The polarized skylight detecting ability that is inherent in many animals is an example of natural systems or processes that could be used as an inspiration for the development of a bioinstrumentation for a human “navigational sense”. Many animals are capable of traveling both long and short distances before using their natural navigation abilities to return to a given location [11]. Insects that navigate short distances, normally for essential daily activities such as locating food sources, include desert ants, honey bees, dung beetles and crickets; while animals and insects that use in-built navigational systems over long distances, normally seasonally for breeding or migratory purposes, include species such as sea turtles, migratory birds and pigeons and monarch butterflies. Insects use information cues to guide their navigation and one source of these is polarized skylight, which they utilize for the purposes of navigation and orientation [11]. Examples of insects that use polarized skylight in this manner include crickets [12], locusts [13], flies [14] and dung beetles [15]. The majority of members of the bee species are diurnal, while some are nocturnal [16]. Central place foragers such as bees and ants use a natural polarization compass both to measure and to adjust their traveling direction in the context of path integration [17–19]. Long-range migrators, such as locusts [20] and some lepidoptera, including monarch butterflies, are also believed to orient themselves by exploiting skylight polarization during their journeys [21]. However, this does remain a subject of debate, as in an alternative behavioral study monarch butterflies failed to respond to e-vector orientation [22]. Desert ants and honeybees use three types of information to navigate. The first of which is based on a path integration memory of the position of the food site with respect to the nest. Path integration is based on egocentric information and enables foraging ants to return to the nest from any position at any time using the shortest direct track without help of terrestrial cues such as landmarks or panoramic views [23–25]. The second type of information used to navigate is based on visual snapshot memories of features that were viewed in the vicinity of or on the way to the food site. Thirdly, these insects also use local vector memories of the direction and length of habitual route segments. Landmark guidance is based on learning and memorizing the positions of terrestrial landmarks, such as bushes and trees, as well as the panorama and skyline along their route and enables the ants to relocate at precise earth-based absolute location [25–27]. The odometer process is handled differently by honeybees and ants: the honeybee uses an optic flow (pattern of apparent motion) during the flight experience, while the desert ant measures its travelling distance by using proprioreceptive cues (ability to sense stimuli regarding its own position, motion and equilibrium) [18]. E-vector information collection in crickets and ants is done by the polarization-sensitive photoreceptors in DRA and neurons in optic lobe [28,29].

Direct light from the sun is unpolarized; it is the scattering of skylight in the atmosphere and the reflection of skylight on water or wet soil surfaces, rocks and vegetation that produces polarized reflected light [30]. Skylight is partially plane-polarized, i.e., in any direction of the sky a particular orientation of the electric field (e-vector) of skylight dominates [31,32]. Malus and Strutt (Lord Rayleigh) in 1871 first discovered the polarization of the skylight. They provided a first explanation and mathematical description of celestial polarization. Subsequently, Chandrasekhar [33] established the full modern theory of this phenomenon. Two decades later, in the 1970s, Sekera [34] developed a computer program that was capable of describing both the direction and the degree of polarization [35]. The scattering of skylight within the Earth's atmosphere creates not only intensity and spectral gradients, but also polarization. Polarization can be described with two parameters: the orientation of the plane in which the electric vector or e-vector of skylight vibrates (direction of polarization or e-vector orientation), and the strength of this phenomenon (degree of polarization). The pattern of e-vector orientation is also recognized as a polarization pattern. The e-vector pattern in the sky uniformly changes its orientation by 15° per hour and marks the position of the pole of the daytime sky [36]. Because the skylight collecting powers of insect facet lenses are too small to catch enough quanta from even the brightest stars, they are not even able to see the stars [37,38]. Polarization sensitivity differs according to species: honeybees, desert ants and flies use UV-receptors [39–41], while crickets use blue receptors [41–43] in polarization vision [40].

Most arthropods, including insects and crustaceans, have a pair of compound eyes, either apposed (mostly insects active at day) or superposed (mostly insects active at night), that provide a wide field of vision. Compound eyes consist of three parts: a dorsal rim area (DRA), remaining dorsal area (DA) and a ventral area (VA). Figure 1(a) shows the compound eye of cricket. Each compound eye contains similar optical units, the ommatidia, of which the number varies in different insect species. There are up to 10,000 in dragonflies. Each ommatidium has its own cornea, lens and photoreceptor cells, which are used for distinguishing brightness and color. A single ommatidium guides skylight through a lens and cone into a channel called a rhabdom, which contains skylight-sensitive cells, as indicated in Figure 1(b). One ommatidium has one rhabdom. The rhabdom is the combined set of rhabdomeres. A rhabdomere is that part of the photoreceptor cell that consists of microvilli, which is a tube-like extension of the photoreceptor cell membrane. In this part of the cell membrane the rhodopsin molecules are embedded. Each rhodopsin has a characteristic absorption spectrum and thus determines the photoreceptor's sensitivity spectrum. When the rhabdomeres of the set of photoreceptors in one ommatidium are fused, the rhabdom is fused; when the rhabdomeres are spatially separated, the rhabdom is open. When the rhabdomeres of a fused rhabdom are organized in layers, the fused rhabdom is tiered. Rhabdom shapes and types are different according to the species. There are two types of rhabdoms: open (in flies, [44]) and fused (in bees and desert ants [45]). The fused rhabdom of butterflies is tiered [46,47]. The type of rhabdom directly impacts the shape of the spectral sensitivity curves of the respective photoreceptor cell [48]. In most arthropods and molluscs each ommatidium consists of retinula cells that are arranged in groups (seven photoreceptor cells in the dung beetle [15]; eight photoreceptor cells in flies [14]; nine photoreceptor cells in ants and honeybees [44]).

Behavioral studies show that the detection of polarized skylight in insect eyes is mediated by the photoreceptors in ommatidia of the DRA (bees [49,50], ants [51], crickets [12]). Each ommatidium of the DRA contains two sets of homochromatic and highly polarization-sensitive photoreceptors with orthogonal microvilli (bee [45]; ants [52]; fly [41,53]; cricket [42,54,55]).

The behavioral studies were confirmed through intracellular electrophysiological investigations that revealed that the photoreceptors in the DRA have much larger polarization sensitivity than in other parts of the eye (bees [45]; ants [52]; cricket [42]). In ants, Cataglyphis, the microvilli of DRA photoreceptors are aligned in parallel along the entire length of the cell, from the distal tip of the rhabdom down to its proximal end, near the basement membrane. The microvilli of the photoreceptor cells R1 and R5 (see Figure 2(a)) are always parallel to each other and perfectly perpendicular to the microvilli in the other photoreceptor cells [56]. Each ommatidium in the dorsal rim area has two photoreceptors, each of which is strongly sensitive to the e-vector orientation of plane-polarized skylight, with axes of polarization at right angles to one another. The axes of polarization direction of these ommatidia provide a fan-shaped orientation [57]. A cross-section of an ommatidium, a detailed rhabdom shape and the alignment of microvilli are shown in Figure 2.

In various insect species the detection of the oscillation plane of polarized skylight is mediated by a group of anatomically and physiologically specialized ommatidia that are situated in the DRA of the compound eye (see Section 2.4). Table 1 shows a mechanism of polarized skylight detection in various animals. In order to effectively use the information for the purposes of navigation, insects need to analyze the input signal of polarized skylight. This involves central processing stages, which include the lamina and medulla, the anterior lobe of the lobula, the anterior optic tract and tubercle, the lateral accessory lobe, and the central complex [59]. This process has been effectively imitated technologically in order to develop the polarization navigation sensor, which will be described in Section 3).

The mechanism for the detection of polarized skylight (e-vector detection) in the DRA involves the following situations:

Optical axes that are always directed upwards. The visual field of the DRA has an elongated shape that extends from the upper front to the upper back, with the center directed somewhat to the contralateral side (bee [45]; desert ant [51,52]; cricket [42]).

During the central processing stage, the medulla polarization sensitive neurons act as integrators, while the central complex polarization neurons serve the function of a compass neuron that guides insect navigation [2]. The information of e-vector orientation is transferred in a form of sinus wave signal from the photoreceptor in the DRA to the central processing area. E-vector responses of particular photoreceptors in the DRA are pooled by a set of polarization neurons, which act as integrators (a set typically consists of three neurons). Each integrator is aligned to a particular e-vector tuning axis. The e-vector tuning axis varies according to the rotation movement of the insect body axis. The larger these variations are the more accurate is the neuron compass reading. The crossed-analyzer in insect eyes consists of two sets of e-vector analyzers, which are orthogonally arranged to each other, and are antagonistically responding to the e-vector. Each set of analyzer may consist of a polarization sensitive photoreceptor and a neuron [58].

The polarization sensitivity can be increased in insects through the application of several characteristics, including the orthogonal arrangement and high alignment of microvilli. The reduction in length prevents polarization reduction due to self-absorption. The increase in microvillar length favours sensitivity and enlarged the cross-section area. High polarization sensitivity in DRA photoreceptors and degraded optics skylight scattering structures in the cornea or missing screening pigment [18,54].

In 2001 Usher [71] used a normal camera with an external linear polarization filter to take two images, the second image was taken with the polarization filter set orthogonal with respect to its position in the first image. The camera used in this experimental investigation was the SONY XC-EU50CE camera. The blue component of the camera's RGB output was used for analysis, as polarization of sunlight is most apparent at ultraviolet and blue wavelengths (350–450 nm). Linear polarizing film was used as a filter to extract the polarized images. All images were smoothed using a two-dimensional Gaussian function. A mean intensity function as a function of polarization angle was derived from the two images and was comparable to the form proposed by Lambrinos [5].

Carey and Sturzl [72] developed the insect-inspired low-resolution omnidirectional vision system by incorporating a near-UV camera and a linear polarizer. By incorporating an additional RGB camera, the full range of the insect's visual spectrum was obtained, making it possible to capture and investigate the visual cues that insects use in flight control and navigation. This investigation could enhance the understanding of the insect navigation system and lead to the incorporation of a similar system in an autonomous mobile robot. The equipment employed was the SONY camera mentioned above, a video signal digitizing device and a UV transmitting glass filter to maximize UV photoreception at about 340 nm.

The output of polarization sensor is described by the following equation [5]: (1) S ( Ø ) = K I ( 1 + dcos ( 2 Ø − 2 Ø max ) )where I is the total intensity given by I = I_max+ I_min.I_max and I_min being the maximum and minimum intensities, respectively. The degree of polarization is denoted by d, Ø is current orientation with respect to the solar meridian, Ø_max is the value that maximizes S(Ø), and K is a constant. The delogarithmized polarization sensor outputs of the direction 0°, 60° and 120° respectively are expressed by Equations (2–4): (2) p ˜ 1 ( Ø ) = 1 − 2 p ¯ 1 ( Ø ) = d cos ( 2 Ø ) (3) p ˜ 2 ( Ø ) = 1 − 2 p ¯ 2 ( Ø ) = d cos ( 2 Ø − 2 π 3 ) (4) p ˜ 3 ( Ø ) = 1 − 2 p ¯ 3 ( Ø ) = d cos ( 2 Ø − 4 π 3 ) (5) d = 1 − 2 p ¯ 1 ( Ø ) cos ( 2 Ø ) (6) Ø = 1 2 arctan ( p ¯ 1 ( Ø ) + 2 p ¯ 2 ( Ø ) − 3 2 3 ( p ¯ 1 ( Ø ) − 1 2 ) )

The algorithm improvement is important for improving the polarization compass. From Equations (2–4), Zhao [9] eliminated the influence of the polarization degree, d, by presenting a new transform, where after simplification of t_i, they got the value of output angle of the polarization sensor S_i [Equation (10)] that was independent of d: (7) t 1 ( Ø ) = p ˜ 1 ( Ø ) − p ˜ 2 ( Ø ) | p ˜ 1 ( Ø ) | + | p ˜ 2 ( Ø ) | (8) t 2 ( Ø ) = p ˜ 2 ( Ø ) − p ˜ 3 ( Ø ) | p ˜ 2 ( Ø ) | + | p ˜ 3 ( Ø ) | (9) t 3 ( Ø ) = p ˜ 3 ( Ø ) − p ˜ 1 ( Ø ) | p ˜ 3 ( Ø ) | + | p ˜ 1 ( Ø ) | (10) S i ( Ø ) = 1 2 arctan ( t i ( Ø ) 3 )where S_i (I = 1,2,3), Ø₁ = 0, Ø₂ = π/3, Ø₃ = 2π/3. The advantage of the new algorithm is its higher precision with identical input error due to the character of its piecewise function, even though the two algorithms have the same output function style.

The output of the polarization sensor could also be described by using the mathematical description known as Stokes parameters (shown in Equation (11)) [109]. Stokes parameters were first introduced by G.G. Stokes in 1852, who described the polarization state in four quantities. As shown in Equation (11), the four quantities: S₀, S₁, S₂ and S₃ could be assumed to be polarization values obtained from Filters 1, 2, 3 and 4 respectively. Here, the first filter is an isotropic filter, passing all states of polarization, Filter 2 is passing the linear state of polarization in the horizontal direction, while Filter 3 passes the linear state of polarization at a 45° direction. Filter 4 filters the circular polarization state only: (11) S → = [ S 0 S 1 S 2 S 3 ]

Filters of Type 1 and Type 2 are found in the works of Lambrinos et al. and Chu and co-workers, where the polarization states could be represented as Equations (12) and (13): (12) S 0 = I 90 ° + I 0 ° (13) S 1 = I 90 ° − I 0 °

The theoretical basis for the system error model of the polarized-skylight angle measurement model (POLAMM) has been developed by Li et al.[110]. The system error model of POLAMM was derived to improve the calculation accuracy of the polarized light navigation sensor. Through this system error model, the system error source parameters could be recognized, and the system error could be compensated to a major extent. From Equation (1), the output of three POL sensors is shown as follows: (14) S i ( Ø ) = K I ( 1 + dcos ( 2 Ø − 2 Ø max ( i ) ) )where S_i (I = 1,2,3), Ø₁ = 0, Ø₂ = π/3, Ø₃ = 2π/3.

According to the practical meaning, Equation (14) should satisfy (15): (15) { − π 2 < Ø ≤ π 2 0 ≤ d ≤ 1For convenience, S_i(Ø) is denoted as S_i. By substituting Equation (15) into Equation (14), Equation (16) is produced: (16) cos 2 Ø = S 1 − S 2 + S 3 2 Dwhere: (17) D = 1 2 ( S 1 − S 2 ) 2 + 1 2 ( S 2 − S 3 ) 2 + 1 2 ( S 1 − S 3 ) 2 > 0

By substituting Equation (17) into Equation (16), the equation of POLAMM could be obtained, as shown by Equation (18): (18) Ø = { arcsin ( 1 − c o s 2 Ø 2 ) , S 2 ≥ S 3 − arcsin ( 1 − cos 2 Ø 2 ) , S 2 ≥ S 3

Here: (19) KId = 2 D 3 (20) d = 2 D S 1 + S 2 + S 3From the differentiation method, the system error model of POLAMM is shown by: (21) Δ Ø = − 1 3 d ∑ j = 1 3 sin ( 2 Ø − 2 Ø j ) [ 1 + dcos ( 2 Ø − 2 Ø j ) ] ⋅ Δ K j − 2 3 ∑ j = 1 3 s i n 2 ( 2 Ø − 2 Ø j ) ⋅ Δ Ø j

The measurement in CMOS-based sensors is performed by calculating the Stokes parameters [89,94]. The electromagnetic radiation travel is utilized as input signal. The mathematical representation of an electromagnetic wave propagating in the z direction is given by Equation (22): (22) E = E 0 cos ( k z − ω t + φ 0 )where E₀ is the amplitude, k is the wave constant (k = 2ð/ë), ù is the circular frequency (ù = kc = 2ðc/ë) and is the initial phase. As described in Section 4.1, the polarization state of an electromagnetic wave can be conveniently described using a set of parameters that are known as Stokes parameters (G.G. Stokes, 1852) [109]. These are represented in a column vector as in Equation (11), called as Stokes vector. Different with Lambrinos et al. and Chu and co-workers approach, this CMOS-based polarization imaging sensor is developed in three types of filters. By modifying the Stokes parameters described by G.G. Stokes, S₀, S₁ and S₂ can be represented by Equations (23), (24), and (25) [80]: (23) S 0 = I 90 ° + I 0 ° (24) S 1 = I 90 ° − I 0 ° (25) S 2 = I 45 ° − I 0 °where I_0° is the intensity of skylight after passing through a horizontal linear polarizer, I_90° is the intensity after a vertical linear polarizer, and I_45° is the intensity after a linear polarizer is placed at 45°. By calculating Equations (22) to (25), the degree of polarization, which is related to maximum and minimum transmitted intensity values, can be obtained.

Smith [111,112] developed an algorithm for a compass that was applied on robots and drones in light clouds. The working principle of this compass was inspired by the insect compound eye. The algorithm was created by measuring the position of the four points in the sky, where, i.e., the angle χ between the polarized e-vector and the meridian equals ±π/4. The azimuth of these four points is invariant to variable cloud cover, provided that polarized skylight is still detectable below the clouds. The sum of these four azimuth values can be turned into a celestial compass, which is useful for the robot or drone. Compared with the photoreceptor-based design, a compass that uses this design offers a simpler device that offers more accuracy during navigation under cloudy sky.