INSECT NAVIGATION BY POLARIZED LIGHT

F. J. Smith

School of Electronics, Electrical Engineering and Computer Science, Queens University, Belfast, N. Ireland

Keywords: Polarization, Skylight, Navigation, Clouds, POL, Insect celestial map, Robot navigation.

Abstract: Many insects can navigate accurately using the polarized light from the sky. A study of a large number of

experiments on the behaviour and anatomy of insects has led to a simple algorithm for navigation by

skylight, suitable for a robot or drone in lightly clouded skies. The algorithm is based on the special ability

of insect eyes to measure the position of the 4 points in the sky at which the polarization angle, i.e. the angle

χ between the polarized E-vector and the meridian, equals ±π/4. The azimuths of these 4 points are possibly

the only measurable quantities that are invariant to variable cloud cover, provided that polarized light is still

detectable below the clouds. It is shown that the sum of these 4 azimuths can be turned into a celestial

compass in a few short steps. A simulation shows that the compass is accurate as well as simple and well

suited for a robot or drone. It can also explain many of the experimental results published on insect

navigation.

1 INTRODUCTION

Due to the scattering of light within the earth’s

atmosphere, skylight is partially linearly polarized,

discovered by the Irish Scientist Tyndall (1869).

Two years later a mathematical description of the

phenomenon was given by Lord Rayleigh (1871) for

the scattering by small particles (air molecules) in

the atmosphere. That an insect can use this

polarization to navigate was first discovered in

experiments with bees by Karl von Frisch (1949).

It took another 25 years before the nature of the

insect’s celestial compass began to be clarified

(Kirshfeld et al., 1975; Bernard and Wehner, 1977).

It depends primarily on a specialized part of the

insect compound eye, a comparatively small group

of photoreceptors, typically 100 in number, situated

in the dorsal rim area. Further insight on these

photoreceptors came from Wehner and co-workers

working with desert ants and bees (Labhart, 1980;

Rossel and Wehner, 1982; Fent and Wehner, 1985;

Wehner, 1997). It was found that each ommatidium

in the dorsal rim of the compound eye has two

photoreceptors with axes of polarization at right

angles to one another and each strongly sensitive to

the E-vector orientation of plane polarized light. The

axes of polarization of these ommatidia have a fan

shaped orientation that has been claimed from

experiments to provide an approximate map for the

polarized sky, a map which the insect can use as a

compass (Rossel, 1993). The variation in E-vector

orientation has also been traced within the central

complex of the brain of an insect (Heinz and

Homberg, 2007).

Although much is known about this insect

compass little is known about the underlying

physical processes that require these 100

photoreceptors, the subject of this research. One

attempt has been made to design a navigational aid

for a robot based on the compass; this uses 3 pairs of

photoreceptors (Wehner, 1997; Lambrinos et al,

1998), simulating the accumulation of results from

many photoreceptors in three different parts of the

fan of receptors used by an insect. This system is

reported to work well in the desert but it is not clear

that it would be accurate under a variable cloudy

sky. NASA has also built robots navigating by

skylight, but these apparently use a different process

based on 3 photoreceptors with 3 different axes of

polarization on a horizontal plane (NASA, 2005).

Few details have been released publicly on this

system or its performance.

This paper proposes that the fan of

photoreceptors is scanning the sky to find the four

points in the sky where the polarization angle, χ, the

angle between the meridian and the polarized E-

vector in the sky, equals ±π/4. We propose that the

anatomy of the eyes of bees, ants, and many other

363

Smith F. (2009).

INSECT NAVIGATION BY POLARIZED LIGHT.

In Proceedings of the International Conference on Bio-inspired Systems and Signal Processing, pages 363-368

DOI: 10.5220/0001553503630368

Copyright

c

SciTePress

insects are designed precisely to detect these four

points, probably the only measurable quantities

invariant to variable light cloud cover. We also show

that the direction of the sun can be found quickly by

a simple algorithm well within the capacity of the

insect brain for all orientations of the head.

In a previous work-in-progress paper (Smith,

2008) a simulation of this insect compass was

attempted using an algorithm involving 16 elements

in a 4X4 array in which all possible solar elevations

were examined one after the other until the correct

elevation was found. Probably this was too difficult

for the brain of an insect and in further studies of

previous experiments on insects it was found that

when insects view the sky through two different

windows they obtain solar azimuths equal to the

average of the two azimuths obtained from each

window (Wehner, 1997). This could not be

explained as part of the above algorithm. In addition

a mapping of the celestial compass in the insect

brain by Heinz and Homberg (2007), although it too

involved sets of arrays of 16 elements, indicated that

the processing of polarized light in the brain

involved simple pairing of contributions from

different sources. These facts led to the discovery of

a new much simpler algorithm in this paper and to a

better understanding of the invariance of the

algorithm to cloud conditions.

In the following we first summarise the

derivation of mathematical expressions for the light

intensities measured by the insect photoreceptors.

This is brief as more details are given in the previous

paper (Smith, 2008). We then show how these

intensities can give the direction of the sun in the

new algorithm

2 THEORY

2.1 Measured Intensities

In an ideal sky with no cloud, as shown by Rayleigh

(1871), the light observed from any patch of sky is

partially polarized, with an elliptical profile for the

electric vector, E, in which the major axis of the

ellipse is at right angles to both the direction of the

sun, represented by the unit vector, S, and to the

direction of the observed patch of sky, k’. The

electric vector in the direction of the major axis is

the E-vector. In the ideal situation where all of the

light observed is scattered once only, the ratio of E

in the directions of the minor axis to the major axis

is cos (θ) where θ is the scattering angle between S

and k’.

When this partially polarized light enters an

ommatidium in the dorsal rim its intensity is

measured by two photoreceptors, each of which can

measure polarized light with parallel structures

called microvilli. The two directions of the

microvilli are at right angles to one another, and

define two orthogonal axes of polarization for these

X and Y photoreceptors. In Figure 1 we illustrate the

orientations of the microvilli in the dorsal rim of the

honey bee by Sommer (1979), as redrawn by Rossel

(1993). The fan shape of the microvilli is apparent.

Figure 1: The paired orthogonal photoreceptors in the

dorsal rims of a bee. The axes of polarization of the Y

photoreceptors are dark, the X photoreceptors light.

The centres of the patches of sky being

observed by the photoreceptors are on the opposite

side of the head, i.e. contralateral (Sommer, 1979).

An examination of Figure (1) shows the axes of the

X photoreceptors are approximately parallel to the

meridians passing through these patches of sky

contralaterally. The same approximate parallel

pattern was found in desert ants by Wehner and

Raber (1979). It follows that the angle that the X

polarization axis makes with this meridian always

equals zero. The discovery that this angle is zero,

learned from the anatomy of bees and ants, turns out

to be critical, and it greatly simplifies the

expressions for the light intensities, S

X

and S

Y

,

measured by the two receptors, X and Y. But before

writing down these expressions we note that in the

real world the sky is not blue, but has a degree of

haze or cloud differing with direction. The light then

entering the ommatidia can be viewed as made up of

two components, one partly polarized as in

Rayleigh’s equations, and the second totally

BIOSIGNALS 2009 - International Conference on Bio-inspired Systems and Signal Processing

364

unpolarized due to multiple scattering. We let U be

the intensity of unpolarized light measured by both

photoreceptors. Then the 2 light intensities are:

[

]

UPS

X

+−= )(sin)(sin1

22

χθ

(1)

[

]

UPS

Y

+−= )(cos)(sin1

22

χθ

(2)

where the factor P depends on terms derived by

Rayleigh (1871) and on the measuring capability of

the photoreceptors.

It has been shown by Labhart (1988) that the

brain of a cricket records the difference between the

two signals, S

Y

and S

X

or rather the difference

between the log of the two signals; so the recorded

signal is

)/()()(

XYXYYX

SSLogSLogSLogS =−=

(3)

To illustrate the variation in these signals as the

azimuth angles of the ommatidia vary we set P = 1

and U=0 in the top of Figure (2). In the bottom we

include simulated clouds by putting U=0.5 sin

2

(a

o

)

with P=1-U. The curves change with cloud cover

but uniquely the zeros in S

YX

are always the same, as

evident mathematically by equating Equations (1)

and (2).

2.2 Solar Azimuth and Elevation

To proceed further we need the polarization angle, χ,

in terms of the solar azimuth, a

s

, and solar elevation,

h

s

. We need also the known azimuth, a

o

, and

elevation, h

o

, of the centre of the patch of sky being

observed by the photoreceptors.

We also know that the E-vector, in the direction

i’, is at right angles to the plane containing the solar

unit vector, S. So i’.S = 0. In our previous paper

(Smith, 2008) it is shown by substituting for S and i’

that this becomes:

0)cos()sin()sin(

)cos()cos()sin()cos()sin()cos()cos(

=−

−

s

ha

s

ha

o

h

s

h

o

h

χ

χχ

(4)

where a = a

s

– a

o

is the azimuth of the sun relative

to the azimuth of the observed sky. We use this

equation later.

2.3 A Compass for a Cloudy Sky

We need to know why insects are measuring the

difference S

YX

between the signals from the two

orthogonally polarized photoreceptors in each

ommatidium. First, the absolute value of the

difference S

YX

is between logs of intensities as in

Equation (3) and since this equals the log of the ratio

-0.8

-0.4

0

0.4

0.8

1.2

1.6

-180 -120 -60 0 60 120 180

Syx

Sy

Sx

Clouded Sky

ao

Figure 2: Illustration of the signals S

X

S

Y

and S

YX

in a

perfect blue sky [U=0] and a sky with simulated clouds

[U=0.5 sin

2

(ao)], as they vary with the azimuth, a

o

, of the

fan of observations measured from the central axis of the

insect with solar elevation h

s

=30

o

, and azimuth a

s

=60

o

.

Note that there are 4 azimuths a

o

where S

YX

=0 or Sx=Sy,

called zeros, and that these are unchanged by the cloud.

of S

T

to S

X

an examination of Equations (1) and (2)

shows that it reduces the effect of the unknown

unpolarized light intensity U caused by clouds, but it

does not remove it. So little reliable information can

be obtained from the absolute values of S

YX

. Instead,

the ommatidia are probably measuring the positions

of features in the measured intensities rather than the

values of the intensities. As these ommatidia scan

the sky through 360

o

S

YX

goes through a maximum

towards the solar meridian, then a minimum, then a

maximum towards the antisolar meridian and finally

a second minimum. Between these four extrema are

four directions in which S

YX

equals zero. So there are

two possibilities: that the ommatidia are looking for

the positions of the 4 extrema or for the positions of

the 4 zeros.

(a) 4 Extrema

As explained above the difference S

YX

in Equation

(3) goes through maxima in the directions of the

solar meridian, a

s

, and of the antisolar meridian,

a

s

+π. This is because the signal S

Y

goes through a

maximum in these directions while S

X

goes through

a minimum. So the difference between them

enhances the maximum in S

Y

Labhart (1988). If at

-0.8

-0.4

0

0.4

0.8

1.2

1.6

-180 -120 -60 0 60 120 180

ao

o

Blue Sky

Syx

Sy

Sx

INSECT NAVIGATION BY POLARIZED LIGHT

365

least one of these 2 maxima is found then it

immediately gives the direction of the sun or the

opposite direction. Other clues can then tell the

insect which of the two directions is correct.

The positions of the other 2 extrema, the 2

minima cannot be used so easily. Although the

minima of S

Y

occur at a

s

±π/2 the maxima in S

X

do

not, and may be as much as 20

o

different. This is

partly because in these directions the factor sin

2

(θ) in

Equations (1) and (2) is not stationary as it is in the

directions a

s

and a

s

+π. So the 2 minima in S

YX

cannot be used in the celestial compass

Although a maximum in S

YX

gives the positions

of the sun, the finding of the exact position of a

maximum, even enhanced, is not easy; so small

errors are likely. But a bigger problem comes from

cloud. As evident from Figure (2) variable cloud can

shift the position of a maximum or produce false

maxima, causing further errors. The wide window of

observation used by insects minimises this effect but

does not remove it. Simulations with a wide window

and real cloudy skies by Labhart (1999) have shown

that the errors caused by cloud in the positions of the

maxima were small, mostly 3

o

or less, but a few

larger errors occur. Nevertheless this approximate

position of the sun is a valuable check or an

alternative to what we now describe.

(b) 4 Zeros

The second possibility is that S

YX

is measuring the 4

zeros. Firstly zeros can be measured more accurately

than the positions of maxima, and secondly the

positions of zeros are almost completely unaltered

by variable cloud, provided only that there is some

polarized light detectable below the cloud. So the

two sources of errors in the positions of the maxima

are removed. We now show how we can use these 4

accurate measurements of zeros to calculate the

sun’s position.

Putting S

YX

= 0 or S

X

=S

Y

in Equations (1 to 3)

brings about a large simplification eliminating the

unknowns U, P and θ in one step and reduces the

equations to simply:

)(cos)(sin

22

χχ

=

. This

makes χ = ±π/4. So finding the zeros where S

YX

=0

tells us the precise azimuths a

o

= Z where χ = ±π/4.

Examples of zeros for different solar elevations for a

constant window of observation between elevations

45

o

and 89

o

are shown in Figure (3). There are

almost always 4 zeros if the window of observation

is at a constant elevation. However, when the solar

elevation is high there may be no zeros. In this case

a robot or insect might increase the elevation of

observation (although this reduces the accuracy).

-90

90

-90 90

Sun

Figure 3: Projection of the sky showing for each solar

elevation the 4 azimuths relative to the sun, of zeros where

S

YX

= 0 and χ=±π/4. These zeros are invariant to cloud

cover. The circles represent elevations of 0

o

, 30

o

and 60

o

.

Therefore, noting that cos(χ) = 1 and sin(χ) = ±1 at

the zeros, Equation (4) simplifies to:

)tan()cos()sin(

)sin()cos(

soos

oos

hhaa

haa

=−±

−

(5)

Solving this for a

s

, the azimuth of the sun, gives

expressions for a

s

, for the 4 zeros where a

o

=Z:

δ

γ

±

±

=

Za

s

(6)

in which δ = arccos(tan(h

s

)cos(h

o

)/K) and γ =

arcsin(1/K) where K

2

= 1+ sin

2

(h

o

). If the robot

scans the sky at a constant elevation h

o

then the

angles γ and δ are also constant, simplifying the

algorithm (Smith, 2008). The angle γ depends only

on the elevation of the observation, it is large, >π/4,

and known to the insect or robot. The angle δ

depends on the solar elevation and when the sun is

on the horizon it equals π/2. It can be calculated by a

robot from the above equation for δ, which needs the

solar elevation, known from the latitude and time;

fortunately we now show that this difficult

calculation is not needed by an insect.

The 4 alternatives in Equation (6) correspond to

the four zeros as illustrated in Figure (4), which we

write as

a

s

= Z

1

+ γ – δ, a

s

= Z

2

– γ – δ (7)

a

s

= Z

3

– γ + δ, a

s

= Z

4

+ γ+ δ (8)

where the signs are chosen by symmetry in the

geometry in Figure (4). Note that all of these

quantities are angles in [0,2π]; so the sums are all

modulus 2π.

If we sum these 4 expressions the γ and δ terms

cancel and we get 4a

s

= Z

1

+Z

2

+Z

3

+Z

4

, mod 2π.

Dividing by 4 gives a

s

, but because of the cyclic

nature of the summation (350

o

+ 20

o

= 10

o

) an

uncertainty of mπ/2 occurs where m=0, 1, 2 or 3.

BIOSIGNALS 2009 - International Conference on Bio-inspired Systems and Signal Processing

366

This uncertainty can be resolved by noting in Figure

4 that (1) Z

2

– Z

1

= Z

4

– Z

3

and (2) the two zeros Z

1

and Z

4

nearest to the sun are closer together than the

other two. These two conditions are used in the

following algorithm to calculate a

s

from 4 measured

zeros, Y

1

, Y

2

, Y

3

, and Y

4

, where at first the order is

not known, i.e. which one of them is Z

1

in Figure 4.

So the algorithm is simple:

1. find the 4 zeros in [0, 2π] where S

X

= S

Y

;

2. put in order Y

1

, Y

2

, Y

3

, and Y

4

;

3. find the sum: S = Y

1

+ Y

2

,+ Y

3

+ Y

4

;

4. put i=1; m=0;

5. if y

2

– y

1

<> y

4

– y

3

then i=2 and m=1;

6. if y

i

– y

i+3

< y

i+2

– y

i+1

then m=m+2;

7. a

s

= (S/4 + m*π/2) mod 2π.

(Note that differences are cyclical and clockwise.)

For example, for a solar elevation h

s

= 60

o

and

observation elevation h

o

= 70

o

we find 4 zeros at 84

o

,

120

o

, 213

o

, and 351

o

. Following the algorithm we

find that m=3 and the 4

th

one is Z

1

, S=768

o

,

S/4=192

o

and a

s

=192

o

+270

o

=462

o

=102

o

, the correct

solar azimuth. Simulations with about 5000

examples have shown that this algorithm succeeds in

almost every case with no ambiguity within a

tolerance of 1 degree. Errors occur only at low or

high solar elevations ( ≤ 2

o

or > h

o

), as long as four

zeros are found.

2.4 Less than 4 Zeros

If 4 zeros cannot be found because part of the

observed sky is obscured a robot can use the average

of the detectable zeros, along with the known values

of γ and δ, to calculate the solar azimuth. Other

clues, such as light intensity, are needed for the signs

of the corrections, remembering that the corrections

are large. An insect, not being able to calculate δ,

puts δ=π/2, assuming that the sun is on the horizon

(Rossel & Wehner, 1982), and uses the average of

the available values: z ± γ ± π/2. Since γ is known

this leaves an error of ±(δ-π/2) if one zero only is

observed. If 2 zeros are observed then the average of

the two zeros gives an error of ±(δ-π/2) or, if they

are symmetric about the solar meridian, an error of

zero. Calculated values of these errors are in general

agreement with experiments with insects (Wehner,

1989). This supports our proposal that the insect

compass is based primarily on the 4 zeros.

An insect may use only two zeros based on the

equations: 2a

s

= Z

1

+Z

4

and 2a

s

= Z

2

+Z

3

deduced

from Equations (7) and (8). We were lead to this

pairing of zeros and to our algorithm by structures

Z1

Z2Z3

Z4

Sun

Figure 4: Example of the approximate azimuths

(directions) of the 4 zeros where S

YX

= 0 relative to the

direction of the sun for a solar elevation h

s

= 60

o

.

found within the brain of an insect observing

polarized light by Heinz & Homberg (2007).

2.5 Robot Design

For a robot the greatest difficulty in building our

skylight compass is finding the direction of the four

zeros. One design (mimicking an insect) uses about

100 pairs of photoreceptors in a circle round the

robot. A problem is that each pair has to observe a

patch of sky with an accurate azimuth and elevation.

In another design the robot has one pair of

photoreceptors which is rotated through 360

o

(like

radar) measuring the azimuth as it moves at a

constant high elevation (e.g. 70

o

). This single pair of

photoreceptors can be made highly sensitive to small

differences in polarization. This would be made

easier by the use of ultraviolet light which can

penetrate cloud more easily than visible light; it is

the light used by most insects (Pomozi et al., 2001).

Although an insect views the polarized sky

contralaterally (for reasons that are unclear) it is

more straight forward for the robot to scan the sky

ipsilaterally, but with the orientations of the

polarization axes in the same directions as an insect.

3 CONCLUSIONS

We have shown that an accurate celestial compass

for a robot can be built round the principle of finding

in skylight the 4 zeros at a constant elevation. The

algorithm was discovered after studying published

experiments on insect navigation and anatomy. The

algorithm is simple and accurate and well within the

capacity of the insect brain. It also allows an insect

to navigate continuously without turning its head. It

explains many experiments on insect behaviour.

Besides the simplicity and accuracy of the

method its greatest advantage is that it is accurate in

INSECT NAVIGATION BY POLARIZED LIGHT

367

hazy and partially clouded skies, because the

position of the zeros is unchanged by cloud. No

other method shows this invariance.

However, much remains uncertain about insect

navigation. For the algorithm to be accurate the top

of the robot or drone must be pointing towards the

zenith. Insects may do this using the 3 ocelli on the

top of their heads (Goodman, 1970). But how they

would do this is not clear. Many insects also have

ommatidia in sets of three with the polarization axes

of the 3 sets differing by about π/3 (Labhart, 1988;

Wehner, 2001). There are several possibilities. If 3

ommatidia from different sets point at the same

patch of sky they could be used to calculate the

polarization angles χ, even in a cloudy sky.

Alternatively they might help identify in one of them

the signs of the corrections ±γ and ±δ if less than 4

zeros are visible. Yet another possibility is that they

are looking also for the 4 zeros where χ = ±π/4 + π/3

and the 4 zeros where χ = ±π/4 - π/3. The algorithm

above might then be repeated for each of the 3 sets

of zeros, giving 3 different approximations for the

solar azimuth. All these possibilities are being

investigated.

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