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SEA LAMPREYS ORIENT TOWARD A SOURCE OF A
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SEA LAMPREYS ORIENT TOWARD A SOURCE OF A SYNTHESIZED
PHEROMONE USING ODOR-CONDITIONED RHEOTAXIS

By

Azizah Wakeelah Muhammad

A THESIS

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

MASTER OF SCIENCE
Mechanical Engineering

2010

ABSTRACT

SEA LAMPREYS ORIENT TOWARD A SOURCE OF A SYNTHESIZED
PHEROMONE USING ODOR-CONDITIONED RHEOTAXIS

By
Azizah Wakeelah Muhammad
Engineers envision a world where odor-tracking, mobile robots can locate the
source of toxic gas or detect unexploded ordnance, but such tasks cannot be efﬁciently
executed until olfaction is fully understood. Given the poor sense of smell in humans, it is
no wonder that what is known about olfaction is gathered from scientists’ observations of
the odor-mediated behaviors of organisms. However, these observations are often made
in a controlled laboratory environment where animal behaviors are easier to analyze than
in their natural habitats. Engineers then use these behaviors as inspiration for odor source
localization algorithms, but rarely is this done in cooperation with scientists using actual
data from behavioral experiments to test their effectiveness. Perhaps more cooperation
between the two communities is needed to solve the odor source localization problem.
This thesis is a collaborative effort between the Department of Fisheries and
Wildlife and the Department of Mechanical Engineering at Michigan State University to
identify how ovulating female sea lampreys track the male sex pheromone (3kPZS) to its
source. Three control algorithms are presented herein that each test a different orientation
hypothesis. The algorithms are then evaluated in computer simulations of the natural
environment. The resulting trajectories are compared with actual observed trajectories
using statistics. Based on this comparison, the best-performing algorithm is chosen.
Strong evidence suggests that sea lampreys use odor-conditioned rheotaxis to locate the

source of synthesized pheromone, as opposed to chemotaxis or stn'ctly rheotaxis.

TABLE OF CONTENTS

LIST OF TABLES ..................................................................................................... v
LIST OF FIGURES ................................................................................................... vi
LIST OF SYMBOLS ................................................................................................. viii
CHAPTER 1
INTRODUCTION ..................................................................................................... l
1.1 Problem Statement ............................................................................. 1
1.2 Background Information .................................................................... 2
1.2.1 Brief Overview of Experiments ............................................. 3
1.2.2 Orientation Hypotheses .......................................................... 4
1.3 Purpose and Content of Thesis .......................................................... 6
CHAPTER 2
RELATED WORK .................................................................................................... 8
2.1 The Moth Approach ........................................................................... 8
2.2 The Bacterium Approach ................................................................... l l
2.3 The Aquatic Organism Approach ...................................................... 12
CHAPTER 3
METHODS ................................................................................................................ l4
3. 1 Control Algorithms ............................................................................ 14
3.1.1 Control Algorithm 1: Chemotaxis ......................................... 17
3.1.2 Control Algorithm 2: Rheotaxis ............................................. 19
3.1.3 Control Algorithm 3: Odor-conditioned rheotaxis ................ 20
3.2 Determination and Validation of Best-Performing
Control Algorithms ............................................................................ 22
CHAPTER 4
RESULTS .................................................................................................................. 23
4.1 Bifurcated Stream Results .................................................................. 23
4.2 Novel Environment Results ............................................................... 26
CHAPTER 5
CONCLUSIONS AND FUTURE WORK ................................................................ 31
APPENDICES ........................................................................................................... 32
Appendix A .................................................................................................... 32
Appendix B .................................................................................................... 34
Appendix C .................................................................................................... 35

iii

REFERENCES .......................................................................................................... 38

iv

LIST OF TABLES

Table l: Localization comparison of movement patterns of ovulated female sea lampreys
in bifurcated stream .................................................................................................... 24

Table 2: Statistical comparison of movement patterns of ovulated female sea lampreys in
bifurcated stream ........................................................................................................ 24

Table 3: Statistical comparison of movement patterns of ovulated female sea lampreys by
control algorithm 3 in novel environment ................................................................. 28

LIST OF FIGURES

Images in this thesis/dissertation are presented in color.

Figure l: Flowchart for control algorithm 1 (chemotaxis) ........................................ 18
Figure 2: Flowchart for control algorithm 2 (rheotaxis) ............................................ 19
Figure 3: Flowchart for control algorithm 3 (odor-conditioned rheotaxis) ............... 21
Figure 4: Sea lamprey movement trajectories in bifurcated stream .......................... 25
Figure 5: Observed sea lamprey movement trajectories in novel environment ......... 29

Figure 6: Simulated sea lamprey movement trajectories by control algorithm 3 in novel
environment ............................................................................................................... 3O

LIST OF SYMBOLS

ENGLISH SYMBOLS

c = 3kPZS concentration, molar (M)

J = performance cost function, no units

N = number of iterations sea lamprey is allowed to keep its current direction, no units
T = sampling time, seconds (5)

v = magnitude of the velocity of the sea lamprey, meters/second (m/s)

Vﬂow = magnitude of the flow velocity, meters/second (m/s)

Wot, y, or 0) = random variables from the environment that affects movement of sea

lamprey, meters (m) for x and y , degrees (0) for 0
ws = chemo-sensor measurement noise, molar (M)

x = horizontal coordinate of sea lamprey position, meters (m)

is, = horizontal centerline of stream, meters (m)

X 0 = variable storing the horizontal coordinates of all observed trajectories, meters (m)
X S = variable storing the horizontal coordinates of all simulated trajectories, meters (m)
y = vertical coordinate of sea lamprey position, meters (m)

)73, = vertical centerline of stream, meters (m)

Y0 = variable storing the vertical coordinates of all observed trajectories, meters (m)

Y S = variable storing the vertical coordinates of all simulated trajectories, meters (m)

z = noisy measurement of 3kPZS concentration, molar (M)

vii

:0 = the lowest possible concentration of 3kPZS that sea lamprey can detect, molar (M)
Zmax = sea lamprey’s memory of the last measured maximum concentration, molar (M)

2, age, = concentration of 3kPZS that sea lamprey uses to determine if it has located
odor source, molar (M)

2,}, = concentration of 3kPZS that sea lamprey uses to determine if it is within odor
plume, molar (M)

GREEK SYMBOLS

a = amount by which sea lamprey rotates to avoid an obstacle or reacquire odor plume,
degrees (°)

5 = threshold for sea lamprey to determine it is directed toward increasing concentration
of 3kPZS, molar (M)

A0 = amount by which sea lamprey is allowed to deviate from “ﬂow direction”, degrees
(o)
/l = forgetting factor used for sea lamprey to discount Zmax’ no units

6 = direction of sea lamprey relative to the horizontal axis, degrees (0)

9d = desired direction of sea lamprey relative to the horizontal axis, degrees (0)
Bﬂow = ﬂow direction (relative to the horizontal axis) + 180°, degrees (0)

p , = Spearman’s rank correlation coefﬁcient between any two quantities, no units

0’ = standard deviation of any quantity, same unit as that quantity

4’ = gain for odor-conditioned rheotaxis strategy, no unit

viii

CHAPTER 1

INTRODUCTION

1.1 Problem Statement

This thesis addresses the problem of identifying the underlying mechanisms used
by sea lampreys to locate an odor source. The sea lamprey (Petromyzon marinus) is a
primitive ﬁsh that parasitizes other ﬁsh when it reaches adulthood. Afterward, it reaches
a sexually mature stage in which it stops feeding and focuses solely on upstream
migration for the purpose of reproduction [1]. In recent years, scientists have discovered
that 3-keto petromyzonol sulfate (3kPZS) is a pheromone emitted by sexually mature
male sea lampreys to attract ovulating female sea lampreys [2]. Of particular interest is an
experiment in which scientists used a synthesized component of 3kPZS and observed the
behavior of ovulating female sea lampreys in response to the odor [3]. The animals were
able to locate the source of the odor over hundreds of meters.

The development of control algorithms that mimic these behaviors in sea
lampreys provides mutual beneﬁt to both engineers and scientists. Such algorithms can
improve the odor source-seeking capabilities of autonomous mobile robots, lending
themselves to be useful in applications such as locating the source of toxic gas and
detecting unexploded ordnance. There are many biologically-inspired odor source
localization algorithms in literature (see Chapter 2), but most of them are not compared
with actual observed movement data to determine their effectiveness. This thesis goes
one step further and develops algorithms that are based on sea lamprey behavior

speciﬁcally.

Concurrently, sea lamprey chemo-orientation mechanisms can be exploited to
control their population in the Great Lakes region, where they have caused much concern
to scientists due to the stress they place on other ﬁsh populations [1], [4]. Current control
methods include the use of 3-trzﬂuoromethyl-4-nitrophenol (TFM), a larnpricide that is
used to kill sea lampreys in their larval stage [4]. However, TFM is expensive and can be
harmful to other ﬁsh, thus the Great Lakes Fishery Commission (GLFC) has reduced its
use by half [4], [5].

This thesis presents three algorithms that each test three different orientation
hypotheses gathered from behavioral ecology: chemotaxis, rheotaxis, and odor-
conditioned rheotaxis (see Section 1.2.2). These hypotheses are used so that the
algorithms are easily validated, and a scientist who is not familiar with pseudo-codes can
still gain understanding from this thesis. These algorithms are evaluated in computer
simulations and compared to observations of ovulating female sea lamprey behaviors
observed by scientists in a bifurcated stream. Subsequently, the best-performing
algorithm is determined by comparing observed and simulated trajectories by statistics.
Furthermore, the best-performing algorithm is evaluated in computer simulations and

compared to the observed data from a novel environment using time series analysis.

1.2 Background Information
This section contains some useful information that will aid in the discussion and

provide a basis for the modeling methods described in Chapter 3.

1.2.1 Brief Overview of Experiments

Scientists in the Department of Fisheries and Wildlife at Michigan State
University conducted two experiments in 2007 and 2008 (not yet published). In these
experiments, ovulating female sea lampreys were released from cages and their
movements were observed. This section will brieﬂy explain both of these experiments.

To begin, there were some procedures common to both experiments that are
worth mentioning. First, synthesized 3kPZS was dissolved in methanol and river water
before being applied to the stream. Second, rhodamine dye was used to visualize the
3kPZS plume in the stream. The distribution and dilution of the plume were measured at
various dye sampling locations and then interpolated. Third, a ﬂow meter was used to
measure the velocity1 of the stream at each dye sampling location. The velocity was
interpolated Fourth, the plume structure and flow velocity were mappedz. Finally,
scientists visually observed sea lamprey movements which were recorded and plotted to
the plume and ﬂow velocity maps. Since movements were observed at discrete locations,
this data was interpolated as well. The behaviors of the individual females were assumed

to be independent [2], [3].

Experiments Conducted in Biﬁrrcated Stream (200 7)
Synthesized 3kPZS was applied to each channel of a bifurcated stream as in Fig.
4. One-hundred and forty-four ovulating female sea lampreys were released from cages

that were 250 m downstream from the sources of 3kPZS. The scientists began movement

 

1 Although the word “velocity” is used, it refers to a scalar quantity (magnitude with no direction).
2 These maps contain only spatial data, and it is assumed that the plume and ﬂow velocity are relatively
constant over time.

3

observations 110 m downstream from the sources of 3kPZS, and only 44 sea lampreys
entered this segment. The scientists observed the full movement trajectories of 33
ovulating female sea lampreys that entered either the left or right channel. The number

that entered each channel and the number that reached each 3kPZS source were recorded.

Experiments Conducted in Novel Environment (2008)

This experiment was conducted in the minor channel of a bifurcated stream as in
Figs. 5 and 6. The channel was 25 m long. The flow of the stream was controlled by a
sandbag wing-darn to create flow and no-ﬂow conditions. To create the ﬂow condition,
the wing-darn was positioned to reroute ﬂow from the major channel into the minor
channel to increase water velocity. To create the no-ﬂow condition, the wing-dam was
used to prevent water from entering the minor channel. However, in the no-ﬂow case,
ﬂow velocity was not completely negligible due to some leakage of the wing-dam and
subsurface water ﬂow. The scientists alternated between applying 3kPZS and no 3kPZS
(or control solvent) to the channel. In summary, there were four conditions: ﬂow with
3kPZS application, ﬂow with control solvent application, no ﬂow with 3kPZS
application, and no ﬂow with control solvent application. The control solvent consisted of
methanol and river water only. The number of sea lampreys to move upstream was

recorded.

1.2.2 Orientation Hypotheses
Taxis (synonymous to orientation with regard to animal behavior) is “an
organism’s maintaining its body position, changing its body position, or both, with regard

4

to stimulus direction.” There are many types of taxis, most of which are deﬁned in terms
of the kind of stimulus and stimulus reception [6]. For instance, bacteria use chemotaxis
(taxis in which a chemical is the stimulus) to locate glucose, serine, and other sources that
provide them energy [7].

It is easy to hypothesize that sea lampreys use chemotaxis, since the objective of
this thesis is to determine the ovulating female’s response to pheromone. Rheotaxis (taxis
in which the water current is the stimulus) is another assumption to make, because sea
lampreys dwell in water. However, in a natural environment, a single sense can become
unreliable as stimuli become unavailable [8]. Sea lampreys may be able to locate an odor
source if the chemical is always present But if the chemical is unavailable, will they stop
their search or continue using another sense? The former could very well be their demise,
and so they will have to rely on other senses for survival.

Many organisms exhibit chemoreception combined with some other sensory cue.
For instance, foraging blue crabs use chemotropotaxis (a type of chemotaxis in which
multiple sensors are used simultaneously to detect chemicals) and rheotaxis to locate the
odor emitted by their food source [9]. Since chemical molecules are primarily transported
by advection [10], rheotaxis is just as important as chemotropotaxis for the blue crab to
locate its food source. Just as blue crabs rely on mechanoreception for odor source
localization, so do male cockroaches. They modulate their orientation to the wind in
response to the female sex pheromone. This movement pattern is called odor-modulated
anemotaxis (taxis in which the wind is the stimulus) [l 1].

Due to this evidence, this thesis will consider odor-conditioned rheotaxis [12] as a

strong hypothesis for sea lamprey behavior. Odor-conditioned rheotaxis can be described

5

as rheotactic or upstream3 behavior that is initiated or enhanced by an odor, usually at
some threshold. Ifthe perceived odor does not meet the threshold requirement, movement
patterns will be exploratory.

In summary, this thesis will consider the following three hypotheses: chemotaxis,

rheotaxis, and odor-conditioned rheotaxis.

1.3 Purpose and Content of Thesis

The purpose of this study is to explain the underlying mechanisms used by sea
lampreys to locate an odor source. This paper seeks to combine behavioral ecology and
control theory to accomplish this while providing mutual beneﬁt to members of the
scientiﬁc and engineering communities.

Orientation hypotheses will be used as the foundation for sea lamprey control
algorithms. Each algorithm will be evaluated in computer simulations. These simulated
results will be compared with the observed behaviors from the bifurcated stream
experiment using statistics. It is the hope of the author that the simulated behaviors
resemble the observed behaviors within reasonable error.

Based on how well each algorithm performs, one will be chosen to explain the
odor source localization mechanisms of the sea lamprey. This approach will be applied to
the novel environment and evaluated in computer simulations. If the best-performing
algorithm effectively models sea lamprey chemo-orientation, then when applied to a

novel environment, it should still mimic the movements of real sea lamprey.

 

3 Taxes can be either positive (toward the stimulus, i.e. upstream rheotaxis) or negative (away from the
stimulus, i.e. downstream rheotaxis). Since this thesis studies the ovulating female sea lamprey’s
orientation toward the male sex pheromone, all taxes hypothesized here is considered to be positive.

6

The remainder of this thesis is organized as follows. Chapter 2 discusses the
various biologically-inspired odor source localization methods in literature. Chapter 3
describes the theory and logic that leads up to the development of the three control
algorithms. Also in this chapter is the protocol used to determine the best-performing
algorithm. Chapter 4 shows the simulated results and compares them with the observed
data. Chapter 5 derives conclusions from the results presented in Chapter 4 and provides

recommendations for future work.

CHAPTER 2

RELATED WORK

This chapter reviews the current biologically-inspired algorithms and techniques
for odor source localization. It is important to note that there aren’t any algorithms
inspired by the sea lamprey or any other vertebrates yet. This is because invertebrates
have proven themselves to be a worthier model for olfactory research than vertebrates
[8]. Hence not much is known about the underlying chemo-orientation mechanisms of
vertebrates. On the other hand, the odor-mediated behaviors of invertebrates and the
simplest of organisms are relatively well-documented. With that in mind, the approaches
discussed in this chapter may or may not be sufficient to identifying sea lamprey chemo-
orientation mechanisms. This chapter begins with the most-studied of all invertebrates:

the moth.

2.1 The Moth Approach

The female moth releases a sex pheromone to attract the male. Upon reception of
the pheromone, the male turns into the wind by an optomotor reaction to the perceived
movement patterns of the ground (optomotor anemotaxis“) [13]. When the male losses
the scent of the pheromone, he switches to a “casting” behavior in which he ﬂies about

perpendicular to the mean wind direction with left-right reversals.

 

‘ An optomotor reaction or reﬂex is “an individual’s attempting to maintain its entire body, or part of its
body, in a constant position with regard to a moving environment” [6]. Optomotor anemotaxis can only be
deﬁned as an individual’s attempt to maneuver upwind with regard to an environment that only appears to
be moving because the individual is in motion. This mechanism requires the use of visual cues.

8

In [14], an unmanned aerial vehicle called the AMOTH (artiﬁcial moth) was used
to locate a source of ethanol in a wind tunnel (3 x 4 x 0.54 m (width x length x
height)) using moth optomotor anemotaxis. The absence and presence of ethanol caused
the AMOTH to switch between two behaviors, casting and surge, respectively. The left-
right reversals of the casting behavior resulted in a “zigzag” trajectory. The surge
behavior consisted of the AMOTH aligning itself with the wind and traveling against it.
Meanwhile, the AMOTH was constantly checking its path for obstacles using visual cues.
The detection of an obstacle overrode the search strategy and collision was avoided.

This system was able to navigate the wind tunnel with the obstacles and
effectively locate the source of ethanol. It was able to detect the ethanol at almost 4 m
from the source (ﬁrst detection coincides with ﬁrst switch to surge mode), about the
maximal length of the wind tunnel. From this observation, the authors of [14] drew the
conclusion that the model can exploit the full dynamic range of the odor plume.
However, at such a short distance to the odor source, and with the wind direction in the
tunnel being uniform, it is not for certain how effective the model truly is. Even in these
limited conditions, the AMOTH’s trajectory was determined to be suboptimal and
inconsistent with the documented behaviors of real moths. Although the authors did not
compare the trajectories with actual moth trajectories, so it is not known in exactly what
way the AMOTH was suboptimal and inconsistent.

Perhaps it has something to do with the manner in Which real moths anemotax as
opposed to the direct upwind surges described in [14]. According to [15], real moths
make regular turns across the direction of wind ﬂow whether they are in contact with the

odor plume or not. This differs from the “surge” technique described in [14] and most

9

other algorithms. In [15], the performance of two different algorithms was compared
using Digiduca [16], a virtual wind tunnel. The ﬁrst algorithm consisted of a simple
surge-cast switching strategy. The second algorithm was a modiﬁed version of the ﬁrst
that included countertuming across the direction of wind ﬂow while making upwind
progress when the odor was detected

Model parameters for both algorithms were constrained to values obtained from
moth experiments. For the ﬁrst algorithm, less than 5% of the individuals ﬂown into the
wind tunnel located the odor source in the time allowed. The second (modiﬁed) algorithm
had a success rate of about 30%. Success rates for real moths solving the same problem
were at least 50% and often 90%. The improved success rates when using the modiﬁed
algorithm suggests that including a countertuming mechanism at all times (as opposed to
surging when odor is present) could improve the optimality and consistency with real
moth behaviors. However, the modiﬁed approach still fell short of the performance of
real moths. The authors attributed this to the selection of model parameters and used a
genetic algorithm (GA) to determine the best combination of parameters. These optimal
parameter values yielded similar success rates but different behaviors in comparison to
real moths. Therefore, [15] shows that there was a trade-off between performance (i.e.
success rates) and behaviors.

In [17], the silkworm moth’s wing-fanning technique is applied to a robotic
platform. The male silkworm moth, who walks when tracking pheromone rather than ﬂy
like other moths, uses wing vibrations to draw the pheromone to him and determine the
direction of the pheromone source [18]-[20]. A comparison was made between moths

with wings removed and moths with wings intact. Odor source localization took longer

10

and was much more difficult for the moths with removed wings. Hence, the robotic
platform developed in [l 7] has a directional probe that consists of a small fan and a
semiconductor gas sensor, mimicking the moth’s wings and antennae respectively. This
replaced the need for plural gas sensors to measure the concentration gradient and
anemometric sensors used in other systems [21], [22].This localization system was able
to successfully locate the source of ethanol in wind tunnel but in much longer time than it
took real moths to locate the source of pheromone. An improvement to the directional
probe made the system feasible in a clean room with multiple wind sources that created a

non-uniform wind proﬁle.

2.2 The Bacterium Approach

Bacteria such as E. coli use chemotaxis to locate food sources. They are able to
progress up an odor concentration gradient in a series of nm and tumble behaviors. Run is
when a bacterium swims in one direction, and tumble is when it randomly chooses a new
direction. The ﬁ'equency at which it switches between these two behaviors is modulated
with respect to the odor concentration gradient. In other words, it tumbles less frequently
when the odor concentration gradient is found to be positive. When there is no
concentration gradient, the bacterium executes a random walk [23].

Ref. [24] describes how bacterial chemotaxis — termed “biased random walk” —
can be used for the location of gradient sources, whether odor, light, or heat. Simulations
of the biased random walk were conducted in a two-dimensional grid model of the real
world. Robots traveled with a mean free path (MF P) of 10 units of distance before

tumbling whenever the concentration gradient was absent (run and tumble as described

11

and was much more difficult for the moths with removed wings. Hence, the robotic
platform developed in [17] has a directional probe that consists of a small fan and a
semiconductor gas sensor, mimicking the moth’s wings and antennae respectively. This
replaced the need for plural gas sensors to measure the concentration gradient and
anemometric sensors used in other systems [21], [22].This localization system was able
to successﬁrlly locate the source of ethanol in wind tunnel but in much longer time than it
took real moths to locate the source of pheromone. An improvement to the directional
probe made the system feasible in a clean room with multiple wind sources that created a

non-uniform wind proﬁle.

2.2 The Bacterium Approach

Bacteria such as E. coli use chemotaxis to locate food sources. They are able to
progress up an odor concentration gradient in a series of run and tumble behaviors. Run is
when a bacterium swims in one direction, and tumble is when it randomly chooses a new
direction. The frequency at which it switches between these two behaviors is modulated
with respect to the odor concentration gradient. In other words, it tumbles less frequently
when the odor concentration gradient is found to be positive. When there is no
concentration gradient, the bacterium executes a random walk [23].

Ref. [24] describes how bacterial chemotaxis — termed “biased random walk” —
can be used for the location of gradient sources, whether odor, light, or heat. Simulations
of the biased random walk were conducted in a two-dimensional grid model of the real
world. Robots traveled with a mean free path (MF P) of 10 units of distance before

tumbling whenever the concentration gradient was absent (run and tumble as described

11

above). Like real bacteria, if the robot sensed a positive gradient, it would decrease its
tumbling frequency for a greater run length. Once the simulations were complete, the
algorithm was implemented on a robot in a phototaxis (taxis in which light is the
stimulus) experiment and compared with the gradient descent method [25]. The robot had
a much lower chance of being trapped between concentration minima or maxima when

using the biased random walk strategy than the gradient descent strategy.

2.3 The Aquatic Organism Approach

When it comes to aquatic organisms, invertebrates (i.e. crustaceans) are still more
studied by scientists than vertebrates (i.e. ﬁsh). Unlike ﬁsh that swim freely in an ocean,
lake, or river, the aforementioned crustaceans each dwell at the lowest level of water at
the interface of the ocean ﬂoor and ﬂuid [26], [27]. As such, an odor signal takes longer
to reach their antennules and they have to ﬂick them, similar to wing farming in silkworm
moths, to enhance the perception of odors and determine the direction of the source [26].

The authors in [28], believe that comparing the known behavior of a robot to the
unknown behavior of an animal will provide information about that animal’s chemo-
orientation mechanisms. They used a biomimetic robot, Robolobster, to test (and discard)
chemo-orientation hypotheses. It is about the size of a real lobster with comparable
turning ability and speed, but it is not intended to match the lobster in appendages or
mode of locomotion. Rather it is programmed with two simple algorithms to understand
Robolobster’s interaction with the turbulent plume. The ﬁrst algorithm has two rules: the
robot will steer toward the side of the chemo-sensor receiving the higher concentration

signal; otherwise the robot moves forward with a constant speed. The second algorithm is

12

the same as the ﬁrst with the addition of a third rule: if both chemo-sensors do not receive
any concentration signal, the robot will back up at half speed. This third step improved
Robolobster’s accuracy but diminished its speed.

Ref. [29] is a continuation of this study. The Robolobster is made to track a plume
of salt to its source in the same conditions as real lobsters that were observed orienting to
a source of food extract. The Robolobster used the second algorithm in [28]. The authors
compared Robolobster’s behavior to that of real lobster and were able to determine that
lobsters steer toward the antennules receiving the highest concentration signal. Although
the algorithm was not able to fully characterize lobster chemo-orientation, the authors
gained much insight into lobster behavior and can make improvements to the algorithm.

An underwater robot was developed in [30] mimicking the odor-mediated
behavior of the crayﬁsh. It consists of an array of electrochemical sensors and a pair of
farming devices mimicking the antennule ﬂicking of the crayﬁsh. The robot was placed in
an aquarium with stagnant ﬂow and its ability to locate the source of a chemical was
monitored. The robot was able to locate the chemical source successfully, although this

conclusion was not drawn in direct comparison to crayﬁsh behavior.

13

CHAPTER 3
METHODS

3.1 Control Algorithms
All three algorithms are a discrete-time kinematic model, which can be considered
as an averaged or sampled kinematic model, of the sea lamprey behavior. They each test
the three orientation hypotheses below.
1. Chemotaxis: The sea lamprey swims in the direction of increasing 3kPZS
concentration. Otherwise it randomly explores a new direction. Movements are

made independent from 6170“,.

2. Rheotaxis: The sea lamprey swims upstream when it is able to perceive ﬂow.
Movements are made independent from the level of 3kPZS concentration.

3. Odor-conditioned rheotaxis: The sea lamprey swims in an upstream direction
(rheotaxis) when it detects 3kPZS at or above a concentration threshold (i.e. when
it is within the odor plume). Otherwise it executes a countertuming maneuver and
is allowed to swim downstream if necessary to reacquire the odor plume.

Movements are always made with respect to Bﬂow.

The three control algorithms regulate 9 while keeping v constant to direct the
sea lamprey toward the odor source. The decision for 9 depends on one of three

parameters: 2 (chemotaxis and odor-conditioned rheotaxis), Bﬂow (rheotaxis and odor-

conditioned rheotaxis), and obstacles (all control algorithms).

14

During respiration, the sea lamprey samples a plume of odor by moving water
into and out of the olfactory epithelium5 [31]. For each sniff, it takes a noisy
measurement of c

z(k) 2 (30¢) + ws (k)

where ws ~ N (0, 0'3 ) is the measurement noise modeled as Gaussian white noise”. The

control algorithms (except rheotaxis) are applied to spatial 3kPZS plume structure maps
and image processing is used to convert the colors to raw values of 3kPZS concentration
in molar at each location on the maps.

The ﬂow direction 6 ﬂow is important because moving upstream will get the sea

lamprey closer to the source [3]. Image processing is used on the ﬂow velocity maps

(except chemotaxis) to convert the colors to raw values of Vﬂow each location on the
maps. The “ﬂow direction”7 6170“, is assumed to be parallel to the banks of the stream.

Finally, each control algorithm incorporates an obstacle avoidance strategys,
because real sea lampreys are not physically able to go through them. Without such a
strategy, a simulated sea lamprey is able to disobey this rule and a robotic sea lamprey

will get stuck at an obstacle. Given that obstacles constrain where the sea lamprey can

 

5 The olfactory epithelium is a thin protective layer of tissue in the nasal cavity that is used to detect odors.
5 Low-level Gaussian noise effectively models noises due to central and peripheral functions of olfactory
neural systems [32].

7 The “ﬂow direction” aﬂow is analogous to the actual stream ﬂow direction: aﬂow = stream ﬂow

direction +180°. See appendices for more detail.

8 Given the fact that sea lampreys do not use vision during upstream migration [1], their obstacle avoidance
strategy will have to differ from that of animals who avoid colliding with obstacles by relying on optomotor
reﬂexes (i.e. silkworm moths). A sea lamprey may not be able to actually avoid an obstacle but can push
away from it or follow its contour).

15

move and affect the ﬂow, this strategy overrides the main orientation strategy of the
control algorithms [3 3].
All algorithms begin with an “IN ITIALIZE” process in which the initial values

for (x, y) , 6 , and other parameters are set. The remaining processes for the algorithms

are presented below with ﬂowcharts that show the sequence of their execution. Each
process contains steps that are presented in the appendices.

Each control algorithm is developed with “what ifs” in mind: “what if” there is no
ﬂow velocity or “what if” there is no odor present. It is assumed that a sea lamprey would
know what to do in these situations, no matter what strategy it is acting by, because it is
able to swim hundreds of meters to locate a source of 3kPZS. Therefore, the control
algorithms are designed so that when the stimulus or stimuli that each relies on is absent,
the sea lamprey will not just remain at one location and wait. Rather it will seek it out.

Within each algorithm there are parameters whose values affect the movement
trajectories generated in the simulations (see appendices). The utmost care must be given
to the calibration of these parameters, and so J is deﬁned to be used with parameter

optimization [34].

 

 

 

 

 

 

 

J = _ + _
xst yst
+|70 -Xsl +llfo-T'sl
its! is!

+(1- PX0,XS ) + (1 - PX0,XS)

By statistical theory [35], if two trajectories are the same then they have equivalent

standard deviations and equivalent means. Likewise, p with a magnitude of 1 indicates

16

that two trajectories are perfectly monotonically related. Therefore, the parameters that
result in simulated trajectories that best resembles the observed trajectories are those that

will minimize J .

3.1.1 Control Algorithm 1: Chemotaxis
l. DETECT ODOR: This process will begin if the sea lamprey does not detect the

odor at the onset of the simulation (i.e. z(l) < :30 9). First, the sea lamprey picks a

random direction to explore. It continues in that same direction for N 0 iterations

or until an obstacle is detected. When either of these events happens, it will orient
itself in a new random direction. It repeats this sequence until the odor is detected.

2. FIND ODOR SOURCE: At the ﬁrst sign of odor, this process will begin. The sea
lamprey uses chemotaxis to swim in the direction of increasing 3kPZS
concentration. It determines that the concentration is increasing by comparing it
with a stored maximum concentration. If the 3kPZS concentration does not

increase for N1 iterations, it will explore a new direction at random. This

sequence was inspired by the “”run and “tumble” observed in bacterial

chemotaxis [23]. This process ends when the odor source is found (i.e.

z(k) > Ztarget )-

 

9 This value is 20 = 10'14 M, the lowest concentration at which it was observed by scientists that 3kPZS
elicited a response from ovulating female sea lampreys.

l7

’M..___.._. _.... ... .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l/ START \
\ /
IN ITIALIZE
NO ‘\‘\\Z(k) Z zb YES
DETECT
ODOR
l
/y\, \
NO <3“) 2 20>; YES
. \‘\\/
FIND ODOR
SOURCE
YES 1Q) 2: myr¥ NO
\\//
//_.M_-__._jL._._.\
l END \
\ /

Figure 1: Flowchart for control algorithm 1 (chemotaxis)

l8

 

3.1.2 Control Algorithm 2: Rheotaxis
FIND ODOR SOURCE: If there is no ﬂow at the onset of the simulation (i.e.

Vﬂowa) = 0), the sea lamprey will explore a random direction. It will keep that direction

until an obstacle is encountered or it detects ﬂow. Once it detects ﬂow, it will swim
against it. If the ﬂow becomes absent for whatever reason, the sea lamprey will continue
swimming in the same direction until ﬂow is detected again when it will swim against as
before. Since a sea lamprey acting on this control algorithm is not informed of the odor
source location by chemical information, the steps in this algorithm will be executed in

an inﬁnite loop as shown in Figure 2.

/P‘——-\
r START l
\ /

.
‘b——

 

 

INITIALIZE

 

 

 

 

 

 

FIND ODOR
SOURCE

I

Figure 2: Flowchart for control algorithm 2 (rheotaxis)

 

 

 

 

19

 

3.1.3 Control Algorithm 3: Odor-conditioned rheotaxis

1.

FIND PLUME: If at the onset of the simulation the sea lamprey is not within the

odor plume (i.e. z(l) < 2,}, 10), this process will be executed. The sea lamprey

begins in an upstream direction or a random direction if no ﬂow is detected as in
control algorithm 2. It continues in this direction until an obstacle is encountered.
Once the sea lamprey determines it is within the odor plume, it will proceed to the
next step.

TRACK PLUME: The sea lamprey is within the odor plume at the onset of this
process and will try to track the plume to the odor source using rheotaxis. If the
sea lamprey leaves the plume, it will execute the next process. This process ends
when the sea lamprey locates the odor source.

REACQUIRE PLUME: The sea lamprey will countertum with respect to 6/10“,

so that it re-enters the odor plumel 1. This process ends when the sea lamprey

reacquires the odor plume.

 

‘0 On the plume structure maps, the “plume” of 3kPZS is depicted by the yellow region. The lowest
concentration of this region is z”, .

” This uses the assumption that if 9 is greater (less) than aﬂow’ decreasing (increasing) 9 will result in

the sea lamprey re-entering the plume.

20

{ START \
\

/

 

 

 

 

 

 

 

 

 

 

 

 

 

INITIALIZE
O /'., / \;\\ \
—lL————<.4k)2., YES
FIND PLUME
//,/ \ \ ‘
No {any “35

 

 

 

TRACK
PLUME

 

 

 

I \

YES ,.// \‘\ N0
:(k) 2 ap—

 

 

 

 

 

/
// x‘
NO// - YES
\{(\k) > ~(arg/e REACQUIRE
\\V// PLUME

 

 

 

NO /‘L \ YES

\\‘
WE Zgb/

_..._Jt-..____n
F END \.
\ /

 

 

 

 

 

 

 

Figure 3: Flowchart for control algorithm 3 (odor-conditioned rheotaxis)

21

3.2 Determination and Validation of Best-Performing Control Algorithm

The simulated trajectories by control algorithms 1, 2, and 3 are compared to the
observed movements in the bifurcated stream experiments (2007). The comparison is
restricted to the 1 10 m segment of the bifurcated stream where the scientists monitored
for sea lamprey movements. The initial positions recorded by the scientists for the 33
different observed trajectories are chosen as starting points for the 33 simulated runs of
each algorithm. The observed trajectories were recorded at non-uniform sampling times.
For a fair comparison, the trajectory points of the observed data are interpolated to make
their sampling time uniform and equivalent to the sampling time of the simulated
movements.

The number of simulated sea lampreys that entered the left and right channels and
successfully located the 3kPZS source in each channel according to control algorithms 1,
2, and 3 are compared to the observed data. Furthermore, Eq. ( l) is used to quantify the
similarities (or lack thereof) between the simulated and observed trajectories. The best-
performing control algorithm is chosen to be that with the lowest value of J .

Once the best-performing control algorithm is determined, it is evaluated in a
computer simulation of the novel environment (2008). The algorithm parameters
calibrated in the bifurcated stream are used. The simulated trajectories are compared with
the observed trajectories using Eq. (1) to validate the best-performing control algorithms

applicability to any environment. Next the results are presented.

22

CHAPTER 4
RESULTS

4.1 Bifurcated Stream Results

Without the use of ﬂow data, the sea lampreys simulated by control algorithm 1
(chemotaxis) were unable to progress upstream to either source of 3kPZS in the
biﬁrrcated stream”. A reason for this is that in a turbulent environment, the odor plume
becomes intermittent and the concentration gradient has many local maxima and minima
[8], [36]. Even in the absence of turbulence, odor plumes tend to meander large distances
from the source [3 7]. Taking into account that 3 out of 33 (Table l) of the sea lampreys
simulated by control algorithm 1 were able to progress into one of the two channels
suggests a possibility that it could work if the sea lampreys are in proximity to the odor
source, but not over hundreds of meters.

The absence of olfaction in control algorithm 2 (rheotaxis) caused the sea
lampreys to swim past both sources of 3kPZS. However, they were able to progress
upstream into either of the channels. A lesser number of sea lampreys simulated by
control algorithm 2 entered the right channel than the real sea lampreys (12 out of 33
compared to 20 out of 33 respectively; Table 1).

Control algorithm 3 simulated sea lampreys whose movements into the left and
right channel were not different than real sea lampreys. With the combination of chemo-

and mechano-reception, the simulated sea lampreys located the sources of 3kPZS with a

 

‘2 The computer simulations timed out if the sea lampreys did not locate the source of 3kPZS in 10.000
iterations, which corresponds to 5,000 seconds. Running the simulations for longer lengths of time requires
more memory than is available on a computer.

23

success rate comparable to that of real sea lampreys. Tables 1 and 2 both show signiﬁcant
evidence that control algorithm 3 is the best strategy in the bifurcated stream. This is also
true by a qualitative comparison as shown in Fig. 4.

Table l: Localization comparison of movement patterns of ovulated female sea
lampreys in bifurcated stream. The number of ovulated female sea lampreys and
simulated females according to control algorithms 1 (chemotaxis), 2 (rheotaxis), and 3
(odor-conditioned rheotaxis) that entered the left channel (“Left”) and right channel
(“Right”). Percent success left and right is the percent of the sea lampreys that entered the
left or right channel that also entered within 0.5 m2 of the left 3kPZS source or right
3kPZS source respectively.

 

Data Source Observed Left Right % Success Leﬁ % Success Right
Exp. 33 13 20 92% 95%
Alg. l 33 l 2 0% 0%
Alg. 2 33 21 12 <5% 0%
Alg. 3 33 12 21 83% 57%

 

 

 

 

 

Table 2: Statistical comparison of movement patterns of ovulated female sea
lampreys in bifurcated stream. Means and standard deviations of J generated twenty
times for simulated sea lampreys according to control algorithms 1 (chemotaxis), 2
(rheotaxis), and 3(odor-conditioned rheotaxis).

 

Alg. l Alg. 2 Alg. 3
j 6.2240 3.2684 2.3970
0", 2.0583 0.9841 1.5411

 

 

 

24

Figure 4: Sea lamprey movement trajectories in bifurcated stream. Observed
movement trajectories of ovulating female sea lampreys (a) and simulated trajectories by
control algorithms 1 (b), 2(c), and 3 (d). Magenta trajectories are the females that entered
the leﬁ channel and cyan trajectories are females that entered the right channel. Color-
coding indicates the estimated concentration of synthesized 3kPZS (M) through the
stream.

(a) Observed movements (b) Control algorithm 1

 

25

Fig. 4 cont.
(c) Control algorithm 2 ((1) Control algorithm 3

 

4.2 Novel Environment Results

Based on the performance results in Tables 1 and 2, control algorithm 3 was
applied to four environmental data sets described in Section 1.2.1 to validate it in
different physical habitats that were not used for calibrating it (bifurcated stream).
Simulated movements of sea lampreys by control algorithm 3 in the novel environment
match well with observed movements. This strongly supports odor-conditioned rheotaxis,
and more speciﬁcally control algorithm 3, as an underlying mechanism for chemo-

orientation in sea lampreys. It is interesting that the simulated trajectories by control
26

algorithm 1 in the biﬁrrcated stream (Fig. 4 (b)) resemble more closely the observed
movements in the no-ﬂow and 3kPZS case (Fig. 5 (c)) than control algorithm 3 (Fig. 6
(c)). This could be due to any of the following reasons:
1. The sea lamprey switches strategies when there is no ﬂow but a presence of
3kPZS, so it progresses upstream in a purely chemotactic manner.

2. Control algorithm 3 is able to use the most minuscule value of Vﬂow to determine
Qﬂow as long as it is not zero, although real sea lampreys do not seem to be able
to detect such low quantities of ”ﬂow Control algorithm 3 needs to take into
account a threshold value for Vﬂow» i.e. the minimum ﬂow velocity at which the
sea lamprey is able to discern Hﬂow.

3. Rather than have the sea lamprey use the previous measurement of 0 ﬂow when
Vﬂow = 0 , Qﬂow needs to, be a random direction.

These possibilities lead to the conclusions of this thesis and recommendations for

future work.

27

Table 3: Statistical comparison of movement patterns of ovulated female sea
lampreys by control algorithm 3 in novel environment. Means and standard

deviations of J generated twenty times for simulated sea lampreys according to control
algorithm 3.

 

Treatment J 0' J
F low-3kPZS 3.5448 1 .0529
F low-Control 3.6524 1.1987
No-ﬂow-3kPZS 4.1973 1.1235
No—ﬂow-Control 4.2375 1 .2783

 

 

28

Figure 5: Observed sea lamprey movement trajectories in novel environment. (3)
Flow — 3kPZS application. (b) Flow - control solvent application. (c) No-ﬂow - 3kPZS
application. ((1) No-ﬂow — control solvent application. C olor-coding indicates the
estimated concentration of synthesized 3kPZS (M) through the stream.

(a) Flow - 3kPZS (b) Flow —— control solvent

 

(d) No-ﬂow - control solvent

1
if ' .
¢k\
‘" .\\
b

I- \

 

29

Figure 6: Simulated sea lamprey movement trajectories by control algorithm 3 in
novel environment. (a) Flow — 3kPZS application. (b) Flow — control solvent
application. (c) No-ﬂow — 3kPZS application. ((1) No-ﬂow — control solvent application.
Color-coding indicates the estimated concentration of synthesized 3kPZS (M) through the
stream.

(a) Flow — 3kPZS (b) Flow — control solvent

 

(c) No-ﬂow — 3kPZS (d) No-ﬂow - control solvent

 

30

CHAPTER 5

CONCLUSIONS AND FUTURE WORK

In this thesis, three control algorithms founded on behavioral ecology and control
theory are presented. Control algorithm 1 is similar to bacterial chemotaxis [7], [23] and
uses only the concentration of 3kPZS to locate the odor source. Control algorithm 2 uses
only ﬂow information for the sea lamprey to try to locate the odor source. Control
algorithm 3 uses both concentration and ﬂow information The three algorithms were
evaluated in computer simulations and based on the results presented in Chapter 4,
control algorithm 3 was highly successful, and thus, odor-conditioned rheotaxis is a
sufficient explanation for sea lamprey chemo-orientation. Testing control algorithm 3 in a
novel environment validates its robustness. The author believes this thesis to be a helpful
contribution to the study of olfaction for both scientists and engineers, particularly in
vertebrates which are understudied compared to invertebrates. Although there may be
some additional chemo-orientation mechanisms that remain unrevealed, an investigation
into the following may provide more insight: determine if sea lampreys switch between
odor-conditioned rheotaxis and chemotaxis; take into account a minimum threshold ﬂow
velocity at which the sea lamprey can measure the “ﬂow direction”; randomize “ﬂow
direction” when ﬂow velocity is zero. The aforementioned investigation will provide
improvements (if any) to control algorithm 3, after which the algorithm can be
implemented on an autonomous underwater vehicle (AUV) equipped with chemo— and

mechano-sensors to locate an odor source.

31

APPENDICES

A. Control algorithm 1: chemotaxis

INITIALIZE
11. Let the initial position and heading angle be (x(1), y(1)) and 6(1) respectively.
12. Measure 2(1).
13. Let zmax(l) = 0, 05(0) = O and 19d(0) is random direction.

14. Let k = l.
DETECT ODOR

01. while the sea lamprey does not detect any odor (z(k) < 20) do:
02. if obstacle is encountered then: 6d (k) is chosen such that the obstacle is

avoided end if

03. if 6d has been the same direction for the last N 0 iterations then:
6d (k) is random direction and a(k) = 0
else: 6d (k) = 9d (k — l) and a(k) = 0 end if
04. (State update) x(k +1) = x(k) + T v cost9(k) + wx (k)
y(k +1) 2 y(k)+ Tvsint9(k)+ w ,(k)
6(k +1) 2 9d (k) + we (k)
05. zmax (k +1) 2 max{ z(k), zmax(k)}
O6. Measure z(k +1).

07. Increment k by 1 end while

32

FIND ODOR SOURCE

SI. while sea lamprey has not located the odor source (z(k) < z, argeﬁ do:
82. if obstacle is encountered then: 9d (k) is chosen such that the obstacle is avoided.

end if

else: go to step 82 end if

S3. if 3kPZS concentration is not increasing (z(k) S zmax (k) + 5 ) then:
if sea lamprey does not detect odor (z(k) < 20) then:
if 6d has been the same direction for the last N 0 iterations then:
9d (k) is random direction and a(k) = 0

else: 9d (k) = 6d (k -—1) and a(k) = 0 end if

else:
if 6d has been the same direction for the last N1 iterations then:
6d (k) is random direction and a(k) = 0
else: 6d (k) = 6d (k — 1) and a(k) = 0 end if

end if

else: 9d (k) = 0(k — 1) and a(k) = 0 end if

S4. Perform steps 03-05.

S5. if z(k) S zmax + 6 then: zmax(k +1) 2 xizmax(k)
else: zmax(k +1) 2 z(k) end if

S6. Measure z(k +1).

S7. Increment k by 1 end while
33

 

B. Control algorithm 2: rheotaxis

INITIALIZE
11. Let the initial position and heading angle be (x(1), y(1)) and 0(1) respectively.
12. Measure vm ag(1).
I3. if vmag(l) : 0 then: lamprey selects a random direction for Hﬂowa)
else: Bﬂowﬂ) is measured at (x(1), y(1)) end if
14. Let a(0) = 0 and 6d (0) = 6 ﬂow(1)-

15. Let k =1.
FIND ODOR SOURCE

S 1. if obstacle is encountered then: 6d (k) is chosen such that the obstacle is avoided

end if
52. (State update) x(k +1) = x(k) + T9 cosB(k) + w, (k)
y(k + 1) = y(k) + Tvsin6’(k) + w, (k)
9(k +1) = 6,, (k) + wa(k)
S3. Measure vmag(k +1).
S4. 1r vﬂow(k +1) = 0 then: BﬂOMk +1) = aﬂowac)
else: 6ﬂ0w(k +1) is measured at (x(k +1), y(k +1)) end if

S5. Increment k by 1

S6. Perform steps Sl-S5.

34

C. Control algorithm 3: odor-conditioned rheotaxis

INITIALIZE
11. Let the initial position and heading angle be (x(1), y(1)) and 6(1) respectively.
12. Measure 2(1) and vmagﬂ).
13. if vmag(1) = 0 then: Bﬂowﬂ) is random direction
else: Bﬂowﬂ) is measured at (x(1), y(1)) end if
14. Let (1(0) 2 O.
15. if z(l) < 2,), then: 6d (0) is random direction in
[9ﬁ0u.1(1)- A91a9ﬂ0w(1)+ 491]
else: 9d (O) is random direction in [6170».(1) — A62 , 6ﬂ0w(1)+ A62]
16. Let k =1.

FIND ODOR PLUME

P1. while sea lamprey is not in the odor plume (z(k) < 2,}, ) do:
P2. if obstacle is encountered then: 6d (k) is chosen such that the obstacle is avoided

end if

P3. 1r 9(k) < 9,70,,(k) - A6, AND 61(k) > eﬂwrk) + A611 then:
19,, (k) = 91,0“,(k) end it
P4. (State update) x(k +1) = x(k) + Tv 005901?) + w, (k)
y(k +1) = y(k) + Tvsir16(k) + w,,(k)
6(k +1) = 9a (k) + Watk)

35

 

P5. Measure z(k +1) and vmag(k +1).
P6. if vmag(k +1) 2 0 then: 6110“,.(k +1) = 6ﬂ0W(k)

else: Bﬂow(k +1) is measured at (x(k +1), y(k +1)) end if
P7. Increment k by 1 end while

TRACK PLUME

Tl . while sea lamprey has not located the odor source (z(k) < z, alga) do:

T2. if sea lamprey exits odor plume (z(k) < 2,], ) then: go to step R1

r

else: go to step S3 end if

T3. if obstacle is encountered then: 6d (k) is chosen such that the obstacle is avoided
else: 6d (k) = 9d (k — 1) and a(k) = 0 end if

T4. if 6d (k) < 6ﬂ0w(k) —~ A62 0R 6d (k) > 6ﬂ0w(k) + A62 then:
(Z; (k) :2 6ﬂ0w(k) end if

T5. Perform steps P4-P6.

T6.Increment k by 1 end while

REACQUIRE PLUME

R1. while sea lamprey is not in odor plume (z(k) < 21h) do:
R2. if obstacle is encountered then: 9d (k) is chosen such that the obstacle is

avoided
else: go to step R3 end if

R3. if sea lamprey exited odor plume on right side then:

36

 

6d(k) =6d(k—1)+a(k-l) and a(k) = 4x1013|z(k)—z,h|

else: (Z; (k) = 0d(k - ])-a(k ‘1) and “(1‘): {x 1013M“ ‘Zthl

end if
R4. Perform steps P4-P6.

R5. Increment k by 1 end while

37

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