TRILATERATION LOCALIZATION FOR MULTI-ROBOT TEAMS

aul M. Maxim

, Suranga Hettiarachchi

, William M. Spears

, Diana F. Spears

Jerry Hamann

, Thomas Kunkel

and Caleb Speiser

University of Wyoming, Laramie, Wyoming 82070, U.S.A.

Eastern Oregon University, La Grande, Oregon 97850, U.S.A.

Keywords:

Localization, trilateration, formations, distributed, outdoor.

Abstract:

The ability of robots to quickly and accurately localize their neighbors is extremely important for robotic

teams. Prior approaches typically rely either on global information provided by GPS, beacons and landmarks,

or on complex local information provided by vision systems. In this paper we describe our trilateration ap-

proach to multi-robot localization, which is fully distributed, inexpensive, and scalable (Heil, 2004). Our prior

research (Spears et. al, 2006) focused on maintaining multi-robot formations indoors using trilateration. This

paper pushes the limits of our trilateration technology by testing formations of robots in an outdoor setting at

larger inter-robot distances and higher speeds.

1 INTRODUCTION

The main contributions of this paper are: (1) a presen-

tation of our trilateration approach to multi-robot lo-

calization (i.e., each robot locates its neighbors), and

(2) a set of experimental results obtained with our tri-

lateration approach under outdoor conditions. These

experimental results highlight the advantages of our

approach and clarify its limitations. The outdoor ex-

periments are conducted in an environment with vary-

ing terrain (e.g., grass, dirt, and concrete), rocks, pro-

truding tree roots, leaves, pine cones and other ground

protrusions. Also, there was a considerable amount

of dust and wind (over 9 meters per second). Despite

this, the robots are able to maintain high quality for-

mations.

The organization of this paper is as follows. Sec-

tion 2 introduces our trilateration approach to local-

ization, which is fully distributed and assumes that

each robot has its own local coordinate frame (i.e.,

no global information is required). Each robot deter-

mines its neighbors’ range and bearing with respect

to its own egocentric, local coordinate system. After

such localization, sensor values and other data can be

exchanged between robots in a straightforward man-

ner. Next, sections 3, 4 and 5 describe our trilatera-

tion implementation and current robot platforms. Sec-

tions 6 and 7 present results from our experiments.

Section 8 summarizes and concludes the paper.

2 LOCALIZATION VIA

TRILATERATION

The purpose of our trilateration technology is to cre-

ate a plug-in hardware module to accurately local-

ize neighboring robots, without global information

and/or the use of vision systems. Our localiza-

tion technology does not preclude the use of other

technologies. Beacons, landmarks, vision systems,

GPS (Borenstein et. al, 1996), and pheromones

are not necessary, but they can be added if de-

sired. It is important to note that our trilateration

approach is not restricted to one particular class of

control algorithms – it is useful for behavior-based

approaches (Balch and Hybinette, 2000), control-

theoretic approaches (Fax and Murray, 2004; Fierro

et. al, 2002), motor schema algorithms (Brogan and

Hodgins, 1997), and physicomimetics (Spears et. al,

2004; Zarzhitsky et. al, 2005; Hettiarachchi, 2007).

In 2D trilateration, the locations of three base

points are known as well as the distances from each

of these three base points to the object to be localized.

Looked at visually, 2D trilateration involves ﬁnding

the location where three circles intersect. Thus, to lo-

cate a robot using 2D trilateration the sensing robot

must know the locations of three points in its own

coordinate system and be able to measure distances

from these three points to the sensed robot.

301

M. Maxim P., Hettiarachchi S., M. Spears W., F. Spears D., Hamann J., Kunkel T. and Speiser C. (2008).

TRILATERATION LOCALIZATION FOR MULTI-ROBOT TEAMS.

In Proceedings of the Fifth International Conference on Informatics in Control, Automation and Robotics, pages 301-307

DOI: 10.5220/0001508803010307

 SciTePress

2.1 Measuring Distance

Our distance measurement method exploits the fact

that sound travelssigniﬁcantly more slowly than light,

thereby enabling us to employ a Difference in Time of

Arrival technique. To tie this to 2D trilateration, as-

sume that each robot has one radio frequency (RF)

transceiver and three ultrasonic acoustic transducers.

The ultrasonic transducers are the “base points.” Sup-

pose robot 2 simultaneously emits an RF pulse and an

ultrasonic acoustic pulse. When robot 1 receives the

RF pulse (almost instantaneously), a clock on robot 1

starts. When the acoustic pulse is received by each of

the three ultrasonic transducers on robot 1, the elapsed

times are computed. These three times are converted

to distances, according to the speed of sound. Be-

cause the locations of the acoustic transducers are

known, robot 1 is now able to use trilateration to com-

pute the location of robot 2 (precisely, the location of

the emitting acoustic transducer on robot 2). Of the

three acoustic transducers, all three must be capable

of receiving, but only one must be capable of trans-

mitting.

Measuring the elapsed times is not difﬁcult. Since

the speed of sound is roughly 340.2 meters per second

at standard temperature and pressure, it takes approx-

imately 2.9 ms for sound to travel one meter. Times of

this magnitude are easily measured using inexpensive

electronic hardware.

2.2 Channeling Acoustic Energy into a

Plane

Ultrasonic acoustic transducers produce a cone of en-

ergy along a line perpendicular to the surface of the

transducer. The width of this main lobe (for the in-

expensive 40 kHz transducers used in our implemen-

tation) is roughly 30

◦

. To produce acoustic energy

in a 2D plane would require 12 acoustic transducers

in a ring. To get three base points would hence re-

quire 36 transducers. This is expensive and is a large

power drain. We adopted an alternative approach.

Each base point is comprised of one acoustic trans-

ducer pointing downward. A parabolic cone (Heil,

2004) is positioned under the transducer, with its tip

pointing up toward the transducer (see Figure 2 later

in this paper). The parabolic cone acts like a lens.

When the transducer is placed at the virtual “focal

point” the cone “collects” acoustic energy in the hor-

izontal plane, and focuses this energy to the receiving

acoustic transducer. Similarly, a cone also functions

in the reverse, reﬂecting transmitted acoustic energy

into the horizontal plane. This works extremely well

– the acoustic energy is detectable to a distance of 3.5

m. which is adequate for our needs. Greater range

can be obtained with more power (the scaling appears

to be quite manageable).

2.3 Related Work

Trilateration is a well-known technique for robot lo-

calization. Most approaches (including ours) are al-

gebraic, although recently a geometric method was

proposed (Thomas and Ros, 2005). Many localiza-

tion techniques, including those involving trilatera-

tion, use global coordinates (Peasgood et. al, 2005);

however ours relies on local coordinates only.

MacArthur (MacArthur, 2003) presents two dif-

ferent trilateration systems. The ﬁrst uses three acous-

tic transducers, but without RF. Localization is based

on the differences between distances rather than the

distances themselves. The three acoustic transducers

are arranged in a line. The second uses two acoustic

transducers and RF in a method similar to our own.

Unfortunately, both systems can only localize points

“in front” of the line.

Cricket (Nissanka, 2005) is another system that

makes use of RF and ultrasound for localization. It

was developed to be used indoors. Compared to our

system, which does not require ﬁxed beacons, the

Cricket requires beacons attached to ﬁxed locations

in order to function. This is not practical for mobile

robot localization in outdoor environments.

Our particular approach was inspired by the CMU

Millibot project. They also use RF and acoustic trans-

ducers for trilateration. However, due to size lim-

itations, each Millibot has only one acoustic trans-

ducer (coupled with a right-angle cone, rather than

the parabolic cone we use). Hence trilateration is a

collaborative endeavor that involves several robots.

To perform trilateration, a minimum of three Milli-

bots must be stationary and serve as beacons at any

moment in time. The set of three stationary robots

changes as the robot team moves. The minimum team

size is four robots (and is preferably ﬁve). Initializa-

tion generally involves having some robots make L-

shaped maneuvers, in order to disambiguate the local-

ization (Navarro-Serment, 1999). Our approach oper-

ates with as few as two robots (but is scalable to an

arbitrary number), due to the presence of three acous-

tic transducers on each robot (see below).

In terms of functionality, an alternative localiza-

tion method in robotics is to use line-of-sight infra-

red (IR) transceivers. When IR is received, signal

strength provides an estimate of distance. The IR sig-

nal can also be modulated to provide communication.

Multiple IR sensors can be used to provide the bear-

ing to the transmitting robot (e.g., see (Rothermich

ICINCO 2008 - International Conference on Informatics in Control, Automation and Robotics

302

et. al, 2004; Payton et. al, 2004)). We view this

method as complementary to our own; however, our

method is more appropriate for tasks where greater

localization accuracy is required. This is especially

important in outdoor situations where water vapor or

dust could change the IR opacity of air. Similar issues

arise with the use of cameras and omni-directional

mirrors/lenses, which require far more computational

power and a light source.

3 OUR TRILATERATION

APPROACH

Our trilateration approach to localization is illustrated

in Figure 1. Assume two robots, shown as circles. An

RF transceiver is in the center of each robot. Each

robot has three acoustic transducers (also called base

points), labeled A, B, and C. Note that the robot’s

local XY coordinate system is aligned with the L-

shaped conﬁguration of the three acoustic transduc-

ers, as shown in the ﬁgure. Note, Y points to the front

of the robot.

Robot 1



s s

Robot 2

A B



Figure 1: Three base points in an XY coordinate system

pattern.

In Figure 1, robot 2 simultaneously emits an RF

pulse and an acoustic pulse from its transducer B.

Robot 1 then measures the distances a, b, and c. With-

out loss of generality, assume that transceiver B of

robot 1 is located at (x

, y

) = (0, 0) (Heil, 2004).

In other words, let A be at (0, d), B be at (0, 0), and C

be at (d, 0), where d is the distance between A and B,

and between B and C (see Figure 1).

For robot 1 to determine the position of B on robot

2 within its own coordinate system, it needs to ﬁnd the

simultaneous solution of three nonlinear equations,

the intersecting circles with centers located at A, B

and C on robot 1 and respective radii of a, b, and c:

Subscripts denote the robot number and the acous-

tic transducer. The transducer A on robot 1 is located at

, y

− x

)

+ (y

− y

)

= a

(1)

− x

)

+ (y

− y

)

= b

(2)

− x

)

+ (y

− y

)

= c

(3)

Given the transducer conﬁguration shown above, we

get (Heil, 2004):

− c

+ d

− a

+ d

An interesting beneﬁt of these equations is that they

can be simpliﬁed even further, if one wants to trilater-

ate purely in hardware (Spears et. al, 2006).

By allowing robots to share coordinate systems,

robots can communicate their information arbitrarily

far throughout a robotic network. For example, sup-

pose robot 2 can localize robot 3. Robot 1 can local-

ize only robot 2. If robot 2 can also localize robot 1

(a fair assumption), then by passing this information

to robot 1, robot 1 can now determine the position of

robot 3. Furthermore, robot orientations can also be

determined. Naturally, localization errors can com-

pound as the path through the network increases in

length, but multiple paths can be used to alleviate this

problem to some degree. Heil (Heil, 2004) provides

details on these issues.

In addition to localization, our trilateration system

can also be used for data exchange. Instead of emit-

ting an RF pulse that contains no information but only

performs synchronization, we can also append data to

the RF pulse. Simple coordinate transformations al-

low robot 1 to convert the data from robot 2 (which is

in the coordinate frame of robot 2) to its own coordi-

nate frame.

4 TRILATERATION

IMPLEMENTATION

Figure 2 illustrate how our trilateration framework is

currently implemented in hardware. Figure 2 (left)

shows three acoustic transducers pointing down, with

reﬂective parabolic cones. The acoustic transducers

transmit and receive 40 kHz acoustic signals.

Figure 2 (middle) shows our in-house acoustic

sensor boards (denoted as “XSRF” boards, for Ex-

perimental Sonic Range Finder). There is one XSRF

board for each acoustic transducer. The XSRF board

calculates the time difference between receiving the

RF signal and the acoustic pulse. Each XSRF con-

tains 7 integrated circuit chips. A MAX362 chip con-

trols whether the board is in transmit or receive mode.

TRILATERATION LOCALIZATION FOR MULTI-ROBOT TEAMS

303

Figure 2: The acoustic transducers and parabolic cones (left). The XSRF acoustic sensor printed circuit board (middle), and

the completed trilateration module (top-down view, right).

When transmitting, a Microchip PIC microprocessor

generates a 40 kHz signal. This signal is sent to

an ampliﬁer, which then interfaces with the acoustic

transducer. This generates the acoustic signal.

In receive mode, a trigger indicates that an RF sig-

nal has been heard and that an acoustic signal is arriv-

ing. When the RF is received, the XSRF board starts

counting. To enhance the sensitivity of the XSRF

board, three stages of ampliﬁcation occur. Each of

the three stages is accomplished with a LMC6032 op-

erational ampliﬁer, providing a gain of roughly 15

at each stage. Between the second and third stage

there is a 40 kHz bandpass ﬁlter to eliminate out-of-

bound noise that can lead to saturation. The signal is

then passed to two comparators, set at thresholds of

± 2 VDC. When the acoustic energy exceeds either

threshold, the XSRF board ﬁnishes counting, indicat-

ing the arrival of the acoustic signal.

The timing counts provided by each of the XSRF

boards is sent to a MiniDRAGON

Freescale 68HCS12 microprocessor that performs the

trilateration calculations. Figure 2 (right) shows the

completed trilateration module from above. The

MiniDRAGON is outlined near the center and the

three XSRF boards are outlined at the bottom.

4.1 Synchronization Protocol

Trilateration involves at least two robots. One trans-

mits the acoustic-RF pulse combination, while the

others use these pulses to compute (trilaterate) the

coordinates of the transmitting robot. Hence, trilat-

eration is a one-to-many protocol, allowing multiple

robots to simultaneously trilaterate and determine the

position of the transmitting robot. Our “token pass-

ing” scheme to allow robots to take turns transmitting

is not used for the experiments in this paper, to sim-

plify the experimental design. Only leader/follower

experiments are presented herein.

Produced by Wytec (http://www.evbplus.com/)

5 Maxelbot PLATFORMS

Our University of Wyoming “Maxelbot” (named after

the two graduate students who designed and built the

robot) is modular. The platform is an MMP5, made by

The Machine Lab

. Figure 3 (left) shows four Max-

elbots. A primary MiniDRAGON is used for control.

It communicates via an I

C bus to all other peripher-

als, allowing us to plug in new peripherals as needed.

Figure 3 (right) shows the architecture. The primary

MiniDRAGON is the board that drives the motors. It

also monitors proximity sensors and shaft encoders.

The trilateration module is shown at the top of the

diagram. This module controls the RF and acoustic

components of trilateration. Additional modules have

been built for digital compasses, thermometers, and

chemical plume tracing (Spears et. al, 2006). The

PIC processors provide communication with the I

bus.

6 TRILATERATION ACCURACY

AS A FUNCTION OF

VELOCITY AND DISTANCE

In (Spears et. al, 2006) we presented the accuracy of

our trilateration technique on stationary robots, to a

distance of one meter. In this paper we present results

for a moving Maxelbot on a treadmill from 0.5 to 3.5

meters behind a stationary Maxelbot placed ahead of

the treadmill, at two speeds: 0.32 m/s and 0.64 m/s.

Hence we are measuring the accuracy of the perfor-

mance of the whole system, including the trilatera-

tion module and our physicomimetics control algo-

rithm. The results shown are the mean ± range of

the error ≡ ideal distance − measured distance (in

cm), as measured physically with a ruler (see Tables 2

See http://www.themachinelab.com/MMP-5.html

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304

Figure 3: Maxelbots and the architecture.

Table 1: Mean and range of error of the followers’ X position at different distances and velocities.

Ideal Distance (cm)

Velocity

50 100 150 200 250 300 350

0.32 m/s 0.0 ± 0.2 0.3 ± 0.3 -0.3 ± 0.3 -0.6 ± 0.4 0.6 ± 0.5 1.9 ± 0.6 0.6 ± 0.6

0.64 m/s 0.0 ± 0.6 0.3 ± 0.3 -1.3 ± 0.9 -1.0 ± 1.0 1.0 ± 1.0 1.9 ± 1.1 0.0 ± 1.3

and 1). Above 3.5 m the acoustic signal is lost, be-

cause the acoustic energy falls below the threshold of

± 2 VDC and hence is not detected.

The mean error in X is very small (< 1%), which

means that the side-to-side position of the Maxelbot is

very close to the desired position. The mean error in

Y is larger, and at higher distances the Maxelbot lags

more behind the desired position (but the mean error

is < 5%). However, note that the range in error is less

at the higher velocity. The increased momentum of

the robot helps ﬁlter sensor noise.

7 OUTDOOR EXPERIMENTS

This section presents three experiments that test the

trilateration system outside. In particular, the Maxel-

bots are run in a region in the center of the University

of Wyoming campus. This region consists mostly of

grass, of average height 5 cm, interspersed with con-

crete sidewalks, trees, rocks, leaves, and other debris.

The grass hits the bottom of the Maxelbot. Although

generally ﬂat, the ground slope can change rapidly

(within 0.6 m), by up to 20

◦

, at boundaries. Results

presented below are averaged over ﬁve independent

runs, each taken over a 20 minute interval. The speed

of the robots is approximately 0.55 m/s. For these ex-

periments we are forced to use the trilateration read-

ings themselves as an estimate to the quality of the

formation. Given the accuracy of the results in the

prior section, this is a reasonable and practical ap-

proach.

7.1 Accuracy of a Linear Formation

The ﬁrst experiment has three Maxelbots in a linear

formation. The purpose of this experiment was to de-

termine the effect of having the middle robot occlude

the acoustic signal between the ﬁrst and third Maxel-

bots. The ﬁrst follower (middle robot) tries to keep

the leader at (0 cm, 63.5 cm) with respect to its local

coordinate system. The second follower tries to keep

the leader at (0 cm, 145 cm) with respect to its local

coordinate system.

Table 3 summarizes the quality of the results. This

ﬁrst follower maintains position quite well. The sec-

ond follower does exhibit some difﬁculties due to oc-

clusion, since it does lag a bit behind the ideal dis-

tance. However, even in this case the distance is ap-

proximately only 10% off from the ideal distance.

Also, the standard deviation is acceptably low.

7.2 Accuracy of Non-Linear Formations

The second and third experiments examine the effect

of position with respect to the quality of the results

(trilateration accuracy can be affected by the differ-

ence in the bearing of one robot with respect to an-

other (Heil, 2004)). We try two different conﬁgura-

tions of the robots. In the ﬁrst, there are three robots.

The right follower tries to keep the leader at (-48 cm,

91 cm) with respect to its local coordinate system.

The left follower tries to keep the leader at (53 cm,

91 cm) with respect to its local coordinate system. In

the second conﬁguration we use four Maxelbots in a

diamond formation. In this latter experiment the wind

TRILATERATION LOCALIZATION FOR MULTI-ROBOT TEAMS

305

Table 2: Mean and range of error of the followers’ Y position at different distances and velocities.

Ideal Distance (cm)

Velocity

50 100 150 200 250 300 350

0.32 m/s -1.0 ± 1.6 0.0 ± 1.3 -1.3 ± 2.5 -1.9 ± 3.2 -1.3 ± 5.1 -5.0 ± 10.2 -5.0 ± 7.6

0.64 m/s -5.7 ± 0.6 -0.6 ± 1.9 -3.8 ± 2.5 -5.1 ± 3.8 -3.2 ± 4.4 -14.0 ± 3.8 -16.5 ± 3.8

Table 3: Accuracy of the two followers’ X and Y positions

in a linear formation (results in cm).

Ideal Mean Std. dev.

Follower1-X 0 -1.5 0.8

Follower1-Y 63.5 67.1 1.0

Follower2-X 0 0.3 4.8

Follower2-Y 145 159.8 4.1

speed near the ground ranged from 4 to 9 m/s.

Table 4 shows the XY-coordinates derived from

the trilateration readings, for both conﬁgurations.

From this table, it can be seen that the means are very

close to the ideal. The standard deviation is some-

what higher, reﬂecting the more difﬁcult environmen-

tal conditions. However, very good formations are

maintained by the trilateration system despite ground

disturbances, wind, dust, and relatively high speed

(Y has at most an error of roughly 11%). Results

are averaged over ﬁve independent runs. Thus far no

position-dependent effects have been noticed (other

than distance, as is expected).

7.3 Trilateration Reliability Results

A detailed data analysis has been performed on the

reliability of the trilateration system during the out-

door experiments. The RF failure rate is 0.2%. The

rate at which the RF pulse is received but acoustic

pings are not received (at all three receivers) is only

1%. Almost every acoustic failure was isolated, and

not consecutive. Consider the interpretation of these

results. Given that acoustic pings are sent at a rate of

approximately four per second (4.17 Hz), this implies

that 1% of the time, the Maxelbots ran for only 0.25

seconds on old data. Only once were two consecutive

pings in a row not received, yielding one 0.5 second

gap in readings.

Of all of our tests, the factor most important to

success was the temperature. Below roughly 6

◦

C the

electronics failed. Given that our components are not

ruggedized, this is not surprising.

8 SUMMARY

This paper describes a 2D trilateration framework for

the fast, accurate localization of neighboring robots.

The framework uses three acoustic transducers and

one RF transceiver. Our framework is designed to be

modular, so that it can be used on different robotic

platforms, and is not restricted to any particular class

of control algorithms. Although we do not rely

on GPS, stationary beacons, or environmental land-

marks, their use is not precluded. Our framework is

fully distributed, inexpensive, and scalable.

To illustrate the general utility of our framework,

we demonstrated the application of our new robots

in outdoor situations. The results from these ex-

periments highlight the accuracy of our trilateration

framework, as well as its current limitations (range

and environmental temperature). For all of the Maxel-

bots, their X and Y positions are within roughly 11%

of the desired values.

Open Source Project URL

The open source URL http://

www.cs.uwyo.edu/∼wspears/maxelbot provides

schematic details and videos of this project. We thank

the Joint Ground Robotics Enterprise for funding

portions of this work.

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