an arm collision avoidance system. An external collision avoidance system,

which does not depend on internal arm location data from encoders,

provides a redundant method for determining arm locations and therefore a

more reliable method for avoiding collisions.

Using a time-of-flight method for measuring the distances between

ultrasonic transducers mounted on the robot arms, the 3-space location of

those transducers, and hence, the distance between the arms can be

determined. Determining the bounds of the error in the distance

measurements enables the determination of how accurately the 3-space

locations on the arms can be known, and therefore how efficiently and exactly

a collision avoidance system can determine if a collision will occur.

General Problem Solution

Determining the 3-space location of the arm joints in the testbed can be

accomplished through the use of ultrasonic transducers. Each arm joint of

interest has a transmitting unit associated with it which transmits an
ultrasonic tone burst. The tone burst is received by a grouping of ultrasonic

microphone receivers set up in a known configuration. Based on the time

during which the tone burst travels from the transmitter to the receiver, a

slant range can be determined by the simple relation D = ct where D is the

slant range between the transmitter and receiver, c is the speed of the sound

wave in air, and t is the time of flight necessary for the received tone burst to

travel from the transmitter to a given receiver.

Using a 3-space Cartesian coordinate system a minimum of three slant

range measurements are required to determine each 3-space location. For the

most basic collision avoidance system, at least two sets of measurements, each

taken with respect to a given point on each arm, are required. Finding the

distance between these two points determines the proximity of the arms and

can alert a collision avoidance system should the distance fall below a

predetermined threshold. More accurate and robust collision avoidance can

be accomplished by placing an individual transmitter at additional points of

interest on each arm. Determining the 3-space locations of these additional

points of interest, and taking the distances between these points provides

better knowledge of the location of each arm with respect to the other and

allows a more accurate collision avoidance system to be developed.

In order to avoid time- and/or frequency-multiplexing issues, and as

the main intent of this research is to determine error bounds on the accuracy

of ultrasonic 3-space measurements, emphasis is placed on accurately

obtaining only one set of slant ranges. An ability to determine the upper

bound on the measurement error leads to the formulation of methods to

minimize the inaccuracy of ultrasonic distance measurements. The results of

this work can then be extended to the design of a fully functional collision
avoidance system which accurately obtains multiple sets of slant ranges for a

more accurate determination of collision danger.

Outline of Thesis

The presentation of research is provided in the following five chapters.

Chapter 2 presents the necessary background information and definitions for

general work with ultrasound and acoustics. It also discusses the basis for

errors in the slant range measurements. Chapter 3 presents a method of

problem solution and an analysis of the sensitivity of the equations to slant

range measurement errors. It also presents various methods by which the

error in the slant range measurements can be reduced to improve overall

measurement accuracy. Chapter 4 provides a description of a type of

experiment used to test the analytical solution and provides a discussion of its

results. Chapter 5 discusses the setup of a prototype collision avoidance

system, discusses its accuracy, and demonstrates various methods of

improving the accuracy along with the improvements' ramifications.

Finally, Chapter 6 provides a summary of the work and a discussion of

conclusions drawn from it. Additionally, suggestions for further research are

made to improve upon what has been presented here.

3
CHAPTER 2

BACKGROUND

Ultrasound

In order to accurately measure distances for the problem at hand, using

non-mechanical measurement methods, the simple time of flight principle is

utilized

D=ct (2.1)

where t is the time necessary for a wave traveling at speed c to traverse a

distance D.

Several options exist for the design of a non-mechanical measurement

system, and depend solely on the type of wave chosen to be emitted to

measure its time of flight. One option is to use wave emissions in the electrooptical spectrum; however, since they travel at the speed of light, 2.1 shows

that their time of flight over short distances where c _ 3 x 108m/s will be on

the order of 10 -12 seconds for mm accuracy and may be difficult to measure

accurately. On the other hand, if acoustic waves are used, which travel at the

speed of sound (c ,_ 340 m/s), the time of flight measurements are easier to

make as they require timing accuracy on the order of 10 -6 seconds for mm

accuracy.

Acoustic waves consist of both audible and inaudible frequencies. The

human ear is only able to detect sounds in the frequency range of

approximately 20 Hz to 20,000 Hz; humanly inaudible frequencies emitted

above 20 KHz are defined as ultrasonic frequencies. Ultrasound is a useful

4
tool for measuring distances mainly because its frequencies allow time of

flightmeasurements to be made more easily than electro-optical frequencies,

and also because of itsinsusceptibility to ambient noise within its acoustic

environment. This study proposes to determine how accurately ultrasound

can be used to compute distances via the time of flightmeasurement method.

While multiple factors affect the propagation of a sound wave through a

given medium, emphasis isplaced on those factors which most directly limit

the accuracy of ultrasound-based distance measurements.

Speed of Sound in a Medium

The expression for the speed of sound in air iscommonly given as

c = (_/ p)i/2 (2.2)

where p isthe density of the medium, and _ is the adiabatic bulk modulus of

the medium[l]. _ is defined as:

= (Y- 0.1e)P (2.3)

where 7 isthe ratio of specific heats (1.402in air)and e is the vapor pressure

within the medium[l][2].

The speed of sound depends on many factors. Changes in temperature,

relative humidity, and pressure all affect the speed of sound within an

environment. For the purposes of this study, pressure changes will not be

studied as the assumption is made that pressure is constant throughout the

medium and therefore induces no changes in the speed of sound.
Changes in the speed of sound due to changes in the medium's

temperature arise from the following interpretation of equations 2.2 and 2.3.

The relation for p which states that

P = P0 273 P (2.4)

K P0

where p and P0 are the instantaneous and constant equilibrium densities of

the medium; K is the temperature of the medium in °K; and P and P0 are the

instantaneous and constant equilibrium pressures of the medium shows that

2.2 can be rewritten as

c=(.¢z. K ..Eo_)112

(Po 273 p )112

(2.5)

Equation 2.5then reduces to

c= co {T+273) 1/2 (2.6)

(273) 112

after noting that K can be written as T + 273, where T is the temperature of the

environment in °G, and c0 is the speed of sound in air at 0°C.

Humidity plays a role in changing the speed of sound within a

medium. Changes in the speed of sound due to changes in the medium's

humidity arise from another interpretation of equations 2.2 and 2.3.

Substituting 2.3 into 2.2 gives

c = ({7- 0.1e) P) 112

p1/2

(2.7)

6
Changes due to humidity directly affect the vapor pressure, e,which, in turn,

affectthe speed of sound. The relation between humidity and vapor pressure

is given as

H (%) = (100%) (,P_p_) (2.8)

e

where H is the relative humidity, Pp is the partial pressure of the water vapor

in the medium, and e is the vapor pressure at the same temperature [31. The

relation between Pp and e is such that at any given temperature

0<Pp<e.

Knowing this relation enables a substitution of Pp for e in equation 2.7. Table

2.1 shows a partial listingof measured vapor pressures for water at different

temperatures.

T (°C) e (10 5 Pa)

0.0 0.00610

5.0 0.00868

10.0 0.0119

15.0 0.0169

20.0 0.0233

40.0 0.0734

Table 2.1 -- Vapor Pressure of Water

Holding all other factors constant and varying Pp from [0, e] allows an

examination of how c changes with Pp over the given range. From the

7
values listed in Table 2.1 it is obvious that in the temperature range of

interest,[15°C, 25°C], contributions of Pp are minimal and the 0.1e term can be

neglected; thus reducing equation 2.7 to the results of 2.6. This means that

changes in c depend much more on changes in temperature than on changes

in humidity. Additionally, the changes in c due to its dependency on

temperature means that many temperature changes along the acoustic wave's

path will affect the speed of the wave at many points from emission to

detection and will require finding ways of handling the error-inducing

temperature gradients.

Further information concerning the derivations of the speed of sound

within a given medium, and the effects of a dynamic environment on that

speed can be found in references [IIto [9].

8
CHAPTER 3

SOLUTION AND ANALYSIS

Froblem Solution

The solution to the collision avoidance problem requires accurate

determination of a specific point in 3-space using ultrasonic distance

measurements (slant ranges). While a minimum of three slant ranges is

required to determine the 3-space location in a Cartesian coordinate system, a

fourth slant range is included in this analysis. The additional slant range is

not necessary to solve the system; however, its addition reduces the number

of variables in the expression of one of the 3-space location components (Dz

for example) to fewer than the number of variables in the expression for DZ

which is not based on the additional slant range measurement. A reduction

in the number of variables in the expression reduces the number of variables

in the corresponding derivative expression, thereby simplifying the analysis

of error propagation in that 3-space location component.

Figure 3.1 demonstrates the geometrical setup of the solution. The

location of the ultrasonic transducer source is given at point S along the Z

axis of the coordinate system. The ultrasonic receivers are located at points P1

through P4. The distances between the transducer and the ultrasonic

receivers are given by LI through L4. Finally,the distances d12, d23, and d24

indicate the known distances between the receiver points PI and P2,P2 and P3,

and P2 and P4 respectively. The distances DX, Dy, and DZ are the X, Y, and Z

components of S's displacement from point PI,which can be found as shown.
Z

S

L1

Y

v

P1

d24 Dx

d12

Dy

X

d23

Figure 3.1 -- Geometrical Setup of Solution

From Figure 3.1, Dx can be determined from the relationship:

Dx = L2 cos o_2 (3.1)

additionally, by the law of cosines:

cos (z2 = L22+_.__c1232 -_._2

2 L2 d23

(3.2)

10
so that IDx can be found as:

DX = L22 + d232 ----_ 2

2 d23

(3.3)

In the same manner, similar equations for both Dy and DZ can be

found as:

Dy = L1 cos 0_I (3.4)

Dz = 1,2cos a3 (3.5)

and

These equations further reduce to:

Dy = L12 + d122 -__L22

2 d12

(3.6)

DZ = L_2 +__..._42 - L42 (3.7)

2 d24

and

Sensitivity Analysis of Solution

The accuracy of allof the distance equations above depends solely on

the accuracy with which the slant ranges are measured. The use of the

relation D = ct implies that the slant range measurement errors will occur due

to errors in the measurements of sound velocity or flighttime. In practical

use, the flighttime measurement error is minimal and can be ignored. The

speed of sound, however issubject to more substantial measurement errors

since itvaries with any temperature changes along the acoustic wave's path.

II
Recall that from previous work in Chapter 2

c= co (T+273)112

(273) I12

(3.8)

where co is the velocity of sound in air at 0 °C; that is co _ 331.18 m/s.

Taking the derivative of 3.8 gives the change in speed of sound due to

the change in temperature of the environment as:

dc= 9__ (273) 112 (:IT (3.9)

2 (T + 273)I/2 273

If the assumption is made that the temperature in the environment

falls in the range [59°F, 77°F] or [15°C, 25°CI then in the worst case the

measurement of T could be in error by 18°F or 10°C. Ifthis isthe case then the

graph of dc becomes as Figure 3.2:

(..)

E

v

8--

i

4

3

2

I

T = 15°C

1 I I I I i I I I I

1 2 3 4 5 6 7 8 9 10

dT (°C)

I

11

I'-

12

Figure 3.2 -- Relation Between dc and dT

12
Therefore, for

dc= 331.18 (279) 112 dT (3.10)

546.00 (T + 273) 1/ 2

with dT = I0°C and T = 15°C, for the assumptions given, the maximum value

for dc (dcmax) = 5.9055 m/s.

Thus the maximum relative error for c is:

dc= 5.9055m/s _- 0.1736 (3.11)

c 340.1567 m/s

and the maximum percentage error for c is:

dc (100%) = 1.7361% error in c

C

(3.12)

Now, to find the error in the slant range measurements, Li,recall the

relationship used to find L_ namely equation 2.1 which is rewritten here as:

Li = tic (3.13)

where the distance, D, has been rewritten as the ith slant range, Li,and the

time of flight,t,as the ith time of flightti,since each slant range has its own

unique time of flight measurement with which it is associated. Further,

taking the derivative of 3.13,the change in the ith slant range, dLi, due to a

change in the velocity of sound in the environment can be found as:

dLi = tidc (3.14)

where ti is treated as an exact constant under the assumption that any

equipment that would be used to measure the time of flightwould do so with

a high enough accuracy so that any error in ticould be neglected.

13
Note that a maximum dL (dLmax) will occur at a large distance. This is

because the time of flight,t,will be greater for a larger distance, and the dcmax

term as shown above will not change over distance, and therefore will only

affect dL as a constant. Now, ifthe ranges of measurement are limited to

distances of L < 3 meters then the time, ti,will be determined by the equation

ti < 3.0 (3.15)

c

which has a plot of:

t (s)

1.00

.875

.750

.625

.500

.375

.250

.125

I

60

90'1'1'1'1'1" 150 210 270 330

120 180 240 300 360

c(m/s)

Figure 3.3 -- Relation between Time of Flight and Speed of Sound

Note that at the values of c in question -- that is from -340.16 m/s to

-346.01 m/s -- the values of tiin the plot of Figure 3.3 range from -8.8194 ms

to -8.6703 ms so that the value of tiacts as a magnitude-reducing scale factor

in 3.14. Thus, for the sound velocity range in question tmax _-8.8194 ms.

14
Now, recalling that the value for dcmax occurs when T = 15°C, the

maximum dL (dLmax) will occur under the same conditions with the time of

flight at tmax. Thus, if the value of dc in 3.14is expanded dL becomes

dL=ti 331.18m/s (273) 112 dT (3.16)

546°C (T+273)II 2

and replacing ti with tmax and dc with dcmax as given above gives dLmax as:

dLmax _ (0.008819 s) (5.9055 m/s) (3.17)

SO,

dLmax _ 0.05208 m

Thus, the maximum relative error in L is:

dL = 0.05208 m _- 0.01736

L 3m

(3.18)

and the maximum percentage error in L is:

dL (100%) * 1.7361% error in L

L

(3.19)

which is the same percentage error for c, and is not surprising since 3.14

shows that dL is related to dc by a constant time value.

Values for dL and the corresponding percentage errors for a distance of

L = 3 meters at given temperatures of 15°C, 20°C, and 25°C are given in Table

3.1. Note that as the temperature changes in increments of 0.5°C, dL changes

in a linear fashion as predicted by 3.16. It can be seen that for every 1°C

change in temperature, the calculated error in the distance measurement will

be, at most, approximately 0.1736% of the true distance. Thus, for example, at

15
a temperature of 20°C over a distance of 1 meter with a temperature change

over that distance of 1°C, the measurement error should be approximately

0.1721 cm, or 1.721 ram.

dT (°C)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

8.0

8.5

9.0

9.5

10.0

T = 15°C

dL % error

0.0000 0.0000%

0.0026 0.0868%

0.0052 0.1736%

0.0078 0.2604%

0.0104 0.3472%

0.0130 0.4340%

0.0156 0.5208%

0.0182 0.6076%

0.0208 0.6944%

0.0234 0.7812%

0.0260 0.8681%

0.0286 0.9549%

0.0312 1.0417%

0.0339 1.1285%

0.0365 1.2153%

0.0391 1.3021%

0.0417 1.3889%

0.0443 1.4757%

0.0469 1.5625%

0.0495 1.6493%

0.0521 1.7361%

T = 20°C

dL % error dL

T = 25°C

% error

0.0000 0.0000%

0.0026 0.0861%

0.0052 0.1721%

0.0077 0.2582%

0.0103 0.3442%

0.0129 0.4303%

0.0155 0.5164%

0.0181 0.8024%

0.0207 0.6885%

0.0232 0.7746%

0.0258 0.8606%

0.0284 0.9467%

0.0310 1.0327%

0.0336 1.1188%

0.0361 1.2049%

0.0387 1.2909%

0.0413 1.3770%

0.0439 1.4630%

0.0465 1.5491%

0.0491 1.6352%

0.0516 1.7212%

0.0000 0.0000%

0.0026 0.0853%

0.0051 0.1707%

0.0077 0.2560%

0.0102 0.3413%

0.0128 0.4267%

0.0154 0.5120%

0.0179 0.5974%

0.0205 0.6827%

0.0230 0.7680%

0.0256 0.8534%

0.0282 0.9387%

0.0307 1.0240%

0.0333 1.1094%

0,0358 1.1947%

0.0384 1.2800%

0.0410 1.3654%

0.0435 1.4507%

0.0461 1.5361%

0.0486 1.6214%

0.0512 1.7067%

Table 3.1 -- Relation Between dT and dL Showing the % Error in L

The preceding results for error in the slant range can be extended to

find the errors in the distance measurements required by the problem

solution. Recalling the equation for Dx from 3,3 ODX is found as:

_Dx =_ (3.20)

d23

16
which, after inserting expanded terms for dL2 and dL3, and simplifying the

result,becomes:

aDx = (0.6065) (273) I12 dT L__ (3.21)

(T + 273)112 d23

Taking the derivatives of equations 3.6 and 3.7 yield equations similar to 3.21:

aDy = (0.6065) (273) 112 dT LI_.I-L2J.2 (3.22)

(T + 273)I/2 d12

and

aDz = (0.6065) (273) 112 dT t_.z.r..__4t_4 (3.23)

(T + 273)1/2 d24

Note that in this case, the error in each of the distances DX, Dy, and DZ

depends solely on the error in the temperature measurement. Thus, again

using the values T : 15°C and dT : 10°C, as before, the following maximum

errors arise in the distance measurements:

ODx = (5.9055) Lt2_LL.m_.t.3 (3.24)

d23

_)Dy = (5.9055) Llh -_:..In_ (3.25)

d12

and

adz = (5.9055) L2._...z._4t4 (3.26)

d24

aDx, _Dy, and 0Dz are maximized when the fight hand side of each of

their equations is maximized. Again, limiting the maximum measured

17
distances to 3 meters, and setting d23-- d12= d24 --I,the maximum values are

found to be:

aDx = 3Dy = _Dz = (5.9055) (0.01411) _ 0.08333 m (3.27)

and the maximum percentage errors in Dx, Dy, and DZ to be:

_Dx (100%) = _IDy (100%)= _IDz (100%) _.2.8735% error

Dx Dy Dz

(3.28)

F,ecl_.l¢ing Slant Range Measurement Errors

It has been shown that the measurement accuracy of the slant ranges is

most affected by changes in temperature along the flight path of the ultrasonic

wave front. Therefore, the method to reduce error in the slant range

measurements must be able to determine the temperature changes along the

flight path and apply those changes to the known speed of sound.

Unfortunately, detecting such temperature changes is a difficult task and

therefore requires an attempt to find a simpler method.

Recall that changes in an environment's temperature directly change

the speed of sound within that environment. This being the case, if the speed

of sound can be accurately measured dynamically (while slant range

measurements are being taken) instead of defining it to be a static value for

the environment, then the effects of any temperature changes can be applied

to the next set of slant range measurements. A simple method of

determining the speed of sound is to set up another ultrasound transmitter

and receiver unit at a known distance and then measure the speed of sound

over that known distance by solving for c from equation 2.1 since D and t are

18
known; thus, c = D/t. Ifthis calibration measurement is made just prior to

measuring the slant ranges then the dynamically measured speed of sound

can be used to measure the slant ranges more accurately than ifa statically

defined speed of sound isused.

This method for dynamically measuring the speed of sound has a

drawback in that itgives a measure of c in only one dimension. Ifthe speed

of sound can be accurately measured in more dimensions, then the

individual speed of sound values can be averaged to find an increasingly

accurate value for c. If,for example, two receiver units are used instead of

one for the calibration measurement then the value for c can be found as the

average of two individual measures of c (cl and c2) and thus take into account

fluctuations in temperature over two dimensions:

C=Cl+C_ (3.29)

2

An ideal setup for the calibration receivers is to use one receiver, RI, to

measure the horizontal component of the speed of sound (ci for example)

and the other receiver, R_ to measure the verticalcomponent of the speed of

sound (c2)as shown in Figure 3.4a.

R2 IIR2

R3

L l m

• I m

S R1 S R1

(a) (b)

Figure 3.4 -- Setup to Calibrate Speed of Sound Value in 2- and 3-Space

19
Since the two receivers are positioned at right angles, setup (a) takes

into account changes in the speed of sound over two dimensions. To include

the changes in the speed of sound over the third dimension, a third receiver,

R3, can be added orthogonally to the firsttwo as shown in Figure 3.4b. This

additional receiver adds another term to 3.29 to give 3.30.

c =-Cl + c_ + ¢'_ (3.30)

3

Therefore, the third term allows fluctuations in temperature over a third

dimension to be taken into account. When averaged with the Cl and c2

values from the first two dimensions, the c3 value allows determination of

the value of c over the entire 3-space environment.

20
CHAPTER 4

EXPERIMENTAL RESULTS

Description of Experiment

A foremost concern associated with making ultrasonic time of flight

distance measurements is the accuracy of the results based on temperature

fluctuations along the path of the ultrasound wave. The more unstable the

temperature along the path of flight for the ultrasonic wave front, the more

inaccurate the distance measurements will be. The previous chapter

demonstrates that the maximum expected percentage error in distance

measurements is approximately 1.7361% at T = 15°C and dT = 10°C.

In order to determine the amount of measurement error which might

be encountered under lab conditions, a set of distance measurement samples

is found for a known distance. The sample mean is calculated from the

samples and then used to determine the average measurement error from

the data. Additionally, the maximum sample value is used to determine the

maximum percentage error in the data. Thus, the average and maximum

bounds of measurement error are determined for the sample data.

The experimental setup consists simply of a Polaroid ultrasonic

transducer set up at a known distance of approximately 3 meters from a target.

The transducer is fired and the distance then calculated from the time of

flight of the received target echo. For this experiment, 180 distance

measurement samples of the known distance were gathered at the rate of one

sample per second to avoid any reverberation errors. The mean was then

21
determined from the data and used to determine the average percentage

measurement error as:

Eavg= dL (100%) _- AL (100%)

L L

(4.1)

where AL = (L - M) with M as the sample mean of the data and L = 3m.

Additionally, the maximum percentage error from the data was determined

by replacing M with the maximum sampled value for the distance so that AL

can be rewritten as AL = (3 - max). The magnitude of the percentage error can

simply be found by using I ALl.

Results of Ex-periment

From the 180 samples taken, the mean was found to be approximately

3.0016 meters and the maximum sampled value to be 3.007 meters. The

calculation of eavg showed that the magnitude of the average percentage error

in the measurement samples was approximately

Savg = AL (100%) = I ($ - $.0016) 1 (1007o) _ 0.0537% (4.2)

L 3

and the magnitude of the maximum percentage error was approximately

Emax = AL (100%) = 1(3 - 3.00711 (100%) _ 0.2333% (4.3)

L 3

The temperature in the room during the time interval in which the

sample measurements were taken was approximately 71°F, or 21.6667°C.

Consulting the T = 20°C column of Table 3.1 shows that for each of the error

22
measurements the value of dT over the path of the wave was approximately

0.4°C for the average measured sample, and approximately 1.4°C for the

maximum measured sample.

In retrospect, the values for the average and maximum percentage

errors seem fairly reasonable, given the temperature of the room and the

fairly static temperature conditions therein. Thus, if equipment was set up as

in Figure 3.1 to measure the DX, Dy, and DZ 3-space component distance

values it would be reasonable to expect that the percentage error values of

those components would be considerably less than the predicted maximum

percentage error value of 2.8735%.

23
CHAPTER 5

PROTOTYPECOLLISION AVOIDANCE SYSTEM

Prototype System Design

The problem of avoiding collisions between two independently

operating, closely-situated robotic arms can be handled by a two-step process.

The first step is the detection of an impending collision and the second is the

actual avoidance of the collision. Detection of an impending collision can be

handled with the following sequence of steps:

(1) Determine the 3-space locations of important points on the arms

(namely the locations of the ultrasound transmitters).

(2) Calculate the distances between the arms using the location of the

points found in step 1.

(3) Compare the calculated distances against preselected nearness

thresholds to determine if the arms are too close.

(4) Based on the nearness comparison, signal the collision avoidance

system if the arms are too close.

The collision avoidance step can be handled by simply taking some evasive

action to avoid the collision. The type of evasive action taken could range

from simply halting the movement of one or both of the arms to more

complicated maneuvering actions to bring the arms to safe distances from

each other.

A minimal setup of ultrasound transmitters and receivers necessary

for the detection of impending collisions is shown in Figure 5.1. The figure

24
shows the two robot arms, each with shoulder, elbow, and wrist joints,

situated along a rail which allows lateral base movement of both arms.

m

I Rai, I

Figure 5.1 -- Minimal Setup of Ultrasound Transmitters and Receivers

Connected to the base of arm I is the array of the four receivers, P1 to P4

(P1 is not shown as it lies directly behind receiver P2). The four receivers are

set up as discussed in Chapter 3 at distances d12, d23, and d24 apart from each

other and are used to determine the location of the two ultrasound

transmitters (denoted $1 and $2) which are connected at the base of each arm's

wrist. The transmitters are placed at the bases of the wrists since the

assumption is made that each arm's wrist location represents, in general, the

maximum extension of the arm into 3-space from its corresponding arm base.

Knowledge of the 3-space locations of the wrists from the transmissions of $1

and $2 enables the calculation of the distances between them and gives a

general estimate of the proximity of the arms. Comparison of the wrists'

proximity based on some predetermined nearness thresholds allows the

system to determine if a collision between the wrists could occur and

therefore whether or not evasive action should be taken.

25
_ototylze System Accuracy

Reducing the total amount of error in the DX, Dy, and Dz

measurements is important for the accurate operation of the system. Once

the distances between the important points on the arms (the wrists in this

simple case) have been calculated, they are compared against nearness

thresholds to determine ifthey fallwithin a range of distances regarded as too

close for safe operation. Inaccuracies in the distance measurements can,

therefore, affect the robots in one of two ways. The inaccuracies can trigger

the collision avoidance system when the actual distances between the

important arm points are not within the range of nearness thresholds,

leading to reduced arm movement capabilities. On the other hand,

inaccuracies can allow the arms to function even when the actual distances

between the important arm points are within the range of nearness

thresholds, possibly leading to an undetectable collision.

As previously demonstrated, the percentage error in any of the 3-space

component distance measurements should be less than approximately 2.874%

with receiver offsets d12 --d23 = d24 = I meter. However, ifoffsets of one

meter are too large for practical use then their reduction means a reduction in

measurement accuracy as shown by equations 3.24 to 3.26. The general form

of those equations is given here for convenience.

_)DA = (5.9055) Lit_..-_ijt_.j (5.1)

dij

where A is the X, Y, or Z axis and i and j are used to index the slant ranges,

time of flights, and offset value. The ratio 5.9055 / dij is simply the ratio of the

constant value 5.9055 divided by an offset value dij. Therefore, if the value

26
for the offsets is chosen to be less than one meter, it has a multiplicative effect

on the error. For this reason, it is advantageous to allow the offset distances

between the receivers to be as large aspossible.

Additional inaccuracies arise from the use of only two ultrasound

transmitters in the original design. Problems can arise because other of the

locations on each of the arms, are not able to be determined. It is possible, for

example, for the shoulder joint of arm I to be closer than the wrist joint of

arm 1 to a point on arm 2. If this is the case,then the simple two-transmitter

design is limited in its ability to accurately determine if a collision will occur.

One method to overcome this limitation is to add additional transmitters at

each major joint (shoulder, elbow, and wrist) on each arm so that the

locations of these joints can be determined. Since the arms consist of rigid

segments of known lengths, a simple point-to-point line algorithm can be

used to draw a line segment between every joint on either of the arms so that

the full 3-space description of each arm location can be determined.

Ultimately, then, the distance between any point on arm I to any point on

arm 2 can be tested against the nearness thresholds to determine if a collision

might occur.

Another concern with the minimal setup of Figure 5.1 arises from the

realization that when either arm 1 or 2's transmitter is placed near 180" from

the receiver setup the possibility exists that the ultrasound transmissions

from $I or $2 will be lost, or only detected after reflection of the transmitted

wave occurs. One method of handling a lost transmission is to simply

assume that the arm's transmitter is currently facing away from the receivers.

Unfortunately, when only one transmitter per arm is used, no method exists

for determining if the entire arm is facing away from the receivers, or only

27
the wrist. If multiple transmitters are utilized on an arm then the detection

of none, some, or all of the multiple transmissions allows the collision

detection system to determine which parts of the arm are facing away from

the receivers (and therefore are not in danger of colliding with the other

arm).

Determining whether a transmission is lost inherently requires some

type of timeout procedure which takes into account the reverberation time of

the transmission in the environment. If timeout is not considered, then the

possibility exists of a reverberated transmission either reaching the receivers

and being counted erroneously as a valid distance or hanging up the collision

detection system indefinitely while the system waits for it to arrive.

Knowledge of the reverberation time within a room is necessary, as well, so

that the collision detection system can determine the proper time to fire the

next ultrasound transmitter so that separate transmissions will not interfere

with each other (assuming that each transmitter operates at the same

frequency).

The reverberation time of a room can be calculated from the following

formula[4l:

tr = 4 V ln(I0 / I) (5.2)

8c_V - S ln(1 - a) c

where V is the volume of the room in m 3, I0 / I is the reduction in intensity

of sound by 106 db, S is the total wall surface of the room in m 2, c is the speed

of sound in the room, and o_ and a are the absorption coefficients of the

gaseous medium and room walls respectively.

28
Using 5.2,tr can be found for a several different room sizes. If 0_and a

are held constant at c__,0.06for air at 20"C (based on classical attenuation at a

frequency of about 40 KHz) and a _,0.2for average room structure attenuation

then a table of reverberation time values can be obtained.

Room Dimensions (m) Reverberation Time (ms)

Length Width Height c_= 0.06 o_= 0.08 Average

5

10

15

10

15

15

3

3

3

3

3

3

198.99

210.64

214.83

223.73

228.46

233.41

166.08

174.11

176.96

182.96

186.12

189.38

5

5

5

10

10

15

182.54

192.38

195.90

203.35

207.29

211.40

Table 5.1 -- Reverberation Time For Rooms of Different Sizes

Table 5.1 demonstrates that in a room heated to 20°C, the time required

to pass before a transmission can be considered as lost is approximately 218 ms

for 0_ = 0.06 and 179 ms for cz = 0.08. Obviously, if the collision detection

system must wait, on average, approximately 198.8 ms for each transmission

to die out, it behooves the designer to keep the number of transmitters in the

system low so that the time required to find all the necessary arm locations is

low. If the time to find all the arm locations is too high, a collision may occur

before the system has a chance to determine the locations of all the

transmitters.

One workaround to this problem is to use ultrasound transmitters of

different frequencies so that all of the transmitter locations can be found in

29
parallel. The disadvantage with parallelness is that all of the receiver

circuitry must be reproduced n times for n transmitters. Additionally, the

frequencies to be used for transmitting must not contain overtones which

match another transmitting frequency or else ambiguity may occur when

determining transmitter locations.

Improving System Desi_-n

Based on the considerations presented for reducing the inaccuracies of

the two-transmitter system, several improvements can be made to the

original prototype design. Clearly, for more accurate arm location

determination, additional transmitters are necessary. If another transmitter is

placed at the elbow joint of each arm it becomes possible to mathematically

determine the location of all points on the arms using the point-to-point line

algorithm discussed previously.

The addition of two more transmitters augments the total time

necessary to find all the necessary arm locations by twofold. To reduce the

total reverberation time, anywhere from one to four distinct transmitting

frequencies can be used so that transmission can occur in parallel. This adds

to the total cost of the system due to redundant hardware, but reduces the

time of reverberation to a more acceptable value.

Finally, improvements in the calibration of the speed of sound can be

utilized to reduce errors in the slant range measurements. Calibration of c

requires at least one more ultrasound transmitter and receiver and possibly

more as previously discussed in Chapter 3. The additional transmitters'

reverberation time must be considered, and used to determine how many, if

3O
any, speed of sound calibration transmitters should be used. Clearly, ifthe

total reverberation time for the system is fairly high then the distance

measurements will not be gathered often enough to be of any timely use, and

the accuracy of the distance measurements will be inconsequential.

31
CHAPTER 6

CONCLUSIONS

_ummary and Conclusions

A method of measuring distances using an ultrasound time of flight

measurement method was examined in order to determine the bounds on

measurement error due to temperature fluctuations along the path of the

ultrasonic wave front. An upper bound on measurement error was

determined to be maximally 1.7361% of the distance when a temperature

change of 10°C was encountered over the path of the wave front. When

applied to the collision detection and avoidance solution presented, this

measurement error for slant ranges translates to an overall maximum error

of approximately 2.8735% for each of the 3-space location components DX, Dy,

and DZ used to determine the 3-space locations of ultrasound transmitters on

the robot arms.

The key method presented to minimize the measurement errors

suggests that a more exact measurement of the speed of sound can reduce the

error. By using multiple ultrasound transmitters and receivers, set up at

known distances, a better estimate of the speed of sound can be determined

over two or three dimensions. This improved estimate of the current speed

of sound in the environment can be used to improve the overall accuracy of

the slant range measurements.

Experimental evidence suggests that the distance measurement errors

do seem to follow the relationship which states that changes in distance

measurements are directly proportional to changes in the temperature along

32
the path of the ultrasonic wave front. Additional experimental evidence

suggests that itispossible to make ultrasound distance measurements within

0.25% of the exact value at common room conditions. This would indeed

appear to support the idea that distance measurements made via the

ultrasound time-of-flight method are suitable for a basic collision detection

and avoidance system.

Suggestions for Further Research

The following list suggests steps which can be taken to further the

efforts of the research completed thus far, Further research time spent

examining the areas listed below will provide a better understanding of the

feasibility of ultrasound to solve the collision avoidance problem.

(1) Construction of a prototype collision avoidance system. This would

require the construction of ultrasound transmitting and receiving

hardware.

(2) Interface of hardware with robot arms and a computer to control allof the

ultrasound transmissions as well as compute the necessary calculations of

3-space location components.

(3) Determination of the 3-space location component (Dx, Dy, DZ) errors in

the system.

(4) Attempted reduction of errors by speed of sound calibration techniques

using additional ultrasound transmitters as discussed in Chapter 3.

(5) Determination of the effectsof adding additional transmitters to system to

implement point-to-point line segment algorithm discussed in Chapter 5.

(6) Determination of the feasibilityof using multiple transmission

frequencies for parallelultrasound transmission as discussed in Chapter 5.

33
(7) Interface of collision avoidance system with existing control software for

robot arms to determine the geometrical and time limits of ability of the

system.

(8) Extension of work to an RF domain solution for use in activities where

ultrasound is not feasible; use in space for example. Information on one

possible technique using Polhemus 3SPACE three-dimensional position

and orientation sensors is given in [10] and [111.

Steps I through 7 deal with implementing and then improving the

accuracy and overall timeout requirements of the ultrasound solution.

Conceivably, the implementation of step 8 would entirely eliminate the

reliance on time of flight measurements for the current design and thus the

effects of timeout discussed in the previous chapter. The parallel monitoring

of the locations of the transmitters on the arms would allow for faster

determination of the arm positions and, in doing so, possibly yield results

suitable for a realtime solution.

34
Cll

[2l

[3]

[41

[51

[6]

[7]

[81

[91

[101

[11]

CHAPTER 7

LITERATURE CITED

Brown, B., Goodman, J., High-Intensity Ultrasonics. Princeton, N]:

Nostrand, 1965.

Humphreys, W., Physics of the Air. New York, NY: McGraw-Hill,

1940.

Sears, F., Young, H., and Zemansky, M., University Physics. Reading,

MA: Addison-Wesley, 1982.

Beyer, R. and Letcher, S., Physical Ultrasonics. New York, NY:

Academic, 1969.

Coppens, A., Frey, A., Kinsler, L., and Sanders, J., Fundamentals of

Acoustics. New York, NY: Wiley, 1982.

Rossi, M., Acoustics and Electroacoustics. Norwood, MA: Artech

House, 1986.

Beranek, D, Acoustic Measurements. New York, NY: Wiley, 1949.

Lindsay, R., Physical Acoustics. Stroudsburg` PA: Dowden,

Hutchinson & Ross, 1974.

_.I'Piner, I.,Ultrasound: Physical, Chemical, and Biological Effects.

New York, NY: Consultants Bureau, 1964.

Foley, J.,van Dam, A., Feiner, S.,and Hughes, J.,Computer Graphics:

Principles and Practice, Second Edition. Reading, MA: Addison

Wesley, 1990, 355.

Zimmerman, T., Lanier, J.,Blanchard, C., Bryson, S.,and Harvill,Y., "A

Hand Gesture Interface Device," Proceedings of the CHI and GI 1987

Conference, New York, NY: ACM, 1987, 189-192.

35