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Miniature flight recorders
Honey bees regularly carry a payload of 55–65 mg6, but a mounted flight recorder should consume only a small fraction of this allowable weight to avoid significantly affecting bee behavior. The dimensions of the recorder must also be small, as the available mounting area on the bee thorax is limited. We propose a flight recorder consisting of a (2times 2times 0.3,text {mm}^3) ASIC mounted on a (3times 3times 0.4,text {mm}^3) printed circuit board, which is similar in size to 3 mm diameter, 1.5 mm tall conventional bee tags (“Queen number set,” Betterbee) and is on par with previous studies using honey bee tagging methods30. The ASIC provides most core functionality including signal detection, memory, power harvesting, and communications circuitry, and the PCB provides a magnetic backscatter coil for near-field wireless communication. The combined weight of our chip-PCB assembly is expected to be at most 10 mg, which is a small fraction of the honey bee payload (Fig. 1c). Based on previous studies, we expect this may slightly reduce foraging trip time but not significantly impact flight characteristics, which is the focus of our system31. Furthermore, future iterations of our flight recorder will have smaller size and weight, minimizing the overall impact on honey bees. Power will be harvested from sunlight, which can provide intensity greater than 1 mW/mm(^2). On-chip photovoltaics can offer power conversion efficiency on the order of 5%, supplying 50 μW of electrical power for the chip. This power budget, while low, is sufficient for the chip, since solar angle measurement and storage of data in memory are not energy-intensive operations and need only occur a few times per second. Furthermore, wireless communication for data upload is only used when the recorder is at the base station, thus allowing the base station to fully power the near-field wireless link. Additionally, IC technology is generally robust to the environmental factors likely to be encountered by honey bees. For instance, the variations in humidity level and temperature experienced by honey bees are not expected to affect chip operation. Our proposed design is a fully power-autonomous, environmentally-robust, miniature flight recorder well-suited to the task of recording honey bee activity.
The orientation of a bee during flight can be described by the yaw ((gamma)), which represents the absolute heading relative to the sun, and the angle-of-incidence (AOI) ((psi)), which represents the overhead angle between the sun and the sensor (Fig. 1b). To record flights, the chip uses ASPs to measure the AOI of sunlight and stores these measurements in on-chip memory. ASPs achieve AOI-sensitivity via a pair of diffraction gratings stacked over a photodiode (Fig. 1b), wherein the first grating induces a diffraction pattern that shifts laterally across the second grating as AOI is swept, thus passing a periodically-varying intensity of light to the photodiode. The stored AOI measurements can be downloaded to a base station upon return to the hive, and from these data the heading throughout the flight can be extracted. Assuming a constant speed of 6.5 m/s32, we can use the sequence of recorded headings to reconstruct the honey bee’s trajectory in post-processing.
The data taken by the flight recorder will be subject to measurement errors, and these errors will manifest in the reconstructed trajectory. Here, we identify and explore methods to mitigate the primary sources of error. We posit that errors will stem primarily from finite heading measurement resolution, finite sampling rate, and random fluctuations in sampling rate (jitter). Each of these error sources can be suppressed through careful chip design, but improvements in these performance variables can only be made at the expense of larger chip area. For instance, the heading measurement resolution will increase if more ASPs are used to measure AOI, but each pixel consumes significant silicon area and contributes additional data that must be stored in memory. Increasing the sampling rate and decreasing sampling rate jitter requires that more measurements be stored if flight time is unchanged, thus increasing the chip area required for memory. The size of the chip is thus inversely related to the severity of the expected measurement error, and trade-offs between chip size and achievable precision should be examined. A smaller sensor is feasible if trajectory reconstruction performance requirements are relaxed; more stringent requirements will necessitate a larger chip that will increase the burden on the bee. If the relationship between final uncertainty in the reconstructed position of the bee and the core sensor specifications is understood, then chip-level performance goals can be formed based on trajectory-level precision requirements.
To determine required heading resolution, sampling rate, and sampling rate jitter, we first describe the procedure to reconstruct trajectories. We define the timestep estimate (hat{Delta t}) as the inverse of the sampling rate, and for known flight speed v the sequence of measured heading estimates ({hat{gamma }_0, hat{gamma }_1, …, hat{gamma }_{n-1}}) can be mapped to position estimate (hat{mathbf {p}}_n = [hat{x}_n, , hat{y}_n]^T) as a function of discrete time index n via the motion model

$$begin{aligned} hat{mathbf {p}}_n = sum _{i=0}^{n-1} v, mathbf {h}(hat{gamma }_i) hat{Delta t} end{aligned}$$
(1)

where h is a unit vector pointing along the heading of the bee. Each sensor estimate of heading and timestep will be subject to errors, and by modeling these errors as additive white noise, we can evaluate a confidence region for each position estimate (hat{mathbf {p}}_n) (Fig. 2a,b). Each analog heading measurement (hat{gamma }_i) must be digitized for storage on-chip and will therefore suffer from quantization error, a form of rounding. We denote this error (epsilon _{gamma ,i}) and model it as a uniform random variable with variance (sigma ^2_gamma) on the interval (pm frac{Delta gamma }{2}), where (Delta gamma) is the heading bin width. Furthermore, the timestep estimate (hat{Delta t}) will be subject to random clock jitter that can be modeled as a Gaussian random variable (epsilon _{Delta t,i}) with variance (sigma ^2_{Delta t}). In this model, we posit for simplicity that the random error in timing scales linearly in proportion to oscillator frequency, thus maintaining a fixed ratio of (sigma _{Delta t}/hat{Delta t}). These measurement errors contribute random error to position estimate ({hat{mathbf {p}}}_n), and thus each position estimate should be viewed as a random variable (mathbf {p}_n). The confidence region surrounding ({hat{mathbf {p}}}_n) depends on the covariance matrix of ({mathbf {p}_n}), and we evaluate these terms by first using the small-angle approximation to linearize the motion model with respect to (epsilon _{gamma ,i}):

$$begin{aligned} mathbf {p}_n = sum _{i=0}^{n-1} v , mathbf {h}(hat{gamma }_i + epsilon _{gamma ,i}) (hat{Delta t}+epsilon _{Delta t, i}) &approx sum _{i=0}^{n-1} v , left( mathbf {h}(hat{gamma }_i) + mathbf {h}^perp (hat{gamma }_i)epsilon _{gamma ,i}right) (hat{Delta t}+epsilon _{Delta t, i}) = sum _{i=0}^{n-1} v , mathbf {h}(hat{gamma }_i)(hat{Delta t}+epsilon _{Delta t,i}) + sum _{i=0}^{n-1} v , mathbf {h}^perp (hat{gamma }_i)epsilon _{gamma ,i} (hat{Delta t} + epsilon _{Delta t, i}) end{aligned}$$
(2)

Figure 2

(a) Sensor measurement errors produce confidence regions surrounding each estimated position shown in an example circular trajectory. (b) Zoomed-in view of confidence region at end of flight, with principle components shown. (c) The two directed standard deviations grow throughout the duration of the trajectory, and are bounded above and below by the directed standard deviations computed from the case of the straight-line trajectory. (d–f) Both directional standard deviations characterizing the final error region will depend on all three of the core sensor specs ({Delta gamma , hat{Delta t}, sigma _{Delta t}}), and the max confidence region dimension will grow if these specs are relaxed. Plots were created in MATLAB33.

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This approximation is valid if (epsilon _{gamma ,i}) is kept small, which can be guaranteed by keeping heading bin width (Delta gamma) small. The covariance matrix of (mathbf {p}_n) is then given by

$$begin{aligned} mathrm {Cov}(mathbf {p}_n, mathbf {p}_n) = varvec{ Sigma }_n approx sum _{i=0}^{n-1} R(hat{{gamma }}_{i}) begin{bmatrix} v^2sigma ^2_{Delta t} &{} 0 \ 0 &{} v^2sigma ^2_{gamma }(sigma _{Delta t}^2 + (hat{Delta t})^2) end{bmatrix} {R^T({hat{gamma }}_i)} end{aligned}$$
(3)

where R is the standard 2 × 2 rotation matrix (Appendix: Derivation of Trajectory Precision Equation). By the Central Limit Theorem, after a sufficient number of timesteps (mathbf {p}_n) will become Gaussian distributed. Thus, the confidence region will be an ellipse with major and minor axes spanned by the eigenvectors ({mathbf {v}_1, mathbf {v}_2}) of ({varvec{Sigma }}_n). The covariances of the confidence region along each of these axes are given by the eigenvalues ({lambda _1, lambda _2}) of (varvec{Sigma }_n), and directed standard deviations can then be defined as ({sigma _1,sigma _2}). A 99% confidence region for position estimate ({mathbf {hat{p}}}_{n}) is given by an ellipse with major and minor axes lengths ({3sigma _1mathbf {v}_1, 3sigma _2mathbf {v}_2}). We therefore conclude that (3sigma _1) and (3sigma _2) are critical values defining achievable trajectory reconstruction precision.
These (3sigma)-bounds can be computed for any measured sequence of headings, but general upper and lower bounds for (3sigma _1) and (3sigma _2) across all possible trajectories can be derived from the (3sigma _1) and (3sigma _2) given by the case in which the bee flies in a straight line. In the straight-line case, the eigenvalues of (varvec{Sigma }_n) are given by n times the diagonal entries of the diagonal matrix in Eq. (3). The directed (3sigma)-bounds are then

$$begin{aligned} 3sigma _{1,n}^*= & {} 3sqrt{n} v sigma _{Delta t} end{aligned}$$
(4)

$$begin{aligned} 3sigma _{2,n}^*= & {} 3sqrt{n} v sigma _{gamma }sqrt{sigma _{Delta t}^2 + (hat{Delta t})^2} end{aligned}$$
(5)

For some values of ({sigma _{gamma }, hat{Delta t}, sigma _{Delta t}}), (3sigma _{1,n}^*) will be larger than (3sigma _{2,n}^*); for others, the converse will be true. These equations provide simple bounds on achievable reconstruction precision that are valid for any trajectory and can be computed from sensor characteristics. Since heading error (epsilon _{gamma _i}) is uniformly distributed, heading variance (sigma ^2_gamma) is defined by heading bin width (Delta gamma), and thus the reconstruction precision is defined by a core suite of sensor specifications: ({Delta gamma , hat{Delta t}, sigma _{Delta t}}). An illustration of the trajectory reconstruction process, along with confidence regions, is shown in Fig. 2, as well as the relationship between reconstruction precision and each of the core chip specs.
For a maximum specified chip sensing area, trajectory precision should be optimized through balanced allocation of area to solar AOI detection and to memory (Fig. 3). We evaluate the optimal area allocation by defining the standard deviation upper bound (3sigma ^*_{max} = max (3sigma _{1,f}^*, 3sigma _{2,f}^*)), where (3sigma _{1,f}^*) and (3sigma _{2,f}^*) are the directed (3sigma)-values computed from a straight-line trajectory that is long enough to completely fill the memory. If more area is spent on pixels for AOI detection, heading resolution can be increased, thus causing (sigma _gamma) to be reduced and correspondingly lowering (3sigma ^*_{2,f}). Conversely, if more area is spent on memory, measurements can be taken more frequently, and timestep and clock jitter can be reduced, thus reducing (hat{Delta t}) and (sigma _{Delta t}). This will cause (3sigma ^*_{1,f}) to decrease, but may cause an increase in (3sigma ^*_{2,f}) since less area is now available for heading sensors. As shown in Fig. 3, the upper bound (3sigma ^*_{max}) minimizes when the two counteracting variables (3sigma ^*_{1,f}) and (3sigma ^*_{2,f}) are equal, and this intersection point prescribes the optimal allocation of sensor area. Our proposed flight recorder features a 4 (mathrm {mm}^2) chip, which can offer sensing area of approximately 3 (mathrm {mm}^2). When this area is allocated optimally, the heading resolution is (2^circ) and timestep is approximately 240 ms at timestep jitter of (3sigma _{Delta t}/Delta t = 0.03). With these specifications, the maximum recordable trajectory length is approximately 4 km, with (3sigma) uncertainty of (pm 2.4) m.
Figure 3

(a) Area usage is optimized where (3sigma ^*_{1,f}) and (3sigma ^*_{2,f}) cross, as this design point co-minimizes the two directional standard deviations characterizing trajectory precision. (b) This design point specifies the heading resolution and number of data words in memory that minimize trajectory uncertainty given a fixed sensor area constraint. Plots were created in MATLAB33.

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Sensor calibration and noise modeling
We next examined the output from existing ASP array sensors in order to create a model for future flight recorders. These particular sensors have 96 pixels, or 24 sets of 4 oriented in 90° angles. Specifically, we designed a calibration apparatus that consists of a platform holding the ASP array driven by a custom microcontroller PCB and an arm with a light emitting diode (LED) to imitate the sun. The platform rotates to mimic a change in yaw, while the arm rotates to mimic different AOI solar light at different times of the day (Fig. 4a). Using this apparatus, we measured light input in 0.9° increments across the entire hemisphere and record the ASP array response (Fig. 4a inset) for a total resolution of 40,000 measurements in a single sweep with 200 AOI angles and 200 yaw angles. Measurements sampled by the microcontroller were transmitted to a desktop computer for logging and processing via a custom MATLAB33 script. Each ASP array response consists of a 48-bit sequence. To interpret the output, we created a lookup table from the unique 48-bit sequence that is stored for each yaw-AOI angle pair. Future data was then compared to these stored sequences in parallel using an XOR operation and the pair with the least difference in bit values was returned.
Figure 4

(a) Calibration apparatus with inset example of a measured ASP quadrature response. (b) ASP array repeatability over a single sensor (left) and multiple sensors (right). The magenta lines denote the expected operating region. (c) Expected system operating region (45°–75°) shown in magenta given the peak foraging hours (10 a.m.–4 p.m.) shown in grey and the AOI solar light during the Summer in Ithaca, NY. Plots in (a-inset), (b) and (c) were created in MATLAB33.

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We characterized sensor repeatability by repeating a sweep three times with a single ASP array and, similarly, characterized precision by comparing sweeps from two additional ASP arrays. A sample curve from the calibration sweep seen in the inset in Fig. 4a shows the characteristic angle dependence. The sensor exhibits poor response uniqueness when the sun approaches zenith and when it nears the horizon; upwards of 100(^circ) error in yaw near 90(^circ) AOI (zenith) and up to 180(^circ) error in yaw at 0°–25° AOI (horizon) (Fig. 4b). The former occurs because the position of a light source directly overhead is ambiguous to the sensor across yaw and consequently, indeterminate. We find that the sensor simply does not operate well in the latter region where light is arriving nearly parallel to the surface of the chip. Furthermore, as is expected, the difference in response is generally greater when comparing different sensors. The horizontal dark bars in the right graph in Fig. 4b are examples of this increased error. We expect our system to operate under favorable foraging conditions. Based on previously published data, we estimate this operating region to be May through September at an example location of the authors’ hometown of Ithaca, New York, USA, with the most active foraging hours being from 10 a.m. to 4 p.m.34. During this time, the AOI spans (45^circ) to (75^circ) (Fig. 4c). Within this region, we see a significantly reduced same-chip error in yaw with a mean and standard deviation of (1.52^circ pm 1.23^circ). We compensate for the remaining error within the operating region as discussed in the following sections.
In order to realistically simulate sensor output, we create a lookup table with an error model for each individual AOI value. Similar to our theoretical model, we fit a normal distribution to error in the yaw angle measured at each AOI. We use this error model to inform reconstruction of recorded bee paths as described in the following sections. For this work, we assume that we have access to calibration data for each particular sensor, however, given the low discrepancy between sensors (Fig. 4b right), we believe that it is possible to avoid individual calibration with more sophisticated data processing. We leave this aspect for future work.
Honey bee foraging simulation
To properly develop our methodology for using instrumented bees to monitor the state of pollination and bloom, we designed a colony foraging simulator with an example apple orchard. Central to our approach is an understanding of the behavior and environmental conditions surrounding honey bee foraging, summarized in Fig. 1c. The following subsections detail orchard, honey bee motion, and colony foraging models.
Orchard model
We modelled the orchard based on common characteristics seen in real orchards (Fig. 5a) as well as those reported by the University of Vermont Cooperative Extension for Growing Fruit Trees35. Specifically, these include a tree trunk radius of 0.15 m, separation between individual trees in a given planted row as 2.4 m, and separation between rows as 5 m. To make the model realistic to a variety of orchards, we add randomness to the trunk radius (0.15–0.30 m) and to the tree locations (up to 0.5 m in any direction). We further use a 60 × 60 m2 area with 200 trees, as is representative of the common grower practice utilizing a single colony per acre36. We account for the fact that trees can be in different stages of bloom by assigning each a randomly generated quality factor between 1 and 10; this quality factor affects the number of feeding events in a flight.
Colony foraging model
A high-quality honey bee colony for commercial apple pollination contains a laying queen, developing brood, and 20,000–40,000 worker bees, of which approximately 25% are “foragers”, or those that leave the hive to collect pollen, nectar, resin, and water29,37,38. Since resin and water foragers are a small proportion of the forager workforce, we expect our flight dataset to be largely from bees that are visiting flowers39. For this work, we assume favorable foraging conditions as previously described and a colony size of 35,000, 25% of which are foragers for a total of 8750 foragers conducting ~ 36,000 flights per day (an average of 4 flights per forager)37,40. In our model, a forager can perform either a learning-, return-, or scout flight. Note that we exclude orientation flights, which are conducted by new foragers, under the assumption that these can be easily classified given their tortuous nature23. A learning flight is when a bee orients to a feeding site it has not previously visited after learning the bearing and distance from one of its sisters in the hive41,42. A return flight occurs when a bee orients to a feeding site it has previously visited, and can be thought of as an optimized version of the learning flight in terms of distance flown43. A scout flight occurs when a forager leaves the hive to search independently for new feeding sites. In our model, we make the assumption based on published behavioral research that 20% of foragers are acting as scout foragers and the remaining 80% perform an initial learning flight followed by return flights to the same source41,42,43.
We incorporate that return flights will frequent the same feeding sites and that neighboring trees are likely to bloom together by randomly assigning initial goal locations (trees that bees advertise in the colony as high quality food sources) to 5 neighboring trees. Bees will randomly choose between these 5, then continue feeding on neighboring trees until they have visited trees with quality factors accumulating to at least 10 before returning to the hive. Scout foragers randomly visit trees in the orchard. Realistically, not all bees will be tagged and some tags will be lost. Here, we consider a conservative estimate that at least 430 or 5% of all foragers will be tagged, leading us to 1750 recorded flights per day, and use accumulated data to overcome the loss of tagged bees, which we expect to occur as a result of predation, senescence, stress, and other factors. Note that honey bees have a pronounced division of labor associated with worker age41, making it easy to tag a cohort and wait for them to become foragers, or to identify foragers and tag them specifically. Tagging 430 bees would take our honey bee technician approximately a day; speeding up this process is an area of future investigation.
Honey bee motion model
To better illustrate the characteristics of foraging flights, we recorded activity between a queenright colony with about 10,000 workers and a nearby feeder station (Fig. 5b–d). Three distinct phases of the bee flights were recorded: an initial orientation flight upon leaving the hive, flights between the hive and the feeder, and search flights near the feeder. Flights near the entrance and the feeder were characterized by rapid turning, whereas flights in between the hive and the feeder were nearly straight “bee lines”. While at the feeder, bees crawl around at a significantly reduced velocity.
Figure 5

(a) Photo of a honey bee in a conventional apple orchard. (b–d) Setup to showcase different types of honey bee flights. Still images from videos recorded at the hive entrance, in between, and at the feeder station, with tracked paths overlaid. (e–f) Recorded heading over the course of a simulated foraging flight and feeding event. Straight line flights are marked in grey, turns in blue, and feeding events in green. Image overlays in (b), (c) and (d) were created in MATLAB33. Plots in (e) and (f) were created in Python 3.7.

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We use this study to inform our foraging simulation. We assume generally straight paths in obstacle-free environments, slow turns when avoiding obstacles, and rapid turns in AOI and yaw when nearing and crawling on a food source. We further base our flight model on the following assumptions, summarized in Fig. 1c. (1) We assume the starting location is well known since the flight path will always originate and terminate at the hive entrance. (2) Based on past studies and the fact that our simulation takes place in a dense apple orchard, we assume most flights will be within the 4 km range of our flight recorder44,45,46. When leaving the hive, bees will fly an average of ~ 7.5 m/s, but once loaded with nectar, flight speed is reduced to ~ 6.5 m/s32. Here, we assume a constant velocity of ~ 6.5 m/s. (3) Based on prior honey bee tracking studies23,47, we represent flight in only two dimensions. The apple trees in the orchards we model are not tall and bees will therefore experience much greater motion in the horizontal plane than the vertical. Regardless of whether a bee in reality will fly over or under the canopy, we can model this issue in two dimensions. Turns around tree trunks represent the largest source of error for flight reconstruction, therefore by modeling flight under the canopy, we model the “worst case” scenario. (4) We estimate that the AOI of sunlight with respect to the orchard will remain within a quantifiable margin throughout the duration of the simulated flights, as bees have been found to spend an average of 20–45 min on foraging flights48. (5) We model the yaw of a bee as constant during bee-line flights, i.e. given no nearby obstacles. When the bee changes its heading to circumvent obstacles, this causes a change in yaw. (6) Once implemented on bees, we expect to be able to add sensors near the hive which would help us acquire current temperature and weather patterns as well as other dynamic factors specific to a particular environment for calibrating our model.
To simulate scouting, learning, and return foraging flights, we combine the honey bee motion model previously described, with the Bug2 algorithm and grid-based path planning49. Grid based path planning uses a discrete grid of points over which an agent searches to find obstacle-free path segments. We compute scout and learning flights as follows. Using the Bug2 algorithm, honey bee paths are generated by first assuming direct flight along a known heading from the hive. Once an obstacle is encountered, the bee searches for a path around it, until it can once more move unhindered toward the goal. The process is repeated until all goals are reached and the bee has returned to the hive. Since no two paths are identical in nature, we plan obstacle navigation with a randomly generated grid. Return flights are found by forming a graph of all the points visited during a learning flight and using a Dijkstra’s search algorithm49 to find the shortest path through these points from the hive to the goal. Once paths are generated, we compute the sequence of headings given the 240 ms sensor sampling frequency reported earlier. An example flight is shown in Fig. 5e,f. These headings are then discretized based on the ASP calibration data discussed earlier, and noise is added given the error shown in Fig. 4c left.
When bees land at a feeding site, they tend to crawl on and among flowers to gather nectar and pollen. We simulate this by generating random motion centered around the feeding site. The average feeding time was reported to be 1–2 min per feeding site6. To make the simulation more realistic, we randomly generate a feeding time between 60 and 120 s for foragers, and between 20 and 130 s for scouts. We furthermore assume that the AOI of sunlight changes as the honey bee tilts up and down while crawling on flowers.
Path reconstruction and generation of foraging activity maps
Path reconstruction inherently depends on the accuracy with which our sensor is able to describe the motion of an instrumented bee. Beyond limited angle resolution, errors related to the sampling rate accumulate when turns occur, at worst (v_{loaded} Delta t) = 1.6 m. To increase the accuracy of our foraging activity maps, we use models of sensor noise and flight speed, and leverage all recorded flights. The full workflow is shown in Fig. 6, where the foraging simulation portion generates the data we expect if our sensors are placed on actual bees, and the remaining flow is the data processing portion of our methodology.
Figure 6

Flow chart combining the colony foraging simulation, simulated flight recorder, path reconstruction, and data processing to output the final foraging activity map. Recorded heading plot made in Python 3.7. Other plots were created in MATLAB33.

Full size image

We first identify feeding and turn features in our path data that stand out above the noise floor. Feeding features consist of crawling behavior in which the bee is moving at much lower speed compared to flying, but with rapidly varying yaw and AOI, inducing an elevated rate of change in measured AOI. A turn is marked by a significant change in yaw. Given the time of day and location, we can find the expected AOI on the orchard and use this to find the yaw from our lookup table. Our algorithm then indexes the sensor noise lookup table to find the related mean and standard deviation and uses this to estimate turns and feeding status. Our method marks a turn for a change greater than three standard deviations in yaw ((6.06^circ)). A feeding site is marked if the detected AOI deviates more than three standard deviations ((3.72^circ)) from the one expected, or if two consecutive AOI samples deviate more than 3 standard deviations, adjustable depending on the total flight time. The average detection accuracy of a turn is 99% with a standard deviation of 0.28% and average detection accuracy of a feeding site is 99% with a standard deviation of 0.29%.
We explore three methods to generate activity maps: from the raw data, we classify feeding features using the aforementioned statistical approach and produce maps based on accumulated path reconstructions; in “iteration 1” and “iteration 4” we take a particle filter approach to improve path localization based on knowledge of the hive and frequented feeding sites respectively. Particle filters are used to track a variable of interest over time by creating many representative particles, generating predictions according to dynamics and error models, and then updating them according to observation models50. In this case, each particle forms a candidate trajectory and predictions are based on speed, sensor readings, and the sensor error model. Assuming that we start without knowledge of the orchard, we build up an observation model by reconstructing and accumulating the raw flight paths (Fig. 6, raw data). To account for the fact that bees may turn at any point between sensor readings, we then upsample our sensor readings by a factor of 8, essentially producing 8 guesses for where the bee actually turned. Based on the artificially upsampled data and a random sample from the sensor noise distribution at a given AOI, we then generate the displacement in each path segment as follows:

$$begin{aligned} begin{bmatrix} Delta x_{t} \ Delta y_{t} end{bmatrix} = v Delta t begin{bmatrix} cos {(gamma _{t} + gamma _{noise})} \ sin {(gamma _{t} + gamma _{noise})} end{bmatrix} end{aligned}$$
(6)

The final path is found as a cumulative sum of these displacements. We repeat this process to generate 5000 particles for each flight. The choice of 5000 is guided by our variable dimensions; the upsampling rate of 8 was the highest we could handle on a quadcore desktop computer with 16 GB RAM—to truly represent all potential turns we would need a number of particles equal to 8 to the power of the number of turns per flight. For reference, the average number of turns per flight is 13, thus the true representation in our sampling approach would require (8^{13}) particles.
In iteration 1, we choose ten of these particles according to proximity to the hive upon return, and use the feeding features from these to form an initial foraging activity map, represented by a discrete grid with computed visit numbers. Note that in a typical localization approach a single particle, or average of several particles, is chosen as the final reconstruction. In our approach, we retain 10 different reconstructions for each individual path in order to better represent the distribution of points in a path due to sensor noise.
After this first pass, we repeat the process, but now filter particles by using the initial activity map as an observation model for the particle filter (Fig. 6, iteration 2–4). Specifically, we update particle probability at each detected feeding site by assigning the probability of the nearest grid cell in the map to the particle, and then sampling the particles by weight. At the end of the process, the ten particles with the highest probability product are used to construct an updated activity map. More

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